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Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian

Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian 1IntroductionIn this article, we study the following nonlinear second-order p(t)p\left(t)-Laplacian system with nonsmooth potential (1.1)ddt(∣u˙(t)∣p(t)−2u˙(t))−a(t)∣u(t)∣p(t)−2u(t)∈∂f(t,u(t)),u(t)→0,as∣t∣→∞,\left\{\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial f\left(t,u\left(t)),\\ u\left(t)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \to \infty ,\end{array}\right.where p,a:R→R+p,a:{\mathbb{R}}\to {{\mathbb{R}}}^{+}, f:R×RN→Rf:{\mathbb{R}}\times {{\mathbb{R}}}^{N}\to {\mathbb{R}}, u↦f(t,u)u\mapsto f\left(t,u)is locally Lipschitz. Here ∂f(t,x)\partial f\left(t,x)denotes the subdifferential of the locally Lipschitz function u↦f(t,u)u\mapsto f\left(t,u).In recent years, the study on p(t)p\left(t)-Laplacian problems has attracted more and more attention. The p(t)p\left(t)-Laplacian possesses more complicated phenomena than the pp-Laplacian. For example, it is inhomogeneous, which causes many difficulties, and some classical theories and methods, such as the theory of Sobolev spaces, are not applicable. The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years; see [26,30,50,52]. One of the most studied models leading to problems of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [59]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the filtration processes of an ideal barotropic gas through a porous medium [2,3]. Another field of application of equations with variable exponent growth conditions is image processing [10]. We refer the reader to [23,55, 56,57,58,60,61] for an overview and references on this subject, and to [12,13, 14,15,28,29,41,45,53,54] for the study of the p(t)p\left(t)-Laplacian equations and the corresponding variational problems.Since many free boundary problems and obstacle problems may be reduced to partial differential equations (PDEs) with discontinuous nonlinearities, the existence of solutions for the problems with discontinuous nonlinearities has been widely investigated in recent years. Chang [4] extended the variational methods to a class of nondifferentiable functionals. In 2000, Kourogenis and Papageorgiou [35] obtained some nonsmooth critical point theorems. Subsequently, the nonsmooth version of the three critical points theorem and the nonsmooth Ricceri-type variational principle was established by Marano and Motreanu [36], who gave an application to elliptic problems involving the pp-Laplacian with discontinuous nonlinearities. Kandilakis et al. [34] obtained the local linking theorem for locally Lipschitz functions. Dai [16] elaborated a nonsmooth version of the fountain theorem and gave an application to a Dirichlet-type differential inclusion. In 2019, Ge and Rădulescu [27] obtained infinitely many solutions for a nonhomogeneous differential inclusion with lack of compactness involving the p(x)p\left(x)-Laplacian.It is well known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems. If a system has transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation, and its perturbed system probably produces chaotic phenomena. Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic orbits of problem (1.1). When p(t)≡pp\left(t)\equiv p, (1.1) reduces to pp-Laplacian system: (1.2)ddt(∣u˙(t)∣p−2u˙(t))−a(t)∣u(t)∣p−2u(t)∈∂f(t,u(t)),\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p-2}u\left(t)\in \partial f\left(t,u\left(t)),Hu and Papageorgious studied the existence of homoclinic solution using the theory of nonsmooth critical points and the idea of approximation [31,32] in the case of periodic nonsmooth potential with scalar equation. However, they did not prove the existence of homoclinic solutions and approached the problem differently from ours. Particularly, none of the works addressed the issues in the case of nonperiodic nonsmooth potential. With regard to the results of (1.1) in PDE, please refer to the literature [6,7,8, 9,17,19,22,33,42,47,48].To the best of our knowledge, there is few paper discussing the homoclinic solutions of problem (1.1) with nonsmooth potential via nonsmooth critical point theory can be found in the existing literature. In order to fill in this gap, inspired by [28,37,41,57], we study problem (1.1) from a more extensive viewpoint. More precisely, we would study the existence of nontrivial homoclinic solutions of problem (1.1) with the generalized subquadratic and superquadratic in two cases of the nonsmooth potential: periodic and nonperiodic, respectively. Moreover, our results generalize and improve the ones in (1.2). The resulting problem engages two major difficulties: first, due to the appearance of the variable exponent, which is not homogeneous, some special techniques and sharp estimation of inequality will be needed to study this type of problem (1.1). Another difficulty we must overcome is verifying the link geometry and certifying boundedness of the sequence of solutions {un}\left\{{u}_{n}\right\}associated with problem (1.1). It is worth to point out that commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents. In these cases, we have to use techniques which are simpler and more direct in this article.Throughout this article, we formulate the hypotheses on p(t)p\left(t), a(t)a\left(t)and basic assumptions on f(t,u)f\left(t,u): H(p)p∈C(R,R+)p\in C\left({\mathbb{R}},{{\mathbb{R}}}^{+})and 1<p−≔inft∈Rp(t)≤supt∈Rp(t)≔p+<∞;1\lt {p}^{-}:= \mathop{\inf }\limits_{t\in {\mathbb{R}}}p\left(t)\le \mathop{\sup }\limits_{t\in {\mathbb{R}}}p\left(t):= {p}^{+}\lt \infty ;H(a)a∈C(R,R+)a\in C\left({\mathbb{R}},{{\mathbb{R}}}^{+})and there exists a0>0{a}_{0}\gt 0such that a(t)≥a0>0a\left(t)\ge {a}_{0}\gt 0for t∈Rt\in {\mathbb{R}};H(f)(i)the function f(t,⋅):R→Rf\left(t,\cdot ):{\mathbb{R}}\to {\mathbb{R}}is measurable for all u∈RNu\in {{\mathbb{R}}}^{N}and f(t,0)=0f\left(t,0)=0;(ii)the function f(⋅,u):RN→Rf\left(\cdot ,u):{{\mathbb{R}}}^{N}\to {\mathbb{R}}is locally Lipschitz for a.e. t∈Rt\in {\mathbb{R}}.Our approach is variationally based on the nonsmooth critical point theory (see Rădulescu and Repovš [51], Diening et al. [18] and the papers by Chang, Fan, Rădulescu, Papageorgiou, Papageorgiou and Zhao et al. [4,21,38,40,44]). For the convenience of the reader, in the next section we recall some basic definitions and facts from the theory, which we shall use in the sequel.This article is organized as follows. In Section 2, we present some necessary preliminary knowledge on the generalized gradient of the locally Lipschitz function and variable exponent Sobolev spaces. In Section 3, we establish and prove the existence of nontrivial homoclinic solution related to periodic problem (1.1). In Section 4, we establish and prove the existence of nontrivial homoclinic solution corresponding to nonperiodic problems (1.1) and (4.2), respectively.Throughout the article, we make use of the following notations: Ls(R)(1≤s<∞){L}^{s}\left({\mathbb{R}})\left(1\le s\lt \infty )denotes the Lebesgue space with the norm ‖u‖s=∫R∣u∣sdt1/s\Vert u{\Vert }_{s}={\left({\int }_{{\mathbb{R}}}| u{| }^{s}{\rm{d}}t\right)}^{1\text{/}s};For any x∈Rx\in {\mathbb{R}}and r>0r\gt 0, Br(x)≔{y∈R:∣y−x∣<r}{B}_{r}\left(x):= \{y\in {\mathbb{R}}:| y-x| \lt r\}and Br=Br(0){B}_{r}={B}_{r}\left(0);C1,C2,…{C}_{1},{C}_{2},\ldots denote positive constants possibly different in different places.2PreliminariesWe start with some preliminary basic results on variable exponent Sobolev spaces. For more details we refer the readers to the book of Rădulescu and Repovš [51], Diening et al. [18] and the papers by Chang, Fan, Rădulescu, Papageorgiou, Papageorgiou and Zhao et al. [4,21,38,40,44].2.1Weighted variable exponential Wa1,p(t){W}_{a}^{1,p\left(t)}spaceIn order to discuss problem (1.1), we recall some known results from critical point theory and the properties of space Wa1,p(t){W}_{a}^{1,p\left(t)}for the convenience of the readers.Let Ω\Omega be a subset of R{\mathbb{R}}, S(Ω,RN)≔{u:the functionu:Ω→RNis measurable}S(\Omega ,{{\mathbb{R}}}^{N}):= \{u:\hspace{0.33em}\hspace{0.1em}\text{the function}\hspace{0.1em}\hspace{0.33em}u:\Omega \to {{\mathbb{R}}}^{N}\hspace{0.1em}\text{is measurable}\hspace{0.1em}\}and any two elements in S(Ω,RN)S(\Omega ,{{\mathbb{R}}}^{N})which are almost equal are considered the same element. Let p,ap,asatisfy assumptions HH(p), HH(a), respectively.Define Lap(t)(Ω,RN){L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})(denoted by Lap(t){L}_{a}^{p\left(t)}) as follows: Lap(t)(Ω,RN)=u∈S(Ω,RN):∫Ωa(t)∣u(t)∣p(t)dt<∞{L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})=\left\{u\in S(\Omega ,{{\mathbb{R}}}^{N}):\mathop{\int }\limits_{\Omega }a\left(t)| u\left(t){| }^{p\left(t)}{\rm{d}}t\lt \infty \right\}endowed with the norm ∣u∣p(t),a=infλ>0:∫Ωa(t)uλp(t)dt≤1.| u{| }_{p\left(t),a}=\inf \left\{\lambda \gt 0:\mathop{\int }\limits_{\Omega }a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}{\rm{d}}t\le 1\right\}.If a(t)≡1a\left(t)\equiv 1, Lap(t){L}_{a}^{p\left(t)}and the corresponding norm ∣u∣p(t),a| u{| }_{p\left(t),a}are written simply by Lp(t){L}^{p\left(t)}, ∣u∣p(t)| u{| }_{p\left(t)}.Define Wa1,p(t)(Ω,RN){W}_{a}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})(denoted by Wa1,p(t){W}_{a}^{1,p\left(t)}) as follows: Wa1,p(t)(Ω,RN)={u∈Lap(t)(Ω,RN):u˙∈Lap(t)(Ω,RN)}{W}_{a}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}):\dot{u}\in {L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})\}with the norm ‖u‖=infλ>0:∫Ωu˙λp(t)+a(t)uλp(t)dt≤1.\Vert u\Vert =\inf \left\{\lambda \gt 0:\mathop{\int }\limits_{\Omega }\left({\left|,\frac{\dot{u}}{\lambda },\right|}^{p\left(t)}+a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}\right){\rm{d}}t\le 1\right\}.In particular, if a(t)≡1a\left(t)\equiv 1, Wa1,p(t){W}_{a}^{1,p\left(t)}is reduced to W1,p(t)(Ω,RN)={u∈Lp(t)(Ω,RN):u˙∈Lp(t)(Ω,RN)}{W}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}):\dot{u}\in {L}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})\}and the norm ‖u‖=∣u∣p(t)+∣u˙∣p(t).\Vert u\Vert =| u{| }_{p\left(t)}+| \dot{u}{| }_{p\left(t)}.We use W01,p(t){W}_{0}^{1,p\left(t)}to represent the space of C0∞(Ω,RN){C}_{0}^{\infty }(\Omega ,{{\mathbb{R}}}^{N})consisting of infinitely continuous differentiable functions with compact supports on Ω\Omega completion in W1,p(t){W}^{1,p\left(t)}. We call the space Lp(t){L}^{p\left(t)}a generalized Lebesgue space, and it is a special kind of generalized Orlicz spaces. The space W1,p(t){W}^{1,p\left(t)}is called a generalized Sobolev space, it is a special kind of generalized Orlicz-Sobolev spaces. For more details on the general theory of generalized Orlicz spaces and generalized Orlicz-Sobolev spaces, see [18,20,51] and references therein.The following propositions summarize the main properties of this norm (see Alves and Liu [1], Rădulescu and Repovš [51] and Fan and Zhao [21]).Proposition 2.1Lap(t){L}_{a}^{p\left(t)}, Wa1,p(t){W}_{a}^{1,p\left(t)}, W01,p(t){W}_{0}^{1,p\left(t)}are reflexive Banach spaces with norms defined above when p−>1{p}^{-}\gt 1.Proposition 2.2Let ρ(u)=∫Ωa(t)∣u(t)∣p(t)dt\rho \left(u)={\int }_{\Omega }a\left(t)| u\left(t){| }^{p\left(t)}{\rm{d}}tfor any u,v∈Lap(t)u,v\in {L}_{a}^{p\left(t)}, then the following properties hold: (i)ρ(u)=0⇔u=0;\rho \left(u)=0\iff u=0;(ii)ρ(u)=ρ(−u);\rho \left(u)=\rho \left(-u);(iii)ρ(αu+βv)≤αρ(u)+βρ(v)\rho (\alpha u+\beta v)\le \alpha \rho \left(u)+\beta \rho \left(v)for any α,β≥0\alpha ,\beta \ge 0, α+β=1;\alpha +\beta =1;(iv)ρ(u+v)≤2p+(ρ(u)+ρ(v));\rho \left(u+v)\le {2}^{{p}^{+}}(\rho \left(u)+\rho \left(v));(v)If λ>1\lambda \gt 1, thenλp+ρ(u)≤ρ(λu)≤λp−ρ(u)≤λρ(u)≤ρ(u);{\lambda }^{{p}^{+}}\rho \left(u)\le \rho \left(\lambda u)\le {\lambda }^{{p}^{-}}\rho \left(u)\le \lambda \rho \left(u)\le \rho \left(u);(vi)‖u‖p(t),a=1\Vert u{\Vert }_{p\left(t),a}=1if and only if ρuλ=1\rho \left(\frac{u}{\lambda }\right)=1, for any u∈Lap(t)⧹{0}u\in {L}_{a}^{p\left(t)}\setminus \left\{0\right\}.Proposition 2.3For any u∈Lap(t)u\in {L}_{a}^{p\left(t)}, the following properties hold: (i)∣u∣p(t),a<1(=1;>1)⇔ρ(u)<1(=1;>1)| u{| }_{p\left(t),a}\lt 1\hspace{0.33em}\left(=1;\gt 1)\iff \rho \left(u)\lt 1\hspace{0.33em}\left(=1;\hspace{0.33em}\gt 1);(ii)If ∣u∣p(t),a>1| u{| }_{p\left(t),a}\gt 1, then ∣u∣p(t),ap−≤ρ(u)≤∣u∣p(t),ap+| u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{-}}\le \rho \left(u)\le | u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{+}};(iii)If ∣u∣p(t),a<1| u{| }_{p\left(t),a}\lt 1, then ∣u∣p(t),ap+≤ρ(u)≤∣u∣p(t),ap−| u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{+}}\le \rho \left(u)\le | u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{-}};(iv)∣u∣p(t),a→0⇔ρ(u)→0| u{| }_{p\left(t),a}\to 0\iff \rho \left(u)\to 0;(v)∣u∣p(t),a→∞⇔ρ(u)→∞| u{| }_{p\left(t),a}\to \infty \iff \rho \left(u)\to \infty .Proposition 2.4Let ϕ(u)=∫Ω(∣u˙∣p(t)+a(t)∣u∣p(t))dt\phi \left(u)={\int }_{\Omega }(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}tfor any u∈Wa1,p(t)u\in {W}_{a}^{1,p\left(t)}, then the following properties hold: (i)‖u‖<1(=1;>1)⇔ϕ(u)<1(=1;>1)\Vert u\Vert \lt 1\hspace{0.25em}\left(=1;\gt 1)\iff \phi \left(u)\lt 1\hspace{0.25em}\left(=\hspace{0.25em}1;\gt 1);(ii)If ‖u‖>1\Vert u\Vert \gt 1, then ‖u‖p−≤ϕ(u)≤‖u‖p+\Vert u{\Vert }^{{p}^{-}}\le \phi \left(u)\le \Vert u{\Vert }^{{p}^{+}};(iii)If ‖u‖<1\Vert u\Vert \lt 1, then ‖u‖p+≤ϕ(u)≤‖u‖p−\Vert u{\Vert }^{{p}^{+}}\le \phi \left(u)\le \Vert u{\Vert }^{{p}^{-}};(iv)‖u‖→0⇔ϕ(u)→0\Vert u\Vert \to 0\iff \phi \left(u)\to 0;(v)‖u‖→∞⇔ϕ(u)→∞\Vert u\Vert \to \infty \iff \phi \left(u)\to \infty .Proposition 2.5Let ρ(u)=∫Ωa(t)∣u∣p(t)dt\rho \left(u)={\int }_{\Omega }a\left(t)| u{| }^{p\left(t)}{\rm{d}}tfor any u∈Lap(t)u\in {L}_{a}^{p\left(t)}, {un}⊂Lap(t)\{{u}_{n}\}\subset {L}_{a}^{p\left(t)}, then the following properties are equivalent: (i)limn→∞∣un−u∣p(t),a=0{\mathrm{lim}}_{n\to \infty }| {u}_{n}-u{| }_{p\left(t),a}=0;(ii)limn→∞ρ(un−u)=0{\mathrm{lim}}_{n\to \infty }\rho \left({u}_{n}-u)=0;(iii)un→u{u}_{n}\to ua.e. t∈Ωt\in \Omega and limn→∞ρ(un)=ρ(u){\mathrm{lim}}_{n\to \infty }\rho \left({u}_{n})=\rho \left(u).Proposition 2.6(Lp(t))∗=Lq(t){({L}^{p\left(t)})}^{\ast }={L}^{q\left(t)}with 1/p(t)+1/q(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}q\left(t)=1and∫Ωu(t)v(t)dt≤2∣u∣p(t)∣v∣q(t),∀u∈Lp(t),v∈Lq(t),\left|\mathop{\int }\limits_{\Omega }u\left(t)v\left(t){\rm{d}}t\right|\le 2| u{| }_{p\left(t)}| v{| }_{q\left(t)},\hspace{1em}\forall u\in {L}^{p\left(t)},\hspace{1em}v\in {L}^{q\left(t)},where (Lp(t))∗{\left({L}^{p\left(t)})}^{\ast }is the dual space of Lp(t){L}^{p\left(t)}.Proposition 2.7C0∞(R,RN){C}_{0}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})is dense in space Wa1,p(t){W}_{a}^{1,p\left(t)}.Proposition 2.8Let u∈Wa1,p(t)u\in {W}_{a}^{1,p\left(t)}, then(i)u∈C(R,RN)u\in C({\mathbb{R}},{{\mathbb{R}}}^{N})and u(t)→0u\left(t)\to 0as ∣t∣→∞| t| \to \infty . Moreover, the embedding Wa1,p(t)↪L∞(R,RN){W}_{a}^{1,p\left(t)}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})is continuous, and there exists a constant κ>0\kappa \gt 0such that‖u‖L∞≤κ‖u‖,∀u∈Wa1,p(t);\Vert u{\Vert }_{{L}^{\infty }}\le \kappa \Vert u\Vert ,\hspace{1em}\forall u\in {W}_{a}^{1,p\left(t)};(ii)If H(p),H(a)\hspace{0.1em}\text{H(p),H(a)}\hspace{0.1em}hold and a(t)→+∞a\left(t)\to +\infty as ∣t∣→∞| t| \to \infty , then the embedding Wa1,p(t)↪L∞(R,RN){W}_{a}^{1,p\left(t)}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})is compact.Consider the following functional: I(u)=∫Ω1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt,∀u∈Wa1,p(t).I\left(u)=\mathop{\int }\limits_{\Omega }\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t,\hspace{1em}\forall u\in {W}_{a}^{1,p\left(t)}.We know that I∈C1(Wa1,p(t),R)I\in {C}^{1}\left({W}_{a}^{1,p\left(t)},{\mathbb{R}})under condition HH(a). Moreover, ⟨I′(u),v⟩=∫R(∣u˙∣p(t)−2u˙v˙+a(t)∣u∣p(t)−2uv)dt,∀u,v∈Wa1,p(t).\langle I^{\prime} \left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}(| \dot{u}{| }^{p\left(t)-2}\dot{u}\dot{v}+a\left(t)| u{| }^{p\left(t)-2}uv){\rm{d}}t,\hspace{1.0em}\forall u,v\in {W}_{a}^{1,p\left(t)}.Proposition 2.9I′I^{\prime} is a mapping of type (S)+{\left(S)}_{+}, i.e., ifun⇀uandlimn→∞(I′(un)−I′(u),un−u)≤0,{u}_{n}\rightharpoonup u\hspace{1em}and\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }(I^{\prime} \left({u}_{n})-I^{\prime} \left(u),{u}_{n}-u)\le 0,then un{u}_{n}has a convergent subsequence in Wa1,p(t){W}_{a}^{1,p\left(t)}.Denote (2.1)A=J′:Wa1,p(t)→(Wa1,p(t))∗,A=J^{\prime} :{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast },then we have ⟨A(u),v⟩=∫Ω(∣u′(t)∣p(t)−2u′(t)v′(t)+a(x)∣u(t)∣p(t)−2uv)dt\langle A\left(u),v\rangle =\mathop{\int }\limits_{\Omega }(| u^{\prime} \left(t){| }^{p\left(t)-2}u^{\prime} \left(t)v^{\prime} \left(t)+a\left(x)| u\left(t){| }^{p\left(t)-2}uv){\rm{d}}tfor all u,v∈Wa1,p(t).u,v\in {W}_{a}^{1,p\left(t)}.Proposition 2.10The mapping A is a strictly monotone, bounded homeomorphism and is of type (S)+{\left(S)}_{+}in Wa1,p(t){W}_{a}^{1,p\left(t)}.2.1.1Periodic variable exponential W2nb1,p(t){W}_{2nb}^{1,p\left(t)}spaceFor any b>0b\gt 0, n≥1n\ge 1, let Tn≐[−nb,nb]{T}_{n}\doteq \left[-nb,nb]. Define L2nbp(t)(Tn,RN){L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})(denoted by L2nbp(t){L}_{2nb}^{p\left(t)}) as follows: L2nbp(t)(Tn,RN)=u∈S(Tn,RN):∫−nbnba(t)∣u(t)∣p(t)dt<∞{L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})=\left\{u\in S({T}_{n},{{\mathbb{R}}}^{N}):\underset{-nb}{\overset{nb}{\int }}a\left(t)| u\left(t){| }^{p\left(t)}{\rm{d}}t\lt \infty \right\}endowed with the norm ∣u∣p(t)=infλ>0:∫−nbnba(t)uλp(t)dt≤1.| u{| }_{p\left(t)}=\inf \left\{\lambda \gt 0:\underset{-nb}{\overset{nb}{\int }}a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}{\rm{d}}t\le 1\right\}.Moreover, L2nb∞(Tn,RN){L}_{2nb}^{\infty }({T}_{n},{{\mathbb{R}}}^{N})(denoted by L2nb∞{L}_{2nb}^{\infty }) be a Banach space with the norm ‖u‖L2nb∞=esssup{∣u(t)∣:t∈[−nb,nb]}.\Vert u{\Vert }_{{L}_{2nb}^{\infty }}=\hspace{0.1em}\text{ess}\hspace{0.1em}\sup \{| u\left(t)| :t\in \left[-nb,nb]\}.Define W2nb1,p(t)(Tn,Rn){W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{n})(denoted by W2nb1,p(t){W}_{2nb}^{1,p\left(t)}) as follows: W2nb1,p(t)(Tn,Rn)={u∈L2nbp(t)(Tn,RN):u˙∈L2nbp(t)(Tn,RN)}{W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{n})=\{u\in {L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}):\dot{u}\in {L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})\}endowed with the norm ‖u‖1=infλ>0:∫−nbnbu˙λp(t)+a(t)uλp(t)dt≤1.\Vert u{\Vert }_{1}=\inf \left\{\lambda \gt 0:\underset{-nb}{\overset{nb}{\int }}\left({\left|,\frac{\dot{u}}{\lambda },\right|}^{p\left(t)}+a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}\right){\rm{d}}t\le 1\right\}.In particular, if b→+∞b\to +\infty , L2nbp(t)(Tn,RN){L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}), L2nb∞(Tn,RN){L}_{2nb}^{\infty }({T}_{n},{{\mathbb{R}}}^{N}), W2nb1,p(t)(Tn,Rn){W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{n})are written simply by Lap(t)(R,RN){L}_{a}^{p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), La∞(R,RN){L}_{a}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N}), Wa1,p(t)(R,Rn){W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{n}), respectively.2.2Nonsmooth analysis theoryThe nonsmooth critical point theory for locally Lipschitz functionals is based on the subdifferential theory of Clark [11], Rădulescu [49], Gasiński and Papageorgiou [25].Definition 2.11Let XXbe a Banach space and let X∗{X}^{\ast }be its topological dual. By ⟨⋅⟩\langle \cdot \rangle we denote the duality brackets for the pair (X,X∗)(X,{X}^{\ast }). A function ϕ:X→R\phi :X\to {\mathbb{R}}is said to be locally Lipschitz, if for every x∈Xx\in Xthere exist U∈N(x)U\in {\mathcal{N}}\left(x)and a constant kU>0{k}_{U}\gt 0, such that ∣ϕ(y)−ϕ(z)∣≤kU‖y−z‖X,∀y,z∈U.| \phi (y)-\phi \left(z)| \le {k}_{U}\Vert y-z{\Vert }_{X},\hspace{1em}\forall y,z\in U.Definition 2.12For a given locally Lipschitz function ϕ:X→R,\phi :X\to {\mathbb{R}},the generalized directional derivative of φ\varphi at x∈Xx\in Xin the direction h∈Xh\in Xis defined by ϕ0(x;h)≐limy→x;λ↓0supϕ(y+λh)−ϕ(y)λ=infε,δ>0sup‖x−y‖X<δ,0<λ<δϕ(y+λh)−ϕ(y)λ.\begin{array}{rcl}{\phi }^{0}\left(x;h)& \doteq & \mathop{\mathrm{lim}}\limits_{y\to x;\lambda \downarrow 0}\sup \frac{\phi (y+\lambda h)-\phi (y)}{\lambda }\\ & =& \mathop{\inf }\limits_{\varepsilon ,\delta \gt 0}\mathop{\sup }\limits_{\Vert x-y{\Vert }_{X}\lt \delta ,0\lt \lambda \lt \delta }\frac{\phi (y+\lambda h)-\phi (y)}{\lambda }.\end{array}Based on Definition 2.12, one can easily verify that the function h↦ϕ0(x;h)h\mapsto {\phi }^{0}\left(x;h)is sublinear, Lipschitz continuous (see [11, Proposition 2.1.1]).Definition 2.13Let ϕ:X→R\phi :X\to {\mathbb{R}}be a locally Lipschitz function. Then generalized subdifferential of ϕ\phi at x∈Xx\in Xis the nonempty set ∂ϕ(x)⊆X∗\partial \phi \left(x)\subseteq {X}^{\ast }defined by ∂ϕ(x)={x∗∈X∗:⟨x∗,h⟩≤ϕ0(x;h),∀h∈X}.\partial \phi \left(x)=\{{x}^{\ast }\in {X}^{\ast }:\langle {x}^{\ast },h\rangle \le {\phi }^{0}\left(x;h),\hspace{1em}\forall h\in X\}.The multifunction x→∂ϕ(x)x\to \partial \phi \left(x)is known as the generalized (or Clarke) subdifferential of ϕ\phi . If ϕ,ψ:X→R\phi ,\psi :X\to {\mathbb{R}}are locally Lipschitz functions, then ∂(ϕ+ψ)(x)⊆∂ϕ(x)+∂ψ(x)\partial \left(\phi +\psi )\left(x)\subseteq \partial \phi \left(x)+\partial \psi \left(x)and for every λ∈R\lambda \in {\mathbb{R}}, ∂(λϕ)(x)=λ∂ϕ(x)\partial \left(\lambda \phi )\left(x)=\lambda \partial \phi \left(x).Definition 2.14Let ϕ:X→R\phi :X\to {\mathbb{R}}be a locally Lipschitz function. A point x∈Xx\in Xis said to be a critical point of ϕ\phi if 0∈∂ϕ(x)0\in \partial \phi \left(x).If x∈Xx\in Xis a critical point of ϕ\phi , then c=ϕ(x)c=\phi \left(x)is a critical value of ϕ\phi . It is easy to see that, if x∈Xx\in Xis a local extremum of ϕ\phi , then 0∈∂ϕ(x)0\in \partial \phi \left(x). Moreover, the multifunction x→∂ϕ(x)x\to \partial \phi \left(x)is upper semicontinuous from XXinto X∗{X}^{\ast }equipped with the w∗{w}^{\ast }topology, i.e., for any U⊆X∗U\subseteq {X}^{\ast }w∗{w}^{\ast }-open, the set {x∈X:∂ϕ(x)⊆U}\left\{x\in X:\partial \phi \left(x)\subseteq U\right\}is open in XX. For more details we refer to Clarke [11, Proposition 2.1.2].Definition 2.15The locally Lipschitz function ϕ:X→R\phi :X\to {\mathbb{R}}satisfies the nonsmooth Palais-Smale (PS) condition, if any sequence {xn}n≥1⊆X{\left\{{x}_{n}\right\}}_{n\ge 1}\subseteq Xsuch that {ϕ(xn)}n≥1is boundedandm(xn)→0asn→∞,{\left\{\phi \left({x}_{n})\right\}}_{n\ge 1}\hspace{1em}\hspace{0.1em}\text{is bounded}\hspace{0.1em}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({x}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,has a strongly convergent subsequence, where m(xn)=min[‖x∗‖:x∗∈∂ϕ(xn)]m\left({x}_{n})=\min {[}\Vert {x}^{\ast }\Vert :{x}^{\ast }\in \partial \phi \left({x}_{n})].Lemma 2.16(Lebourg’s mean value Theorem [39]). Given the points x and y in X and a real-valued function ϕ\phi which is Lipschitz continuous on an open set containing the segment [x,y]={(1−t)x+ty:t∈[0,1]}\left[x,y]=\left\{\left(1-t)x+ty:t\in \left[0,1]\right\}, there exist z=x+t0(y−x)z=x+{t}_{0}(y-x), with 0<t0<10\lt {t}_{0}\lt 1, and x∗∈∂ϕ(z){x}^{\ast }\in \partial \phi \left(z)such thatϕ(y)−ϕ(x)=⟨x∗,y−x⟩.\phi (y)-\phi \left(x)=\langle {x}^{\ast },y-x\rangle .If ϕ∈C1(X,R)\phi \in {C}^{1}\left(X,{\mathbb{R}}), then as we already mentioned ∂ϕ(x)={ϕ′(x)}\partial \phi \left(x)=\left\{\phi ^{\prime} \left(x)\right\}and so the above definition of the PS condition coincides with the classical (smooth) one. In the context of the smooth theory, Cerami introduced a weaker compactness condition which in our nonsmooth setting has the following form:Definition 2.17The locally Lipschitz function ϕ:X→R\phi :X\to {\mathbb{R}}satisfies the nonsmooth Cerami condition (C-condition), if any sequence {xn}n≥1⊆X{\left\{{x}_{n}\right\}}_{n\ge 1}\subseteq Xsuch that {ϕ(xn)}n≥1is bounded and(1+‖xn‖)m(xn)→0asn→∞,{\left\{\phi \left({x}_{n})\right\}}_{n\ge 1}\hspace{0.33em}\hspace{0.1em}\text{is bounded and}\hspace{0.1em}\hspace{0.33em}(1+\Vert {x}_{n}\Vert )m\left({x}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,has a strongly convergent subsequence.Lemma 2.18(Weierstrass theorem [43]). Assume that φ\varphi is a locally Lipschitz functional on a Banach space X and φ:X→R\varphi :X\to {\mathbb{R}}satisfies: (i)φ\varphi is weakly lower semicontinuous;(ii)φ\varphi is coercive.Then there exists x∗∈X{x}^{\ast }\in Xsuch that φ(x∗)=minx∈Xφ(x)\varphi \left({x}^{\ast })={\min }_{x\in X}\varphi \left(x).Lemma 2.19(Nonsmooth mountain pass theorem [35]). Let X be a reflexive Banach space, ϕ:X→R\phi :X\to {\mathbb{R}}a locally Lipschitz functional satisfying the PS-condition. Assume that there exist x0,x1∈X,{x}_{0},{x}_{1}\in X,c0∈R{c}_{0}\in {\mathbb{R}}and ϱ>0\varrho \gt 0such that ‖x1−x0‖>ϱ\Vert {x}_{1}-{x}_{0}\Vert \gt \varrho andmax{ϕ(x0),ϕ(x1)}<c0=inf[ϕ(y):‖y−x0‖=ϱ].\max \left\{\phi \left({x}_{0}),\phi \left({x}_{1})\right\}\lt {c}_{0}=\inf {[}\phi (y):\Vert y-{x}_{0}\Vert =\varrho ].Then ϕ\phi has a critical point x∈Xx\in Xwith c=ϕ(x)≥c0c=\phi \left(x)\ge {c}_{0}, where c is given byc=infγ∈Γ1maxt∈Tϕ(γ(t)),c=\mathop{\inf }\limits_{\gamma \in {\Gamma }_{1}}\mathop{\max }\limits_{t\in T}\phi \left(\gamma \left(t)),Γ1={γ∈C([0,1],x):γ(0)=x0,γ(1)=x1}.{\Gamma }_{1}=\left\{\gamma \in C\left(\left[0,1],x):\gamma \left(0)={x}_{0},\gamma \left(1)={x}_{1}\right\}.3Periodic p(t)p\left(t)-Laplacian inclusion systemIn this section, we establish the existence of homoclinic solutions with periodic assumption for problem (1.1). In this situation, our hypotheses on p,ap,aand ffare the following: H(p)1p(t)p\left(t)is 2b2b-periodic;H(a)1a(t)a\left(t)is 2b2b-periodic;H(f)1(i)the function f(t,⋅):R→Rf\left(t,\cdot ):{\mathbb{R}}\to {\mathbb{R}}is 2b2b-periodic;(ii)for almost all t∈T=[−b,b]t\in T=\left[-b,b], there exists a function α(t)∈C(R)∩Lγ(t)γ(t)−α(t)(R)\alpha \left(t)\in C\left({\mathbb{R}})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}\left({\mathbb{R}})such that ∣ω∣≤a(t)(1+∣u∣α(t)−1),∀u∈RN,ω∈∂f(t,u(t)),| \omega | \le a\left(t)\left(1+| u{| }^{\alpha \left(t)-1}),\hspace{1em}\forall u\in {{\mathbb{R}}}^{N},\hspace{0.33em}\omega \in \partial f\left(t,u\left(t)),where a∈L∞(R)a\in {L}^{\infty }\left({\mathbb{R}}), α+<γ−<γ(t)<γ+<p−{\alpha }^{+}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(iii)there exist constants M,α,β>0M,\alpha ,\beta \gt 0such that 0≤p++1α+β∣u∣νf(t,u)≤−f0(t,u;−u)∀t∈T,∣u∣≥M,0\le \left({p}^{+}+\frac{1}{\alpha +\beta | u{| }^{\nu }}\right)f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\forall t\in T,\hspace{1em}\hspace{1em}| u| \ge M,where ν<p−\nu \lt {p}^{-};(iii′)there exist constants μ>p+,M>0\mu \gt {p}^{+},M\gt 0such that μf(t,u)≤−f0(t,u;−u)∀t∈T,∣u∣≥M;\mu f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\forall t\in T,\hspace{0.33em}| u| \ge M;(iv)there exists a function q(t)>0q\left(t)\gt 0such that lim∣u∣→0(w,u)∣u∣p(t)≤0,lim∣u∣→+∞inff(t,u)∣u∣q(t)>0,∀t∈T,w∈∂f(t,u),\mathop{\mathrm{lim}}\limits_{| u| \to 0}\frac{\left(w,u)}{| u{| }^{p\left(t)}}\le 0,\hspace{1em}\mathop{\mathrm{lim}}\limits_{| u| \to +\infty }\inf \frac{f\left(t,u)}{| u{| }^{q\left(t)}}\gt 0,\hspace{1em}\forall t\in T,\hspace{0.33em}w\in \partial f\left(t,u),where p+<q−{p}^{+}\lt {q}^{-}.Our main results can be stated as follows.Theorem 3.1If hypotheses H(p), H(p)1{\text{H(p)}}_{1}, H(a), H(a)1{\text{H(a)}}_{1}, H(f), and H(f)1{\text{H(f)}}_{1}: (i), (ii), (iii), (iv) hold, then problem (1.1) has a nontrivial homoclinic solution.Theorem 3.2If hypotheses H(p), H(p)1{\text{H(p)}}_{1}, H(a), H(a)1{\text{H(a)}}_{1}, H(f), and H(f)1{\text{H(f)}}_{1}: (i), (ii), (iii′^{\prime} ), (iv) hold, then problem (1.1) has a nontrivial homoclinic solution.Proof of Theorem 3.1We consider the following auxiliary periodic problem: (3.1)−ddt(∣u˙(t)∣p(t)−2u˙(t))+a(t)∣u(t)∣p(t)−2u(t)∈∂f(t,u(t)),a.e.t∈Tn,u(−nb)=u(nb),u˙(−nb)=u˙(nb).\left\{\begin{array}{l}-\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))+a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial f\left(t,u\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e.}\hspace{0.1em}\hspace{1em}t\in {T}_{n},\hspace{1.0em}\\ u\left(-nb)=u\left(nb),\dot{u}\left(-nb)=\dot{u}\left(nb).\hspace{1.0em}\end{array}\right.From [5], we know that problem (3.1) has a nontrivial solution un∈C2nb1(Tn,RN){u}_{n}\in {C}_{2nb}^{1}({T}_{n},{{\mathbb{R}}}^{N}). Let φn:W2nb1,p(t)(Tn,RN)→R{\varphi }_{n}:{W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})\to {\mathbb{R}}be defined by (3.2)φn(u)=∫−nbnb1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−∫−nbnbf(t,u(t))dt=φ˜n(u)−∫−nbnbf(t,u(t))dt.{\varphi }_{n}\left(u)=\underset{-nb}{\overset{nb}{\int }}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}f\left(t,u\left(t)){\rm{d}}t={\widetilde{\varphi }}_{n}\left(u)-\underset{-nb}{\overset{nb}{\int }}f\left(t,u\left(t)){\rm{d}}t.We claim that φn{\varphi }_{n}be the locally Lipschitz functional. In fact, for all u1,u2∈W2nb1,p(t)(Tn,RN){u}_{1},{u}_{2}\in {W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}), one has (3.3)∣φ˜n(u1)−φ˜n(u2)∣=∣φ˜n′(u˜)⋅(u1−u2)∣,| {\widetilde{\varphi }}_{n}({u}_{1})-{\widetilde{\varphi }}_{n}({u}_{2})| =| {\widetilde{\varphi }}_{n}^{^{\prime} }\left(\tilde{u})\cdot ({u}_{1}-{u}_{2})| ,where u˜=su1+(1−s)u2,s∈(0,1)\tilde{u}=s{u}_{1}+\left(1-s){u}_{2},s\in \left(0,1). Let Ω⊂Tn\Omega \subset {T}_{n}, fix u0∈W2nb1,p(t)(Ω,RN){u}_{0}\in {W}_{2nb}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})and Br={u∈W2nb1,p(t)(Tn,RN):∥u−u0∥1≤r}.{B}_{r}=\{u\in {W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}):{\parallel u-{u}_{0}\parallel }_{1}\le r\}.Note that Br{B}_{r}is compact, which yields that there exists C1>0{C}_{1}\gt 0such that (3.4)∥φ˜n′(u˜)∥W2nb−1,p(t)(Tn,RN)≤C1,{\parallel {\widetilde{\varphi }}_{n}^{^{\prime} }\left(\tilde{u})\parallel }_{{W}_{2nb}^{-1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})}\le {C}_{1},as r→0r\to 0. Then, it follows from (3.3) and (3.4), we obtain (3.5)φ˜n(u1)−φ˜n(u2)∣=∣φ˜n(u˜)⋅(u1−u2)∣≤∥φ˜n′(u˜)∥W2nb−1,p(t)(Tn,RN)∥u1−u2∥1≤C1∥u1−u2∥1,{\widetilde{\varphi }}_{n}({u}_{1})-{\widetilde{\varphi }}_{n}({u}_{2})| =| {\widetilde{\varphi }}_{n}\left(\tilde{u})\cdot ({u}_{1}-{u}_{2})| \le {\parallel {\widetilde{\varphi }}_{n}^{^{\prime} }\left(\tilde{u})\parallel }_{{W}_{2nb}^{-1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})}{\parallel {u}_{1}-{u}_{2}\parallel }_{1}\le {C}_{1}{\parallel {u}_{1}-{u}_{2}\parallel }_{1},for all u1,u2∈W2nb1,p(t)(Ω,RN){u}_{1},{u}_{2}\in {W}_{2nb}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}).On the other hand, it follows from H(f)1{}_{1}: (ii) and Lemma 2.16, for all u1,u2∈W2nb1,p(t)(Ω,RN){u}_{1},{u}_{2}\in {W}_{2nb}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}), we have (3.6)∣f(t,u1)−f(t,u2)∣≤a(t)(1+∣u˜∣α(t)−1)∣u1−u2∣| f(t,{u}_{1})-f(t,{u}_{2})| \le a\left(t)(1+| \tilde{u}{| }^{\alpha \left(t)-1})| {u}_{1}-{u}_{2}| and a(t)∣u˜∣α(t)−1≤(γ(t)−α(t))∣a(t)∣γ(t)−1γ(t)−α(t)γ(t)−1+α(t)−1γ(t)−1∣u˜∣γ(t)−1,a\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}\le \frac{\left(\gamma \left(t)-\alpha \left(t))| a\left(t){| }^{\tfrac{\gamma \left(t)-1}{\gamma \left(t)-\alpha \left(t)}}}{\gamma \left(t)-1}+\frac{\alpha \left(t)-1}{\gamma \left(t)-1}| \tilde{u}{| }^{\gamma \left(t)-1},which imply that there exist some constants C2{C}_{2}, C3>0{C}_{3}\gt 0such that (3.7)(a(t)∣u˜∣α(t)−1)γ(t)γ(t)−1≤C2∣a(t)∣γ(t)γ(t)−α(t)+C3∣u˜∣γ(t).{(a\left(t)| \tilde{u}{| }^{\alpha \left(t)-1})}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}\le {C}_{2}| a\left(t){| }^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}+{C}_{3}| \tilde{u}{| }^{\gamma \left(t)}.Then, in virtue of (3.6), (3.7) and Hölder inequality, one has (3.8)∫−nbnbf(t,u1)dt−∫−nbnbf(t,u2)dt≤∫−nbnba(x)(1+∣u˜∣α(t)−1)∣u1−u2∣dt≤∣a(t)∣γ(t)γ(t)−1+∣a(t)∣u˜∣α(t)−1∣γ(t)γ(t)−1∣u1−u2∣γ(t)≤C4∣∣u1−u2∣∣1.\begin{array}{rcl}\left|\underset{-nb}{\overset{nb}{\displaystyle \int }}f(t,{u}_{1}){\rm{d}}t-\underset{-nb}{\overset{nb}{\displaystyle \int }}f(t,{u}_{2}){\rm{d}}t\right|& \le & \underset{-nb}{\overset{nb}{\displaystyle \int }}a\left(x)(1+| \tilde{u}{| }^{\alpha \left(t)-1})| {u}_{1}-{u}_{2}| {\rm{d}}t\\ & \le & \left[{| a\left(t)| }_{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}+{| a\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| }_{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}\right]{| {u}_{1}-{u}_{2}| }_{\gamma \left(t)}\\ & \le & {C}_{4}| | {u}_{1}-{u}_{2}| {| }_{1}.\end{array}Hence, from (3.2), (3.5) and (3.8), we obtain φn(u)=φ˜n(u)−∫−nbnbf(t,u(t))dt≤C1∣∣u1−u2∣∣1+C4∣∣u1−u2∣∣1≤C5∣∣u1−u2∣∣1,{\varphi }_{n}\left(u)={\widetilde{\varphi }}_{n}\left(u)-\underset{-nb}{\overset{nb}{\int }}f\left(t,u\left(t)){\rm{d}}t\le {C}_{1}| | {u}_{1}-{u}_{2}| {| }_{1}+{C}_{4}| | {u}_{1}-{u}_{2}| {| }_{1}\le {C}_{5}| | {u}_{1}-{u}_{2}| {| }_{1},which yields that φn{\varphi }_{n}be the nonsmooth locally Lipschitz energy functional corresponding to problem (3.1). Therefore, it follows from (3.2), H(f): (ii) and H(f)1{}_{1}: (iii), (iv), for σ≥1\sigma \ge 1, there exist C6,C7>0{C}_{6},{C}_{7}\gt 0such that φ1(σu)=∫−bb1p(t)(∣σu˙∣p(t)+a(t)∣σu∣p(t))dt−∫−bbf(t,σu)dt≤σp+∫−bb1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−σq−∫−bb∣u∣q(t)dt−C6σp+∫−bb∣u∣p+dt+2bC7.\begin{array}{rcl}{\varphi }_{1}\left(\sigma u)& =& \underset{-b}{\overset{b}{\displaystyle \int }}\frac{1}{p\left(t)}(| \sigma \dot{u}{| }^{p\left(t)}+a\left(t)| \sigma u{| }^{p\left(t)}){\rm{d}}t-\underset{-b}{\overset{b}{\displaystyle \int }}f\left(t,\sigma u){\rm{d}}t\\ & \le & {\sigma }^{{p}^{+}}\underset{-b}{\overset{b}{\displaystyle \int }}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-{\sigma }^{{q}^{-}}\underset{-b}{\overset{b}{\displaystyle \int }}| u{| }^{q\left(t)}{\rm{d}}t-{C}_{6}{\sigma }^{{p}^{+}}\underset{-b}{\overset{b}{\displaystyle \int }}| u{| }^{{p}^{+}}{\rm{d}}t+2b{C}_{7}.\end{array}Since p+<q−{p}^{+}\lt {q}^{-}, there exists a constant σ0>0{\sigma }_{0}\gt 0such that σ>σ0\sigma \gt {\sigma }_{0}and u¯∈W2b1,p(t)(T1,RN)\bar{u}\in {W}_{2b}^{1,p\left(t)}({T}_{1},{{\mathbb{R}}}^{N}), we have φ1(σu¯)<0{\varphi }_{1}(\sigma \bar{u})\lt 0.Let uˆ∈W2b1,p(t)(T1,RN)\hat{u}\in {W}_{2b}^{1,p\left(t)}({T}_{1},{{\mathbb{R}}}^{N})be defined as follows: uˆ(t)=u¯(t),ift∈T1;0,ift∈Tn⧹T1.\hat{u}\left(t)=\left\{\begin{array}{ll}\bar{u}\left(t),\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\in {T}_{1};\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\in {T}_{n}\setminus {T}_{1}.\end{array}\right.Note that f(t,0)=0f\left(t,0)=0, then for all σ≥σ0\sigma \ge {\sigma }_{0}, we deduce that φn(σuˆ)=φ1(σu¯){\varphi }_{n}\left(\sigma \hat{u})={\varphi }_{1}\left(\sigma \bar{u}).As in [5], we see that the solution un∈C2nb1(Tn,RN){u}_{n}\in {C}_{2nb}^{1}\left({T}_{n},{{\mathbb{R}}}^{N})of problem (3.1) is obtained via the nonsmooth mountain pass theorem. One will immediately obtain the fact that there exists ρ>0\rho \gt 0such that cn≔infγ∈Γnsupt∈[0,1]φn(γ(t))=φn(un)≥inf[φn(u):‖u‖=ρ]>0,{c}_{n}:= {\inf }_{\gamma \in {\Gamma }_{n}}\mathop{\sup }\limits_{t\in \left[0,1]}{\varphi }_{n}\left(\gamma \left(t))={\varphi }_{n}\left({u}_{n})\ge \inf {[}{\varphi }_{n}\left(u):\Vert u\Vert =\rho ]\gt 0,where Γn={γ∈C([0,1],W2nb1,p(t)):γ(0)=0,γ(1)=σuˆ}{\Gamma }_{n}=\{\gamma \in C(\left[0,1],{W}_{2nb}^{1,p\left(t)}):\gamma \left(0)=0,\gamma \left(1)=\sigma \hat{u}\}for σ≥σ0\sigma \ge {\sigma }_{0}and 0∈∂φn(un)0\in \partial {\varphi }_{n}\left({u}_{n})for all n≥1n\ge 1. Extending by constant, as n1≤n2{n}_{1}\le {n}_{2}we see that Wn11,p(t)⊆Wn21,p(t)andΓn1⊆Γn2{W}_{{n}_{1}}^{1,p\left(t)}\subseteq {W}_{{n}_{2}}^{1,p\left(t)}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{\Gamma }_{{n}_{1}}\subseteq {\Gamma }_{{n}_{2}}and consequently cn2≤cn1,∀n1<n2.{c}_{{n}_{2}}\le {c}_{{n}_{1}},\hspace{1em}\forall {n}_{1}\lt {n}_{2}.This way we have produced a decreasing sequence {cn}n≥1{\left\{{c}_{n}\right\}}_{n\ge 1}of critical values. For every n≥1n\ge 1, from (3.2), we have (3.9)cn=φn(un)=∫−nbnb1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫−nbnbf(t,un(t))dt≤c1,{c}_{n}={\varphi }_{n}\left({u}_{n})=\underset{-nb}{\overset{nb}{\int }}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}f\left(t,{u}_{n}\left(t)){\rm{d}}t\le {c}_{1},which implies that (3.10)∫−nbnbp+p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫−nbnbp+f(t,un(t))dt≤p+c1.\underset{-nb}{\overset{nb}{\int }}\frac{{p}^{+}}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}{p}^{+}f\left(t,{u}_{n}\left(t)){\rm{d}}t\le {p}^{+}{c}_{1}.Then, it follows from (3.10), we obtain (3.11)∫−nbnb(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫−nbnbp+f(t,un(t))dt≤p+c1.\underset{-nb}{\overset{nb}{\int }}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}{p}^{+}f\left(t,{u}_{n}\left(t)){\rm{d}}t\le {p}^{+}{c}_{1}.Using (2.1), one has A(un)=wn,wn∈L2nb∞,wn(t)∈∂f(t,un(t)),a.e.t∈Tn,\begin{array}{l}A\left({u}_{n})={w}_{n},\hspace{1em}{w}_{n}\in {L}_{2nb}^{\infty },\\ {w}_{n}\left(t)\in \partial f\left(t,{u}_{n}\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e.}\hspace{0.1em}\hspace{0.33em}t\in {T}_{n},\end{array}which yields that (3.12)−∫−nbnb(∣u˙n∣p(t)+a(t)∣un∣p(t))dt=∫−nbnb⟨wn(t),−un(t)⟩dt≤∫−nbnbf0(t,un(t);−un(t))dt.-\underset{-nb}{\overset{nb}{\int }}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t=\underset{-nb}{\overset{nb}{\int }}\langle {w}_{n}\left(t),-{u}_{n}\left(t)\rangle {\rm{d}}t\le \underset{-nb}{\overset{nb}{\int }}{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t)){\rm{d}}t.Combining (3.11) with (3.12), we obtain (3.13)∫−nbnb(−p+f(t,un)−f0(t,un;−un))dt≤p+c1.\underset{-nb}{\overset{nb}{\int }}(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t\le {p}^{+}{c}_{1}.By virtue of H(f)1{}_{1}: (iii), one has f(t,un)≤(α+β∣un∣ν)(−p+f(t,un)−f0(t,un;−un)),∀∣un∣>M.f\left(t,{u}_{n})\le (\alpha +\beta | {u}_{n}{| }^{\nu })(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})),\hspace{1em}\forall | {u}_{n}| \gt M.Hence, from H(f): (ii), (3.13) and Proposition 2.8 (i), there exists a constant ξ0{\xi }_{0}which is independent of nnsuch that ∫−nbnbf(t,un)dt=∫Tn∩{∣un∣>M}f(t,un)dt+∫Tn∩{∣un∣≤M}f(t,un)dt≤∫Tn∩{∣un∣>M}(α+β∣un∣ν)(−p+f(t,un)−f0(t,un;−un))dt+ξ0≤(α+β‖un‖∞ν)∫Tn∩{∣un∣>M}(−p+f(t,un)−f0(t,un;−un))dt+ξ0≤(α+β‖un‖∞ν)∫−nbnb(−p+f(t,un)−f0(t,un;−un))dt+ξ0≤p+c1(α+βκν‖un‖ν)+ξ0,\begin{array}{rcl}\underset{-nb}{\overset{nb}{\displaystyle \int }}f\left(t,{u}_{n}){\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \gt M\right\}}f\left(t,{u}_{n}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \le M\right\}}f\left(t,{u}_{n}){\rm{d}}t\\ & \le & \mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \gt M\right\}}(\alpha +\beta | {u}_{n}{| }^{\nu })(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t+{\xi }_{0}\\ & \le & (\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \gt M\right\}}(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t+{\xi }_{0}\\ & \le & (\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\underset{-nb}{\overset{nb}{\displaystyle \int }}(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t+{\xi }_{0}\\ & \le & {p}^{+}{c}_{1}(\alpha +\beta {\kappa }^{\nu }\Vert {u}_{n}{\Vert }^{\nu })+{\xi }_{0},\end{array}which, together with (3.2), (3.9) and Proposition 2.4 (ii), imply that 1p+‖un‖p−≤1p+∫−nbnb(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤∫−nbnb1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt=φn(un)+∫−nbnbf(t,un)dt≤c1+p+c1(α+βκν‖un‖ν)+ξ0,\begin{array}{rcl}\frac{1}{{p}^{+}}\Vert {u}_{n}{\Vert }^{{p}^{-}}& \le & \frac{1}{{p}^{+}}\underset{-nb}{\overset{nb}{\displaystyle \int }}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & \le & \underset{-nb}{\overset{nb}{\displaystyle \int }}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & =& {\varphi }_{n}\left({u}_{n})+\underset{-nb}{\overset{nb}{\displaystyle \int }}f\left(t,{u}_{n}){\rm{d}}t\\ & \le & {c}_{1}+{p}^{+}{c}_{1}(\alpha +\beta {\kappa }^{\nu }\Vert {u}_{n}{\Vert }^{\nu })+{\xi }_{0},\end{array}for ‖un‖≥1\Vert {u}_{n}\Vert \ge 1. Since ν<p−\nu \lt {p}^{-}, there exists a constant ξ1{\xi }_{1}which is independent of nnsuch that (3.14)∫−nbnb1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤c1+p+c1(α+βκνξ1ν)+ξ0≔ξ2,\underset{-nb}{\overset{nb}{\int }}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\le {c}_{1}+{p}^{+}{c}_{1}(\alpha +\beta {\kappa }^{\nu }{\xi }_{1}^{\nu })+{\xi }_{0}:= {\xi }_{2},where ξ2{\xi }_{2}is a constant which is independent of nn. Thus, by (3.14), we obtain (3.15)‖un‖W2nb1,p(t)≤ξ3,\Vert {u}_{n}{\Vert }_{{W}_{2nb}^{1,p\left(t)}}\le {\xi }_{3},where ξ3>0{\xi }_{3}\gt 0is independent of nn. Moreover, by an argument as in the proof of [46, (2.19)], there exists a constant ξ4{\xi }_{4}which is independent of nnsuch that (3.16)‖un‖L2nb∞≤ξ4.\Vert {u}_{n}{\Vert }_{{L}_{2nb}^{\infty }}\le {\xi }_{4}.In what follows, we extend by periodicity un{u}_{n}and wn{w}_{n}to all of R{\mathbb{R}}. From (3.15) and the fact that the embedding W2nb1,p(t)↪C(Tn,RN){W}_{2nb}^{1,p\left(t)}\hspace{0.33em}\hookrightarrow \hspace{0.33em}C\left({T}_{n},{{\mathbb{R}}}^{N})is compact, we may assume that (3.17)un→uinCloc(R,RN),{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N}),hence u∈C(R,RN)u\in C\left({\mathbb{R}},{{\mathbb{R}}}^{N}). In view of hypothesis H(f)1{}_{1}: (ii), one has ‖wn(t)‖≤‖a‖∞(1+‖un(t)‖α(t)−1)=ξ5,\Vert {w}_{n}\left(t)\Vert \le \Vert a{\Vert }_{\infty }(1+\Vert {u}_{n}\left(t){\Vert }^{\alpha \left(t)-1})={\xi }_{5},where ξ5>0{\xi }_{5}\gt 0is a constant which is independent of nn. Passing to a subsequence if needed, we may assume that wn⇀w{w}_{n}\rightharpoonup win La∞(R,RN){L}_{a}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})and wn⇀w{w}_{n}\rightharpoonup win L2nbq(t)(Tn,RN){L}_{2nb}^{q\left(t)}({T}_{n},{{\mathbb{R}}}^{N}), where 1/p(t)+1/q(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}q\left(t)=1. It is obvious that w∈La∞(R,RN)∩Llocq(t)(R,RN)w\in {L}_{a}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})\cap {L}_{{\rm{loc}}}^{q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), thus w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t))in Tn{T}_{n}for all n≥1n\ge 1and w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t))on R{\mathbb{R}}.For any τ>0\tau \gt 0, we have ∫−ττ∣un(t)−u(t)∣p(t)dt→0,asn→∞,\underset{-\tau }{\overset{\tau }{\int }}| {u}_{n}\left(t)-u\left(t){| }^{p\left(t)}{\rm{d}}t\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,which shows that (3.18)limn→∞∫−ττ∣un(t)∣p(t)dt=∫−ττ∣u(t)∣p(t)dt.\mathop{\mathrm{lim}}\limits_{n\to \infty }\underset{-\tau }{\overset{\tau }{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t=\underset{-\tau }{\overset{\tau }{\int }}| u\left(t){| }^{p\left(t)}{\rm{d}}t.Choose n0≥1{n}_{0}\ge 1such that [−τ,τ]⊆Tn0=[−n0b,n0b]\left[-\tau ,\tau ]\subseteq {T}_{{n}_{0}}=\left[-{n}_{0}b,{n}_{0}b]and for n≥n0n\ge {n}_{0}, from (3.15), we derive (3.19)∫−ττ∣un(t)∣p(t)dt≤∫−n0bn0b∣un(t)∣p(t)dt≤max{ξ3p+,ξ3p−}=ξ30.\underset{-\tau }{\overset{\tau }{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t\le \underset{-{n}_{0}b}{\overset{{n}_{0}b}{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t\le \max \{{\xi }_{3}^{{p}^{+}},{\xi }_{3}^{{p}^{-}}\}={\xi }_{3}^{0}.Thus, from (3.18) and (3.19), one has (3.20)∫−ττ∣u(t)∣p(t)dt≤ξ30.\underset{-\tau }{\overset{\tau }{\int }}| u\left(t){| }^{p\left(t)}{\rm{d}}t\le {\xi }_{3}^{0}.By the arbitrariness of τ>0\tau \gt 0, from (3.20), we deduce that u∈Lp(t)u\in {L}^{p\left(t)}.Let θ∈C0∞(R,RN)\theta \in {C}_{0}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N}), then suppθ⊆Tn\hspace{0.1em}\text{supp}\hspace{0.1em}\theta \subseteq {T}_{n}for large n≥1n\ge 1, which together with (3.15), performs integration by parts and Proposition 2.6, we have ∫R(un(t),θ˙(t))dt=∫R(θ(t),u˙n(t))dt=∫−nbnb(u˙n(t),θ(t))dt≤‖u˙n‖L2nbp(t)‖θ‖L2nbq(t)≤ξ3‖θ‖L2nbq(t).\left|\mathop{\int }\limits_{{\mathbb{R}}}({u}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t\right|=\left|\mathop{\int }\limits_{{\mathbb{R}}}(\theta \left(t),{\dot{u}}_{n}\left(t)){\rm{d}}t\right|=\left|\underset{-nb}{\overset{nb}{\int }}({\dot{u}}_{n}\left(t),\theta \left(t)){\rm{d}}t\right|\le \Vert {\dot{u}}_{n}{\Vert }_{{L}_{2nb}^{p\left(t)}}\Vert \theta {\Vert }_{{L}_{2nb}^{q\left(t)}}\le {\xi }_{3}\Vert \theta {\Vert }_{{L}_{2nb}^{q\left(t)}}.Note that (un(t),θ˙(t))→(u(t),θ˙(t))({u}_{n}\left(t),\dot{\theta }\left(t))\to (u\left(t),\dot{\theta }\left(t))in Cloc(R,RN){C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N}). By (3.16), we have ∣(un(t),θ˙(t))∣≤‖un‖L2nb∞∣θ˙(t)∣≤ξ4∣θ˙(t)∣a.e. onTn=[−nb,nb].| \left({u}_{n}\left(t),\dot{\theta }\left(t))| \le \Vert {u}_{n}{\Vert }_{{L}_{2nb}^{\infty }}| \dot{\theta }\left(t)| \le {\xi }_{4}| \dot{\theta }\left(t)| \hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}{T}_{n}=\left[-nb,nb].Let η(t)=ξ4∣θ˙(t)∣,ift∈suppθ;0,ift∉suppθ.\eta \left(t)=\left\{\begin{array}{ll}{\xi }_{4}| \dot{\theta }\left(t)| ,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\in \hspace{0.1em}\text{supp}\hspace{0.1em}\theta ;\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\notin \hspace{0.1em}\text{supp}\hspace{0.1em}\theta .\end{array}\right.Then, η∈L1(R)\eta \in {L}^{1}\left({\mathbb{R}})and ∣(un(t),θ˙(t))∣≤η(t)| ({u}_{n}\left(t),\dot{\theta }\left(t))| \le \eta \left(t)a.e. on R{\mathbb{R}}for all nn. Therefore, by generalized Lebesgue-dominated convergence theorem, we see that ∫R(un(t),θ˙(t))dt→∫R(u(t),θ˙(t))dt\mathop{\int }\limits_{{\mathbb{R}}}({u}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t\to \mathop{\int }\limits_{{\mathbb{R}}}(u\left(t),\dot{\theta }\left(t)){\rm{d}}tand ∫R(u(t),θ˙(t))dt≤ξ3‖θ‖L2nbq(t).\left|\mathop{\int }\limits_{{\mathbb{R}}}(u\left(t),\dot{\theta }\left(t)){\rm{d}}t\right|\le {\xi }_{3}\Vert \theta {\Vert }_{{L}_{2nb}^{q\left(t)}}.It follows from [5] that u∈Wa1,p(t)(R,RN)u\in {W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}).Recall that wn⇀w∈Llocq(t)(R,RN){w}_{n}\rightharpoonup w\in {L}_{{\rm{loc}}}^{q\left(t)}\left({\mathbb{R}},{{\mathbb{R}}}^{N}), which leads to (3.21)∫R(wn(t),θ(t))dt→∫R(w(t),θ(t))dt.\mathop{\int }\limits_{{\mathbb{R}}}({w}_{n}\left(t),\theta \left(t)){\rm{d}}t\to \mathop{\int }\limits_{{\mathbb{R}}}(w\left(t),\theta \left(t)){\rm{d}}t.Furthermore, by (3.17), one has (3.22)∫R(a(t)∣un(t)∣p(t)−2un(t),θ(t))dt→∫R(a(t)∣u(t)∣p(t)−2u(t),θ(t))dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(a\left(t)| {u}_{n}\left(t){| }^{p\left(t)-2}{u}_{n}\left(t),\theta \left(t)){\rm{d}}t\to \mathop{\int }\limits_{{\mathbb{R}}}\left(a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t),\theta \left(t)){\rm{d}}t.Since un{u}_{n}is a solution of problem (3.1), then we obtain ddt(∣u˙n(t)∣p(t)−2u˙n(t))∈L2nbp(t)\frac{{\rm{d}}}{{\rm{d}}t}(| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t))\in {L}_{2nb}^{p\left(t)}. So it follows that ∣u˙n(t)∣p(t)−2u˙n(t)∈W2nb1,p(t)| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t)\in {W}_{2nb}^{1,p\left(t)}for all n≥1n\ge 1. By (3.22) and Proposition 2.1, let ∣u˙n∣p(t)−2u˙n⇀vinWloc1,p(t)(R,RN),| {\dot{u}}_{n}{| }^{p\left(t)-2}{\dot{u}}_{n}\hspace{0.33em}\rightharpoonup \hspace{0.33em}v\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{W}_{{\rm{loc}}}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}),then we have (3.23)∣u˙n∣p(t)−2u˙n→vinCloc(R,RN).| {\dot{u}}_{n}{| }^{p\left(t)-2}{\dot{u}}_{n}\to v\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{C}_{{\rm{loc}}}\left({\mathbb{R}},{{\mathbb{R}}}^{N}).Define the continuous functional λ:R→R\lambda :{\mathbb{R}}\to {\mathbb{R}}by λ(u)=∣u∣p(t)−2u,∀u∈Cloc(R,RN).\lambda \left(u)=| u{| }^{p\left(t)-2}u,\hspace{1em}\forall u\in {C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N}).Then, from (3.23) that u˙n=λ−1(∣u˙n∣p(t)−2u˙n)→λ−1(v)inLloc1(R,RN),{\dot{u}}_{n}={\lambda }^{-1}(| {\dot{u}}_{n}{| }^{p\left(t)-2}{\dot{u}}_{n})\to {\lambda }^{-1}(v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{L}_{{\rm{loc}}}^{1}\left({\mathbb{R}},{{\mathbb{R}}}^{N}),which shows that λ−1(v)=u˙{\lambda }^{-1}\left(v)=\dot{u}, so we have v=∣u˙∣p(t)−2u˙v=| \dot{u}{| }^{p\left(t)-2}\dot{u}. Also, integration by parts yields that (3.24)∫R(∣u˙(t)∣p(t)−2u˙(t),θ˙(t))dt=−∫Rddt(∣u˙(t)∣p(t)−2u˙(t)),θ(t)dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t),\dot{\theta }\left(t)){\rm{d}}t=-\mathop{\int }\limits_{{\mathbb{R}}}\left(\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t)),\theta \left(t)\right){\rm{d}}t.Hence, it follows from (3.24), we obtain (3.25)∫R(∣u˙n(t)∣p(t)−2u˙n(t),θ˙(t))dt→−∫Rddt(∣u˙(t)∣p(t)−2u˙(t)),θ(t)dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t\to -\mathop{\int }\limits_{{\mathbb{R}}}\left(\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t)),\theta \left(t)\right){\rm{d}}t.For any n≥1n\ge 1, we have (3.26)∫R(∣u˙n(t)∣p(t)−2u˙n(t),θ˙(t))dt+∫R(a(t)∣un(t)∣p(t)−2un(t),θ(t))dt=∫R(wn(t),θ(t))dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}\left(a\left(t)| {u}_{n}\left(t){| }^{p\left(t)-2}{u}_{n}\left(t),\theta \left(t)){\rm{d}}t=\mathop{\int }\limits_{{\mathbb{R}}}\left({w}_{n}\left(t),\theta \left(t)){\rm{d}}t.Letting n→∞n\to \infty , then from (3.21), (3.22), (3.25), and (3.26), we obtain −∫Rddt(∣u˙(t)∣p(t)−2u˙(t))θ(t)dt+∫Ra(t)∣u(t)∣p(t)−2u(t)θ(t)dt=∫R(w(t),θ(t))dt.-\mathop{\int }\limits_{{\mathbb{R}}}\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))\theta \left(t){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\theta \left(t){\rm{d}}t=\mathop{\int }\limits_{{\mathbb{R}}}\left(w\left(t),\theta \left(t)){\rm{d}}t.From the arbitrary of θ∈C0∞(R,RN)\theta \in {C}_{0}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N}), we can deduce that −ddt(∣u˙(t)∣p(t)−2u˙(t))+a(t)∣u(t)∣p(t)−2u(t)=w(t),a.e.t∈R-\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))+a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)=w\left(t),\hspace{1em}\hspace{0.1em}\text{a.e.}\hspace{0.1em}\hspace{0.33em}t\in {\mathbb{R}}and w∈Llocq(t)(R,RN)w\in {L}_{{\rm{loc}}}^{q\left(t)}\left({\mathbb{R}},{{\mathbb{R}}}^{N}), w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t)).Next, we show that u(±∞)=u˙(±∞)=0u\left(\pm \infty )=\dot{u}\left(\pm \infty )=0.Recall that Proposition 2.8 (i), since u∈Wa1,p(t)(R,RN)u\in {W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), we have u(t)→0u\left(t)\to 0as ∣t∣→∞| t| \to \infty . Because w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t)), by virtue of H(f)1{}_{1}: (ii), one has (3.27)‖w(t)‖≤a1(t)(1+‖u(t)‖α(t)−1).\Vert w\left(t)\Vert \le {a}_{1}\left(t)(1+\Vert u\left(t){\Vert }^{\alpha \left(t)-1}).Note that u∈Wa1,p(t)(R,RN)u\in {W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), then ∣u∣p(t)−2u∈Lq(t)(R,RN)| u{| }^{p\left(t)-2}u\in {L}^{q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), so w∈Lq(t)(R,RN)w\in {L}^{q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}). Therefore, ∣u˙∣p(t)−2u˙∈Wa1,q(t)(R,RN)| \dot{u}{| }^{p\left(t)-2}\dot{u}\in {W}_{a}^{1,q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}). Again from Proposition 2.8 (i) that ∣u˙(t)∣p(t)−1→0| \dot{u}\left(t){| }^{p\left(t)-1}\to 0as ∣t∣→∞| t| \to \infty . Thus, we obtain u˙(t)→0\dot{u}\left(t)\to 0, i.e., u˙(±t)=0\dot{u}\left(\pm t)=0as ∣t∣→∞| t| \to \infty .Finally, we show that u(t)≠0u\left(t)\ne 0. For all n≥1n\ge 1, from (3), we have (3.28)a0∫−nbnb∣un∣p(t)dt≤∫−nbnb(wn(t),un(t))dt.{a}_{0}\underset{-nb}{\overset{nb}{\int }}| {u}_{n}{| }^{p\left(t)}{\rm{d}}t\le \underset{-nb}{\overset{nb}{\int }}\left({w}_{n}\left(t),{u}_{n}\left(t)){\rm{d}}t.Let (3.29)hn(t)=(wn(t),un(t))∣un(t)∣p(t),ifun(t)≠0;0,ifun(t)=0.{h}_{n}\left(t)=\left\{\begin{array}{ll}\frac{\left({w}_{n}\left(t),{u}_{n}\left(t))}{| {u}_{n}\left(t){| }^{p\left(t)}},\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{u}_{n}\left(t)\ne 0;\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{u}_{n}\left(t)=0.\end{array}\right.Then, it follows from (3.28) and (3.29) that a0∫−nbnb∣un∣p(t)dt≤∫−nbnbhn(t)∣un(t)∣p(t)dt≤esssupTnhn∫−nbnb∣un(t)∣p(t)dt.{a}_{0}\underset{-nb}{\overset{nb}{\int }}| {u}_{n}{| }^{p\left(t)}{\rm{d}}t\le \underset{-nb}{\overset{nb}{\int }}{h}_{n}\left(t)| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t\le \hspace{0.1em}\text{ess}\hspace{0.1em}{\sup }_{{T}_{n}}{h}_{n}\underset{-nb}{\overset{nb}{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t.By virtue of hypothesis H(f)1{}_{1}: (iv), for a given ε>0\varepsilon \gt 0, we can find δ>0\delta \gt 0such that (3.30)(wn(t),un(t))∣un(t)∣p(t)≤ε,∣un(t)∣p(t)≤δ,∀t∈Tn,ω∈∂f(t,un).\frac{\left({w}_{n}\left(t),{u}_{n}\left(t))}{| {u}_{n}\left(t){| }^{p\left(t)}}\le \varepsilon ,\hspace{1em}| {u}_{n}\left(t){| }^{p\left(t)}\le \delta ,\hspace{1em}\forall t\in {T}_{n},\hspace{0.33em}\omega \in \partial f\left(t,{u}_{n}).If u=0u=0, then it follows from (3.17) that (3.31)un(t)⟶0inCloc(R,RN){u}_{n}\left(t)\hspace{0.33em}\longrightarrow \hspace{0.33em}0\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N})and so we can find n0≥1{n}_{0}\ge 1such that ∣un(t)∣p(t)≤δ,∀t∈Tn,n≥n0.| {u}_{n}\left(t){| }^{p\left(t)}\le \delta ,\hspace{1.0em}\forall t\in {T}_{n},\hspace{0.33em}n\ge {n}_{0}.Thus for all n≥n0n\ge {n}_{0}and almost all t∈Tnt\in {T}_{n}, we have hn(t)≤ε{h}_{n}\left(t)\le \varepsilon and so a0≤esssupTnhn=esssupRhn≤ε,∀n≥n0.{a}_{0}\le \mathop{{\rm{esssup}}}\limits_{{T}_{n}}\hspace{1em}{h}_{n}=\mathop{{\rm{esssup}}}\limits_{{\mathbb{R}}}{h}_{n}\le \varepsilon ,\hspace{1.0em}\forall n\ge {n}_{0}.In the above derivation process, we use the fact that un{u}_{n}and ωn{\omega }_{n}are extended by periodicity to all of R{\mathbb{R}}. Let ε↘0\varepsilon \searrow 0reach a contradiction since 0<a00\lt {a}_{0}. This proves that u≠0u\ne 0.□Proof of Theorem 3.2Similar to the proof of Theorem 3.1, we can prove Theorem 3.2, so we omit its course.□Example 3.1Let p(t)=52+12sintp\left(t)=\frac{5}{2}+\frac{1}{2}\sin tfor t∈Rt\in {\mathbb{R}}, and f(t,u)=a(t)∣u∣3ln(1+∣u∣),∀(t,u)∈[−π,π]×RN,f\left(t,u)=a\left(t)| u{| }^{3}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in \left[-\pi ,\pi ]\times {{\mathbb{R}}}^{N},where a(t)a\left(t)satisfies H(a), H(a)1{}_{1}with b=πb=\pi . It is evident that ffis locally Lipschitz and ∂f(t,u)=a(t)3∣u∣uln(1+∣u∣)+∣u∣2u1+∣u∣.\partial f\left(t,u)=a\left(t)\left[3| u| u\mathrm{ln}(1+| u| )+\frac{| u{| }^{2}u}{1+| u| }\right].Then −f0(t,u;−u)=3+∣u∣(1+∣u∣)ln(1+∣u∣)f(t,u)≥3+11+∣u∣f(t,u)≥p++11+∣u∣f(t,u),\begin{array}{rcl}-{f}^{0}\left(t,u;-u)& =& \left[3+\frac{| u| }{(1+| u| )\mathrm{ln}(1+| u| )}\right]f\left(t,u)\\ & \ge & \left(3+\frac{1}{1+| u| }\right)f\left(t,u)\\ & \ge & \left({p}^{+}+\frac{1}{1+| u| }\right)f\left(t,u),\end{array}which implies that ffsatisfies H(f)1{\text{H(f)}}_{1}:(iii) with α=β=ν=1\alpha =\beta =\nu =1. Hence, from Theorem 3.1, problem (1.1) has a nontrivial homoclinic solution.Example 3.2Let p(t)=52+12sintp\left(t)=\frac{5}{2}+\frac{1}{2}\sin tfor t∈Rt\in {\mathbb{R}}, and f(t,u)=a(t)∣u∣7/2ln(1+∣u∣),∀(t,u)∈[−π,π]×RN.f\left(t,u)=a\left(t)| u{| }^{7\text{/}2}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in \left[-\pi ,\pi ]\times {{\mathbb{R}}}^{N}.Similar to Example 3.1, problem (1.1) has a nontrivial homoclinic solution by Theorem 3.2.4Nonperiodic p(t)p\left(t)-Laplacian inclusion systemIn this section, we investigate the question of existence of homoclinic (to zero) solutions without periodic assumptions. Namely, we examine the following two types of problems: (4.1)ddt(∣u˙(t)∣p(t)−2u˙(t))−a(t)∣u(t)∣p(t)−2u(t)∈∂f(t,u(t)),\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial f\left(t,u\left(t)),and another problem (4.2)ddt(∣u˙(t)∣p(t)−2u˙(t))−a(t)∣u(t)∣p(t)−2u(t)∈∂f1(t,u(t))−∂f2(t,u(t)),\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial {f}_{1}\left(t,u\left(t))-\partial {f}_{2}\left(t,u\left(t)),where aasatisfy the following hypothesis: H(a)2lim∣t∣→+∞a(t)=+∞{\mathrm{lim}}_{| t| \to +\infty }a\left(t)=+\infty .Note that a∈C(R,R+)a\in C({\mathbb{R}},{{\mathbb{R}}}^{+})is coercive, then H′(a)2̲\underline{\hspace{0.1em}\text{H}^{\prime} {\text{(a)}}_{2}}is satisfied, namely, H′(a)2there exists r>0r\gt 0such that lim∣y∣→∞meas({t∈Br(y):a(t)≤b})=0for anyb>0.\mathop{\mathrm{lim}}\limits_{| y| \to \infty }\hspace{0.1em}\text{meas}\hspace{0.1em}(\{t\in {B}_{r}(y):a\left(t)\le b\})=0\hspace{1em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}b\gt 0.For the nonlinearity ff, we suppose the following hypotheses: H(f)2(ii)for almost all t∈Rt\in {\mathbb{R}}, there exist a function α(t)∈C(R)∩Lγ(t)γ(t)−α(t)(R)\alpha \left(t)\in C\left({\mathbb{R}})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}\left({\mathbb{R}})such that ∣ω∣≤a(t)(1+∣u∣α(t)−1),∀u∈RN,ω∈∂f(t,u(t)),| \omega | \le a\left(t)\left(1+| u{| }^{\alpha \left(t)-1}),\hspace{1em}\forall u\in {{\mathbb{R}}}^{N},\hspace{1em}\omega \in \partial f\left(t,u\left(t)),where a∈L∞(R)a\in {L}^{\infty }\left({\mathbb{R}}), α+<γ−<γ(t)<γ+<p−{\alpha }^{+}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(ii′)there exist two functions ai(t)<a(t)(i=1,2){a}_{i}\left(t)\lt a\left(t)\left(i=1,2)such that ∣ω∣≤a1(t)α1(t)∣u∣α1(t)−1,∀t∈R,∣u∣≤1,| \omega | \le {a}_{1}\left(t){\alpha }_{1}\left(t)| u{| }^{{\alpha }_{1}\left(t)-1},\hspace{1em}\forall t\in {\mathbb{R}},\hspace{0.33em}| u| \le 1,and ∣ω∣≤a2(t)α2(t)∣u∣α2(t)−1,∀t∈R,∣u∣≥1,| \omega | \le {a}_{2}\left(t){\alpha }_{2}\left(t)| u{| }^{{\alpha }_{2}\left(t)-1},\hspace{1em}\forall t\in {\mathbb{R}},\hspace{0.33em}| u| \ge 1,where ω∈∂f(t,u(t))\omega \hspace{-0.08em}\in \hspace{-0.08em}\partial f\left(t,u\left(t)), ai(t)∈C(R,R+){a}_{i}\left(t)\hspace{-0.08em}\in \hspace{-0.08em}C\left({\mathbb{R}},{{\mathbb{R}}}^{+})and αi(t)∈C(R,R+)∩Lγ(t)γ(t)−αi(t)(R){\alpha }_{i}\left(t)\hspace{-0.08em}\in \hspace{-0.08em}C\left({\mathbb{R}},{{\mathbb{R}}}^{+})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-{\alpha }_{i}\left(t)}}\left({\mathbb{R}}), αi+<γ−<γ(t)<γ+<p−{\alpha }_{i}^{+}\hspace{-0.08em}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(iii)there exist constants M,α,β>0M,\alpha ,\beta \gt 0such that 0≤p++1α+β∣u∣νf(t,u)≤−f0(t,u;−u)∀(t,u)∈R×RN,0\le \left({p}^{+}+\frac{1}{\alpha +\beta | u{| }^{\nu }}\right)f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where ν<p−\nu \lt {p}^{-};(iii′)there exist constants μ>p+,M>0\mu \gt {p}^{+},M\gt 0such that μf(t,u)≤−f0(t,u;−u)∀(t,u)∈R×RN;\mu f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N};(iv)there exists a function q(t)>0q\left(t)\gt 0such that lim∣u∣→0w∣u∣p+−1=0,lim∣u∣→+∞inff(t,u)∣u∣q(t)>0,∀t∈R,w∈∂f(t,u),\mathop{\mathrm{lim}}\limits_{| u| \to 0}\frac{w}{| u{| }^{{p}^{+}-1}}=0,\hspace{1em}\mathop{\mathrm{lim}}\limits_{| u| \to +\infty }\inf \frac{f\left(t,u)}{| u{| }^{q\left(t)}}\gt 0,\hspace{1em}\forall t\in {\mathbb{R}},w\in \partial f\left(t,u),where p+<q−{p}^{+}\lt {q}^{-}.Theorem 4.1If hypotheses H(p), H(a), H(f) and H(f)2{\text{H(f)}}_{2}: (ii), (iii), (iv) hold, then problem (4.1) has at least a nontrivial homoclinic solution.Theorem 4.2If hypotheses H(p), H(a), H(f)\hspace{0.1em}\text{H(p), H(a), H(f)}\hspace{0.1em}and H(f)2{\text{H(f)}}_{2}: (ii),(iii′),(iv)\left(ii),\hspace{0.33em}\left(iii^{\prime} ),\hspace{0.33em}\left(iv)hold, then problem (4.1) has at least a nontrivial homoclinic solution.Remark 4.3Note that H(f)2̲\underline{{\text{H(f)}}_{2}}: (ii) is weaker than [27, f1{}_{1}], by virtue of hypothesis H′(a)2̲,\underline{\hspace{0.1em}\text{H}^{\prime} {\text{(a)}}_{2}},we cannot immediately obtain [27, Theorem 1.2] for problem (4.1) with symmetrical condition f(t,−u)=f(t,u)f\left(t,-u)=f\left(t,u)for all (t,u)∈R×RN\left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Theorem 4.4If hypotheses H(p), H(a), H(a)2{\text{H(a)}}_{2}, H(f), H(f)2{\text{H(f)}}_{2}: (ii′)\left(ii^{\prime} )and the following condition hold:(v)there exist an open subset Ω⊂R\Omega \subset {\mathbb{R}}and function γ(t)\gamma \left(t)such thatf(t,u)≥η∣u∣γ(t),∀(t,u)∈Ω×RN,∣u∣≤1,f\left(t,u)\ge \eta | u{| }^{\gamma \left(t)},\hspace{1em}\forall \left(t,u)\in \Omega \times {{\mathbb{R}}}^{N},\hspace{1em}| u| \le 1,where γ(t)\gamma \left(t)satisfies H(p)\hspace{0.1em}\text{H(p)}\hspace{0.1em}and γ+<p−{\gamma }^{+}\lt {p}^{-}, η>0\eta \gt 0is a constant.Then problem (4.1) has at least a nontrivial homoclinic solution.With regard to problem (4.2), in this situation, assume that pp, aaand f1{f}_{1}satisfy all the conditions in Theorem 4.2 and f2{f}_{2}satisfies the following conditions: H(f)3(i)the function f2(t,⋅):R→R{f}_{2}\left(t,\cdot ):{\mathbb{R}}\to {\mathbb{R}}is measurable for all u∈RNu\in {{\mathbb{R}}}^{N}and f2(t,0)=0{f}_{2}\left(t,0)=0;(ii)the function f2(⋅,u):RN→R{f}_{2}\left(\cdot ,u):{{\mathbb{R}}}^{N}\to {\mathbb{R}}is locally Lipschitz for a.e. t∈Rt\in {\mathbb{R}};(iii)for almost all t∈Rt\in {\mathbb{R}}, there exists a function α(t)∈C(R)∩Lγ(t)γ(t)−α(t)(R)\alpha \left(t)\in C\left({\mathbb{R}})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}\left({\mathbb{R}})such that ∣ω∣≤a(t)(1+∣u∣α(t)−1),∀u∈RN,ω∈∂f2(t,u(t)),| \omega | \le a\left(t)\left(1+| u{| }^{\alpha \left(t)-1}),\hspace{1em}\forall u\in {{\mathbb{R}}}^{N},\hspace{1em}\omega \in \partial {f}_{2}\left(t,u\left(t)),where a∈L∞(R)a\in {L}^{\infty }\left({\mathbb{R}}), α+<γ−<γ(t)<γ+<p−{\alpha }^{+}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(iv)there exists a constant ϱ∈[p+,μ)\varrho \in {[}{p}^{+},\mu )such that f20(t,u;u)≤ϱf2(t,u),∀(t,u)∈R×RN.{f}_{2}^{0}\left(t,u;\hspace{0.33em}u)\le \varrho {f}_{2}\left(t,u),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Theorem 4.5If hypotheses H(p), H(a)\hspace{0.1em}\text{H(p), H(a)}\hspace{0.1em}, H(f)3{\text{H(f)}}_{3}hold, then problem (4.2) has at least a nontrivial homoclinic solution.Proof of Theorem 4.1In order to prove Theorems 4.1, 4.2 and 4.4, we first define a functional φ:Wa1,p(t)→R\varphi :{W}_{a}^{1,p\left(t)}\to {\mathbb{R}}as follows: (4.3)φ(u)=∫R1p(t)(∣u˙(t)∣p(t)+a(t)∣u(t)∣p(t))dt−∫Rf(t,u(t))dt.\varphi \left(u)=\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}\left(t){| }^{p\left(t)}+a\left(t)| u\left(t){| }^{p\left(t)}){\rm{d}}t-\mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t.Using the same type of reasoning as the proof of Theorem 3.1, it is easy to verify that φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1).Let {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}be such that (4.4)∣φ(un)∣≤M1andm(un)→0asn→+∞,| \varphi \left({u}_{n})| \le {M}_{1}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({u}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to +\infty ,where M1>0{M}_{1}\gt 0is a constant. Our proofs are divided into two steps.Step 1: un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}.Since ∂φ(un)⊆W−1,p(t)\partial \varphi \left({u}_{n})\subseteq {W}^{-1,p\left(t)}is weakly compact and norm is weakly lower semicontinuous, according to Weierstrass theorem (Lemma 2.18). We can find un∗∈∂φ(un){u}_{n}^{\ast }\in \partial \varphi \left({u}_{n})such that m(un)=‖un∗‖m\left({u}_{n})=\Vert {u}_{n}^{\ast }\Vert for n≥1n\ge 1.Define nonlinear operator ℒ:Wa1,p(t)→(Wa1,p(t))∗{\mathcal{ {\mathcal L} }}:{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast }as follows: ⟨ℒ(u),v⟩=∫R∣u˙(t)∣p(t)−2(u˙(t),v˙(t))dt,∀u,v∈Wa1,p(t).\langle {\mathcal{ {\mathcal L} }}\left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}| \dot{u}\left(t){| }^{p\left(t)-2}(\dot{u}\left(t),\dot{v}\left(t)){\rm{d}}t,\hspace{1em}\forall u,v\in {W}_{a}^{1,p\left(t)}.According to the literature [21], ℒ{\mathcal{ {\mathcal L} }}is monotonic and semicontinuous, so it is maximal monotone (see also [24]), therefore, un∗=ℒ(un)−wn{u}_{n}^{\ast }={\mathcal{ {\mathcal L} }}\left({u}_{n})-{w}_{n}for n≥1n\ge 1, and wn∈∂f(t,un){w}_{n}\in \partial f\left(t,{u}_{n}), wn∈Lp′(t){w}_{n}\in {L}^{p^{\prime} \left(t)}, where 1/p′(t)+1/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} \left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1.In another way, by the selection of sequence{un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}, we obtain (4.5)∣⟨un∗,un⟩∣≤εn,εn↓0,| \langle {u}_{n}^{\ast },{u}_{n}\rangle | \le {\varepsilon }_{n},\hspace{1em}{\varepsilon }_{n}\downarrow 0,which yields that (4.6)−∫R(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rωnundt≤εn.-\mathop{\int }\limits_{{\mathbb{R}}}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{\omega }_{n}{u}_{n}{\rm{d}}t\le {\varepsilon }_{n}.Note that ⟨wn,−un⟩≤f0(t,un;−un),\langle {w}_{n},-{u}_{n}\rangle \le {f}^{0}\left(t,{u}_{n};-{u}_{n}),using this fact and by (4.3), (4.4) and (4.6), one has (4.7)p+M1≥p+φ(un)−⟨un∗,un⟩≥∫Rp+p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−p+∫Rf(t,un)dt−∫R(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rwnundt≥∫R[−p+f(t,un)−f0(t,un;−un)]dt.\begin{array}{rcl}{p}^{+}{M}_{1}& \ge & {p}^{+}\varphi \left({u}_{n})-\langle {u}_{n}^{\ast },{u}_{n}\rangle \\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{{p}^{+}}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-{p}^{+}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}){\rm{d}}t\\ & & -\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{w}_{n}{u}_{n}{\rm{d}}t\\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})]{\rm{d}}t.\end{array}For any (t,un)∈R×(RN⧹{0})\left(t,{u}_{n})\in {\mathbb{R}}\times ({{\mathbb{R}}}^{N}\setminus \left\{0\right\}), by H(f)2{}_{2}: (iii), we have (4.8)f(t,un(t))≤(α+β∣un(t)∣ν)[−p+f(t,un)−f0(t,un;−un)].f\left(t,{u}_{n}\left(t))\le (\alpha +\beta | {u}_{n}\left(t){| }^{\nu }){[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})].Hence, without loss of generality, we assume that ‖un‖≥1\Vert {u}_{n}\Vert \ge 1. Otherwise, ‖un‖\Vert {u}_{n}\Vert is bounded. It follows from (4.3), (4.4), (4.7), (4.8), Propositions 2.4 (ii) and 2.8 (i) that (4.9)1p+‖un‖p−≤∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt=φ(un)+∫Rf(t,un(t))dt≤φ(un)+∫R(α+β∣un(t)∣ν)[−p+f(t,un)−f0(t,un;−un)]dt≤M1+(α+β‖un‖∞ν)∫R[−p+f(t,un)−f0(t,un;−un)]dt≤M1+p+M1(α+β‖un‖∞ν)≤M1+p+M1(α+βκν‖un‖ν).\begin{array}{rcl}\frac{1}{{p}^{+}}\Vert {u}_{n}{\Vert }^{{p}^{-}}& \le & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & =& \varphi \left({u}_{n})+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}\left(t)){\rm{d}}t\\ & \le & \varphi \left({u}_{n})+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\alpha +\beta | {u}_{n}\left(t){| }^{\nu }){[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})]{\rm{d}}t\\ & \le & {M}_{1}+\left(\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})]{\rm{d}}t\\ & \le & {M}_{1}+{p}^{+}{M}_{1}(\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\\ & \le & {M}_{1}+{p}^{+}{M}_{1}(\alpha +\beta {\kappa }^{\nu }\Vert {u}_{n}{\Vert }^{\nu }).\end{array}Note that ν<p−\nu \lt {p}^{-}, in light of (4.9), it is easy to show that {‖un‖}\left\{\Vert {u}_{n}\Vert \right\}is bounded. So passing to a subsequence if necessary, it can be assumed that un⇀u{u}_{n}\hspace{0.33em}\rightharpoonup \hspace{0.33em}uin Wa1,p(t){W}_{a}^{1,p\left(t)}, un⇀u{u}_{n}\hspace{0.33em}\rightharpoonup \hspace{0.33em}uin Lp(t){L}^{p\left(t)}. Because (4.5), then ⟨ℒ(un),un−u⟩−∫Rwn(un−u)dt≤εn,∀n≥1.\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle -\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}\left({u}_{n}-u){\rm{d}}t\le {\varepsilon }_{n},\hspace{1em}\forall n\ge 1.By virtue of the fact that {wn}n≥1{\left\{{w}_{n}\right\}}_{n\ge 1}is bounded in Lp′(t){L}^{p^{\prime} \left(t)}, then limsupn→∞⟨ℒ(un),un−u⟩≤0{\mathrm{limsup}}_{n\to \infty }\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle \le 0. By Proposition 2.9, we deduce that un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}.Step 2: φ\varphi satisfies nonsmooth mountain pass theorem.Let ε>0\varepsilon \gt 0be small enough, in view of hypotheses H(f)2{}_{2}: (ii), (iv), one has f(t,u)≤ε∣u∣p++c(ε)∣u∣α(t),∀(t,u)∈R×RN.f\left(t,u)\le \varepsilon | u{| }^{{p}^{+}}+c\left(\varepsilon )| u{| }^{\alpha \left(t)},\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Let ‖u‖=ρ\Vert u\Vert =\rho be small enough, then by Proposition 2.4, we have (4.10)φ(u)≥1p+∫R(∣u˙∣p(t)+a(t)∣u∣p(t))dt−ε∫R∣u∣p+dt−c(ε)∫R∣u∣α(t)dt≥1p+‖u‖p+−εcp+p+‖u‖p+−c(ε)cα−α−‖u‖α−\begin{array}{rcl}\varphi \left(u)& \ge & \frac{1}{{p}^{+}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\varepsilon \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{{p}^{+}}{\rm{d}}t-c\left(\varepsilon )\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{\alpha \left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}\end{array}for the above ε\varepsilon , let εcp+p+<12p+\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\lt \frac{1}{2{p}^{+}}, where cp+(cα−){c}_{{p}^{+}}({c}_{{\alpha }^{-}})is the embedding constant from Wa1,p(t){W}_{a}^{1,p\left(t)}to Lp+(Lα−){L}^{{p}^{+}}({L}^{{\alpha }^{-}}). Then, from (4.10), we obtain φ(u)≥12p+‖u‖p+−c(ε)cα−α−‖u‖α−.\varphi \left(u)\ge \frac{1}{2{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}.Note that p+<α−{p}^{+}\lt {\alpha }^{-}, then there exists a constant r>0r\gt 0such that φ(u)≥r\varphi \left(u)\ge rwhen ‖u‖=ρ\Vert u\Vert =\rho for ρ\rho small enough.By H(f)2{}_{2}: (ii), (iv), we see that (4.11)f(t,u)≥∣u∣q(t),∀t∈R,∣u∣≥Mf\left(t,u)\ge | u{| }^{q\left(t)},\hspace{1em}\forall t\in {\mathbb{R}},\hspace{1em}| u| \ge Mand (4.12)∣f(t,u)∣≤c0a(t),∀t∈R,∣u∣<M,| f\left(t,u)| \le {c}_{0}a\left(t),\hspace{1em}\forall t\in {\mathbb{R}},\hspace{1em}| u| \lt M,respectively. Thus, for any u∈W1,p(t)⧹{0}u\in {W}^{1,p\left(t)}\setminus \left\{0\right\}and σ>1\sigma \gt 1, from (4.3), (4.11) and (4.12), we obtain φ(σu)=∫R1p(t)(∣σu˙∣p(t)+a(t)∣σu∣p(t))dt−∫Rf(t,σu)dt≤σp+∫R1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−σq−∫R∣u∣q(t)dt+c0∫Ra(t)dt.\begin{array}{rcl}\varphi \left(\sigma u)& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \sigma \dot{u}{| }^{p\left(t)}+a\left(t)| \sigma u{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,\sigma u){\rm{d}}t\\ & \le & {\sigma }^{{p}^{+}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-{\sigma }^{{q}^{-}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{q\left(t)}{\rm{d}}t+{c}_{0}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}a\left(t){\rm{d}}t.\end{array}Since p+<q−{p}^{+}\lt {q}^{-}, it is easy to show that φ(σu)→−∞\varphi \left(\sigma u)\to -\infty as σ→+∞\sigma \to +\infty . Because φ(0)=0\varphi \left(0)=0and φ\varphi satisfies Lemma 2.19, hence there exists at least one nontrivial critical point, that is, system (4.1) has at least one homoclinic orbit.□Proof of Theorem 4.2Applying the proof of Theorem 4.1, we deduce that φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1).Let {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}be such that (4.13)∣φ(un)∣≤M1andm(un)→0asn→+∞,| \varphi \left({u}_{n})| \le {M}_{1}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({u}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to +\infty ,where M1>0{M}_{1}\gt 0as given in (4.4). As ∂φ(un)⊆W−1,p(t)\partial \varphi \left({u}_{n})\subseteq {W}^{-1,p\left(t)}is weakly compact and the norm is weakly lower semicontinuous, by Lemma 2.18, we can choose un∗∈∂φ(un){u}_{n}^{\ast }\in \partial \varphi \left({u}_{n})such that m(un)=‖un∗‖m\left({u}_{n})=\Vert {u}_{n}^{\ast }\Vert for n≥1n\ge 1.Define nonlinear operator ℒ:Wa1,p(t)→(Wa1,p(t))∗{\mathcal{ {\mathcal L} }}:{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast }as follows: ⟨ℒ(u),v⟩=∫R∣u˙(t)∣p(t)−2(u˙(t),v˙(t))dt,∀u,v∈Wa1,p(t).\langle {\mathcal{ {\mathcal L} }}\left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}| \dot{u}\left(t){| }^{p\left(t)-2}(\dot{u}\left(t),\dot{v}\left(t)){\rm{d}}t,\hspace{1em}\forall u,v\in {W}_{a}^{1,p\left(t)}.According to the literature [21], ℒ{\mathcal{ {\mathcal L} }}is monotonic and semicontinuous, so it is maximal monotone (see also [24]), therefore un∗=ℒ(un)−wn{u}_{n}^{\ast }={\mathcal{ {\mathcal L} }}\left({u}_{n})-{w}_{n}for n≥1n\ge 1, and wn∈∂f(t,un){w}_{n}\in \partial f\left(t,{u}_{n}), wn∈Lp′(t){w}_{n}\in {L}^{p^{\prime} \left(t)}, where 1/p′(t)+1/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} \left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1.In another way, by the selection of sequence {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}, we obtain (4.14)∣⟨un∗,un⟩∣≤εn,εn↓0,| \langle {u}_{n}^{\ast },{u}_{n}\rangle | \le {\varepsilon }_{n},\hspace{1em}{\varepsilon }_{n}\downarrow 0,which yields (4.15)−p+∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rωnundt≤εn.-{p}^{+}\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{\omega }_{n}{u}_{n}{\rm{d}}t\le {\varepsilon }_{n}.Note that ⟨wn,−un⟩≤f0(t,un;−un),\langle {w}_{n},-{u}_{n}\rangle \le {f}^{0}\left(t,{u}_{n};-{u}_{n}),using this fact and by (4.3), (4.13) and (4.15), one has μM1+εn≥μφ(un)+⟨un∗,−un⟩=μ∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫Rf(t,un)dt+⟨un∗,−un⟩=μ∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫Rf(t,un)dt−p+∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rωnundt≥(μ−p+)∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫R(μf(t,un)+f0(t,un;−un))dt,\begin{array}{rcl}\mu {M}_{1}+{\varepsilon }_{n}& \ge & \mu \varphi \left({u}_{n})+\langle {u}_{n}^{\ast },-{u}_{n}\rangle \\ & =& \mu \left[\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}){\rm{d}}t\right]+\langle {u}_{n}^{\ast },-{u}_{n}\rangle \\ & =& \mu \left[\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}){\rm{d}}t\right]-{p}^{+}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{\omega }_{n}{u}_{n}{\rm{d}}t\\ & \ge & (\mu -{p}^{+})\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\mu f\left(t,{u}_{n})+{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t,\end{array}which leads to (4.16)(μ−p+)∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤μM1+εn+∫R(μf(t,un(t))+f0(t,un(t);−un(t)))dt.(\mu -{p}^{+})\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\le \mu {M}_{1}+{\varepsilon }_{n}+\mathop{\int }\limits_{{\mathbb{R}}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t.By H(f)2{}_{2}: (ii), there exist two functions a1(t),b1(t)∈L∞(R)+{a}_{1}\left(t),{b}_{1}\left(t)\in {L}^{\infty }{\left({\mathbb{R}})}_{+}such that ∣f(t,un(t))∣≤a1(t)+b1(t)∣un(t)∣α(t).| f\left(t,{u}_{n}\left(t))| \le {a}_{1}\left(t)+{b}_{1}\left(t)| {u}_{n}\left(t){| }^{\alpha \left(t)}.Recall that u↦f(t,u)u\mapsto f\left(t,u)is local Lipschitz, there exists c(t)∈L∞(R)+c\left(t)\in {L}^{\infty }{\left({\mathbb{R}})}_{+}such that f0(t,un(t);−un(t))≤c(t)∣un(t)∣,∀u∈RN.{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))\le c\left(t)| {u}_{n}\left(t)| ,\hspace{1em}\forall u\in {{\mathbb{R}}}^{N}.Thus, there exist a2>0,b2∈L∞(R)+{a}_{2}\gt 0,{b}_{2}\in {L}^{\infty }{\left({\mathbb{R}})}_{+}such that μf(t,un(t))+f0(t,un(t);−un(t))≤a2b2(t),∀t∈R,∣un∣<M,\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))\le {a}_{2}{b}_{2}\left(t),\hspace{1em}\forall t\in {\mathbb{R}},\hspace{0.33em}| {u}_{n}| \lt M,which implies that there exists a constant C3>0{C}_{3}\gt 0such that (4.17)∫{∣un∣<M}(μf(t,un(t))+f0(t,un(t);−un(t)))dt≤C3.\mathop{\int }\limits_{\left\{| {u}_{n}| \lt M\right\}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t\le {C}_{3}.Combining with (4.17) and H(f)2{\text{H(f)}}_{2}: (iii′)\left({\rm{iii}}^{\prime} ), we can deduce (4.18)∫R(μf(t,un(t))+f0(t,un(t);−un(t)))dt=∫{∣un∣<M}(μf(t,un(t))+f0(t,un(t);−un(t)))dt+∫{∣un∣≥M}(μf(t,un(t))+f0(t,un(t);−un(t)))dt≤C3.\begin{array}{l}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t\\ \hspace{1.0em}=\mathop{\displaystyle \int }\limits_{\left\{| {u}_{n}| \lt M\right\}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t+\mathop{\displaystyle \int }\limits_{\left\{| {u}_{n}| \ge M\right\}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t\le {C}_{3}.\end{array}Hence, from (4.16) and (4.18), we obtain (4.19)(μ−p+)∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤C4.(\mu -{p}^{+})\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\le {C}_{4}.Note that μ>p+\mu \gt {p}^{+}, it follows from (4.19) that {un}n≥1⊆W1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}^{1,p\left(t)}is bounded. So passing to a subsequence if necessary, it can be assumed that un⇀u{u}_{n}\rightharpoonup uin Wa1,p(t){W}_{a}^{1,p\left(t)}, un⇀u{u}_{n}\rightharpoonup uin Lp(t){L}^{p\left(t)}. Because (4.14), then ⟨ℒ(un),un−u⟩−∫Rwn(un−u)dt≤εn,∀n≥1.\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle -\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}\left({u}_{n}-u){\rm{d}}t\le {\varepsilon }_{n},\hspace{1em}\forall n\ge 1.By virtue of the fact that {wn}n≥1{\left\{{w}_{n}\right\}}_{n\ge 1}is bounded in Lp′(t){L}^{p^{\prime} \left(t)}, then limsupn→∞⟨ℒ(un),un−u⟩≤0.\mathop{\mathrm{limsup}}\limits_{n\to \infty }\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle \le 0.By Proposition 2.9, we obtain un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}.Next, we need only to verify that φ\varphi satisfies nonsmooth mountain pass theorem, i.e., Lemma 2.19, the proof is similar to Theorem 4.1, so we omitted its course.□Proof of Theorem 4.4Consider the functional φ:Wa1,p(t)→R\varphi :{W}_{a}^{1,p\left(t)}\to {\mathbb{R}}be defined as (4.3), i.e., (4.20)φ(u)=∫R1p(t)(∣u˙(t)∣p(t)+a(t)∣u(t)∣p(t))dt−∫Rf(t,u(t))dt≔φ˜(u)−∫Rf(t,u(t))dt.\varphi \left(u)=\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}\left(t){| }^{p\left(t)}+a\left(t)| u\left(t){| }^{p\left(t)}){\rm{d}}t-\mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t:= \widetilde{\varphi }\left(u)-\mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t.First, we prove that φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1). However, similar arguments as Theorem 4.1, it is easy to see that φ˜\widetilde{\varphi }is the locally Lipschitz functional. So we only need to show ∫Rf(t,u(t))dt{\int }_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}tis the locally Lipschitz functional.Let Ω⊂R\Omega \subset {\mathbb{R}}, from H(f)2{}_{2}: (ii)′^{\prime} and Lemma 2.16, for all u1,u2∈Wa1,p(t)(Ω,RN){u}_{1},{u}_{2}\in {W}_{a}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}), one has (4.21)∣f(t,u1)−f(t,u2)∣≤ai(t)αi(t)∣u˜∣α(t)−1∣u1−u2∣,i=1,2| f(t,{u}_{1})-f(t,{u}_{2})| \le {a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| {u}_{1}-{u}_{2}| ,\hspace{1em}i=1,2and ai(t)αi(t)∣u˜∣αi(t)−1≤(γ(t)−αi(t))∣ai(t)αi(t)∣γ(t)−1γ(t)−αi(t)γ(t)−1+αi(t)−1γ(t)−1∣u˜∣γ(t)−1,i=1,2,{a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{{\alpha }_{i}\left(t)-1}\le \frac{\left(\gamma \left(t)-{\alpha }_{i}\left(t))| {a}_{i}\left(t){\alpha }_{i}\left(t){| }^{\tfrac{\gamma \left(t)-1}{\gamma \left(t)-{\alpha }_{i}\left(t)}}}{\gamma \left(t)-1}+\frac{{\alpha }_{i}\left(t)-1}{\gamma \left(t)-1}| \tilde{u}{| }^{\gamma \left(t)-1},\hspace{1em}i=1,2,which yields that (4.22)(ai(t)αi(t)∣u˜∣αi(t)−1)γ(t)γ(t)−1≤C8∣a(t)∣γ(t)γ(t)−αi(t)+C9∣u˜∣γ(t),i=1,2,{({a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{{\alpha }_{i}\left(t)-1})}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}\le {C}_{8}| a\left(t){| }^{\tfrac{\gamma \left(t)}{\gamma \left(t)-{\alpha }_{i}\left(t)}}+{C}_{9}| \tilde{u}{| }^{\gamma \left(t)},\hspace{1em}i=1,2,where u˜=su1+(1−s)u2,s∈(0,1)\tilde{u}=s{u}_{1}+\left(1-s){u}_{2},s\in \left(0,1), C8,C9>0{C}_{8},{C}_{9}\gt 0. Then, from (4.21), (4.22) and Hölder inequality, we obtain ∫Rf(t,u1)dt−∫Rf(t,u2)dt≤∫Rai(t)αi(t)∣u˜∣α(t)−1∣u1−u2∣dt≤∣ai(t)αi(t)∣u˜∣α(t)−1∣γ(t)γ(t)−1∣u1−u2∣γ(t)≤C10∣∣u1−u2∣∣,i=1,2.\begin{array}{rcl}\left|\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f(t,{u}_{1}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f(t,{u}_{2}){\rm{d}}t\right|& \le & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| {u}_{1}-{u}_{2}| {\rm{d}}t\\ & \le & {| {a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| }_{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}{| {u}_{1}-{u}_{2}| }_{\gamma \left(t)}\\ & \le & {C}_{10}| | {u}_{1}-{u}_{2}| | ,\hspace{1em}i=1,2.\end{array}Hence, φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1).Next, our proofs are divided into three steps.Step 1: φ\varphi is coercive.It follows from H(f)2{\text{H(f)}}_{2}: (ii′)\left({\rm{ii}}^{\prime} )that (4.23)f(t,u)≤a1(t)∣u∣α1(t),∣u∣≤1;a2(t)∣u∣α2(t),∣u∣>1.f\left(t,u)\le \left\{\begin{array}{ll}{a}_{1}\left(t)| u{| }^{{\alpha }_{1}\left(t)},\hspace{1.0em}& | u| \le 1;\\ {a}_{2}\left(t)| u{| }^{{\alpha }_{2}\left(t)},\hspace{1.0em}& | u| \gt 1.\end{array}\right.Let ‖u‖≥1\Vert u\Vert \ge 1, it follows from (4.3), Propositions 2.2, 2.4, and 2.6 that (4.24)φ(u)=∫R1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−∫Rf(t,u)dt≥1p+‖u‖p−−∫{t:∣u∣≤1}f(t,u)dt−∫{t:∣u∣>1}f(t,u)dt≥1p+‖u‖p−−∫{t:∣u∣≤1}a1(t)∣u∣α1(t)dt−∫{t:∣u∣>1}a2(t)∣u∣α2(t)dt≥1p+‖u‖p−−C11∫{t:∣u∣≤1}bα1(t)p(t)aα1(t)p(t)∣u∣α1(t)dt−C12∫{t:∣u∣>1}bα2(t)p(t)aα2(t)p(t)∣u∣α2(t)dt≥1p+‖u‖p−−2C11∣bα1(t)p(t)∣Lr1(t)∣u∣p(t),aα˜1−2C12∣bα2(t)p(t)∣Lr2(t)∣u∣p(t),aα˜2≥1p+‖u‖p−−2C11∣bα1(t)p(t)∣Lr1(t)‖u‖α˜1−2C12∣bα2(t)p(t)∣Lr2(t)‖u‖α˜2,\begin{array}{rcl}\varphi \left(u)& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,u){\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \le 1\right\}}f\left(t,u){\rm{d}}t-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \gt 1\right\}}f\left(t,u){\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \le 1\right\}}{a}_{1}\left(t)| u{| }^{{\alpha }_{1}\left(t)}{\rm{d}}t-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \gt 1\right\}}{a}_{2}\left(t)| u{| }^{{\alpha }_{2}\left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-{C}_{11}\mathop{\displaystyle \int }\limits_{\left\{t:| u| \le 1\right\}}{b}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}{a}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}| u{| }^{{\alpha }_{1}\left(t)}{\rm{d}}t-{C}_{12}\mathop{\displaystyle \int }\limits_{\left\{t:| u| \gt 1\right\}}{b}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}{a}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}| u{| }^{{\alpha }_{2}\left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-2{C}_{11}| {b}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{1}\left(t)}}| u{| }_{p\left(t),a}^{{\widetilde{\alpha }}_{1}}-2{C}_{12}| {b}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{2}\left(t)}}| u{| }_{p\left(t),a}^{{\widetilde{\alpha }}_{2}}\hspace{1em}\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-2{C}_{11}| {b}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{1}\left(t)}}\Vert u{\Vert }^{{\widetilde{\alpha }}_{1}}-2{C}_{12}| {b}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{2}\left(t)}}\Vert u{\Vert }^{{\widetilde{\alpha }}_{2}},\end{array}where Ci+10=supt∈Rai(t)(i=1,2.){C}_{i+10}={\sup }_{t\in {\mathbb{R}}}{a}_{i}\left(t)\left(i=1,2.), b(t)=a(t)−1b\left(t)=a{\left(t)}^{-1}, 1/ri(t)+αi(t)/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}{r}_{i}\left(t)+{\alpha }_{i}\left(t)\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1, and α˜i∈[αi−,αi+]{\widetilde{\alpha }}_{i}\in {[}{\alpha }_{i}^{-},{\alpha }_{i}^{+}]is a constant. From H(f)2{}_{2}: (ii′)\left({\rm{ii}}^{\prime} ), we know αi−<αi+<p−{\alpha }_{i}^{-}\lt {\alpha }_{i}^{+}\lt {p}^{-}, so αi˜<p−\widetilde{{\alpha }_{i}}\lt {p}^{-}. Hence, by H(a)2{}_{2}, we have φ(u)→+∞\varphi \left(u)\to +\infty as ‖u‖→+∞\Vert u\Vert \to +\infty . Thus, φ\varphi is bounded below.Step 2: φ\varphi is sequence weakly lower semicontinuous.Without loss of generality, we assume that un⇀u{u}_{n}\rightharpoonup uin Wa1,p(t){W}_{a}^{1,p\left(t)}, so from Proposition 2.8 (ii), we have un→u{u}_{n}\to uin L∞(R){L}^{\infty }\left({\mathbb{R}}), which implies that un→uandf(t,un(t))→f(t,u(t)),∀t∈R.{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}f(t,{u}_{n}\left(t))\to f\left(t,u\left(t)),\hspace{1em}\forall t\in {\mathbb{R}}.By Fatou lemma, we have (4.25)limn→∞sup∫Rf(t,un(t))dt≤∫Rf(t,u(t))dt.\mathop{\mathrm{lim}}\limits_{n\to \infty }\sup \mathop{\int }\limits_{{\mathbb{R}}}f(t,{u}_{n}\left(t)){\rm{d}}t\le \mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t.Hence, from (4.3) and (4.25), we obtain (4.26)liminfn→∞φ(un)≥liminfn→∞∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−limsupn→∞∫Rf(t,un(t))dt≥∫R1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−∫Rf(t,u(t))dt=φ(u),\begin{array}{rcl}\mathop{\mathrm{liminf}}\limits_{n\to \infty }\varphi ({u}_{n})& \ge & \mathop{\mathrm{liminf}}\limits_{n\to \infty }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}({| {\dot{u}}_{n}| }^{p\left(t)}+a\left(t){| {u}_{n}| }^{p\left(t)}){\rm{d}}t-\mathop{\mathrm{limsup}}\limits_{n\to \infty }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f(t,{u}_{n}\left(t)){\rm{d}}t\\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t\\ & =& \varphi \left(u),\end{array}which shows that φ\varphi is sequence weakly lower semicontinuous.Using Lemma 2.18, there is a global minimum point u0∈Wa1,p(t){u}_{0}\in {W}_{a}^{1,p\left(t)}such that φ(u0)=minu∈Wa1,p(t)φ(u).\varphi \left({u}_{0})=\mathop{\min }\limits_{u\in {W}_{a}^{1,p\left(t)}}\varphi \left(u).Step 3: φ(u0)<0\varphi \left({u}_{0})\lt 0.Let u0∈(W01,p(t)⋂Wa1,p(t))⧹{0}{u}_{0}\in ({W}_{0}^{1,p\left(t)}\hspace{0.33em}\bigcap \hspace{0.33em}{W}_{a}^{1,p\left(t)})\setminus \left\{0\right\}with ‖u0‖=1\Vert {u}_{0}\Vert =1, by (4.3) and condition (v), for 0<s<10\lt s\lt 1, we obtain φ(su0)=∫R1p(t)(∣su˙0∣p(t)+a(t)∣su0∣p(t))dt−∫Rf(t,su0(t))dt≤sp−p−−∫Ωf(t,su0(t))dt≤sp−p−−ηsγ+∫Ω∣u0(t)∣γ(t)dt.\begin{array}{rcl}\varphi \left(s{u}_{0})& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| s{\dot{u}}_{0}{| }^{p\left(t)}+a\left(t)| s{u}_{0}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,s{u}_{0}\left(t)){\rm{d}}t\\ & \le & \frac{{s}^{{p}^{-}}}{{p}^{-}}-\mathop{\displaystyle \int }\limits_{\Omega }f\left(t,s{u}_{0}\left(t)){\rm{d}}t\\ & \le & \frac{{s}^{{p}^{-}}}{{p}^{-}}-\eta {s}^{{\gamma }^{+}}\mathop{\displaystyle \int }\limits_{\Omega }| {u}_{0}\left(t){| }^{\gamma \left(t)}{\rm{d}}t.\end{array}Note that 1<γ+<p−1\lt {\gamma }^{+}\lt {p}^{-}, it is easy to show that φ(su0)<0\varphi \left(s{u}_{0})\lt 0as s>0s\gt 0small enough.□Proof of Theorem 4.5Define a functional ψ:Wa1,p(t)→R\psi :{W}_{a}^{1,p\left(t)}\to {\mathbb{R}}as follows: (4.27)ψ(u)=∫R1p(t)(∣u˙(t)∣p(t)+a(t)∣u(t)∣p(t))dt−∫Rf1(t,u(t))dt+∫Rf2(t,u(t))dt.\psi \left(u)=\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}\left(t){| }^{p\left(t)}+a\left(t)| u\left(t){| }^{p\left(t)}){\rm{d}}t-\mathop{\int }\limits_{{\mathbb{R}}}{f}_{1}\left(t,u\left(t)){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{f}_{2}\left(t,u\left(t)){\rm{d}}t.Arguments as in proof of Theorems 4.1 and 4.5, we can easily obtain ψ\psi as the nonsmooth Lipschitz energy functional corresponding to problem (4.2).Let {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}be such that (4.28)∣ψ(un)∣≤M2andm(un)→0asn→+∞,| \psi \left({u}_{n})| \le {M}_{2}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({u}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to +\infty ,where M2>0{M}_{2}\gt 0is a constant. Because ∂ψ(un)⊆W−1,p(t)\partial \psi \left({u}_{n})\subseteq {W}^{-1,p\left(t)}is weakly compact and the norm is weakly lower semicontinuous. By Lemma 2.18, we can choose un∗∈∂ψ(un){u}_{n}^{\ast }\in \partial \psi \left({u}_{n})such that m(un)=‖un∗‖m\left({u}_{n})=\Vert {u}_{n}^{\ast }\Vert for n≥1n\ge 1.Define nonlinear operator ℒ:Wa1,p(t)→(Wa1,p(t))∗{\mathcal{ {\mathcal L} }}:{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast }as follows: ⟨ℒ(u),v⟩=∫R∣u˙(t)∣p(t)−2(u˙(t),v˙(t))dt,∀u,v∈Wa1,p(t).\langle {\mathcal{ {\mathcal L} }}\left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}| \dot{u}\left(t){| }^{p\left(t)-2}(\dot{u}\left(t),\dot{v}\left(t)){\rm{d}}t,\hspace{1em}\forall u,v\in {W}_{a}^{1,p\left(t)}.According to the literature [21], ℒ{\mathcal{ {\mathcal L} }}is monotonic and semicontinuous, so it is maximal monotone (see also [24]), therefore, un∗=ℒ(un)−wn1+wn2{u}_{n}^{\ast }={\mathcal{ {\mathcal L} }}\left({u}_{n})-{w}_{n}^{1}+{w}_{n}^{2}for n≥1n\ge 1, and wn1∈∂f1(t,un){w}_{n}^{1}\in \partial {f}_{1}\left(t,{u}_{n}), wn2∈∂f2(t,un){w}_{n}^{2}\in \partial {f}_{2}\left(t,{u}_{n}), wn1{w}_{n}^{1}, wn2∈Lp′(t){w}_{n}^{2}\in {L}^{p^{\prime} \left(t)}, where 1/p′(t)+1/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} \left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1.In another way, by the selection of sequence {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}, we obtain (4.29)∣⟨un∗,un⟩∣≤εn,εn↓0.| \langle {u}_{n}^{\ast },{u}_{n}\rangle | \le {\varepsilon }_{n},\hspace{1em}{\varepsilon }_{n}\downarrow 0.Then, it follows from (4.27), (4.28), (4.29), H(f)2{}_{2}: (iii′^{\prime} ) and H(f)3{}_{3}: (iv), we can show that (4.30)M2+εnμ≥ψ(un)−1μ⟨un∗,un⟩=∫R1p(t)−1μ(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫Rf1(t,un(t))dt+∫Rf2(t,un(t))dt+1μ∫R(⟨wn1,un⟩−⟨wn2,un⟩)dt≥∫R1p(t)−1μ(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+1μ∫R[−f10(t,un;−un)−μf1(t,un)]dt+1μ∫R[μf2(t,un)−f20(t,un;un)]dt≥1p+−1μ∫R(∣u˙n∣p(t)+a(t)∣un∣p(t))dt.\begin{array}{rcl}{M}_{2}+\frac{{\varepsilon }_{n}}{\mu }& \ge & \psi \left({u}_{n})-\frac{1}{\mu }\langle {u}_{n}^{\ast },{u}_{n}\rangle \\ & =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\left(\frac{1}{p\left(t)}-\frac{1}{\mu }\right)(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & & -\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{f}_{1}\left(t,{u}_{n}\left(t)){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{f}_{2}\left(t,{u}_{n}\left(t)){\rm{d}}t+\frac{1}{\mu }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\langle {w}_{n}^{1},{u}_{n}\rangle -\langle {w}_{n}^{2},{u}_{n}\rangle ){\rm{d}}t\\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\left(\frac{1}{p\left(t)}-\frac{1}{\mu }\right)(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & & +\frac{1}{\mu }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}-{f}_{1}^{0}\left(t,{u}_{n};-{u}_{n})-\mu {f}_{1}\left(t,{u}_{n})]{\rm{d}}t+\frac{1}{\mu }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}\mu {f}_{2}\left(t,{u}_{n})-{f}_{2}^{0}\left(t,{u}_{n};{u}_{n})]{\rm{d}}t\\ & \ge & \left(\frac{1}{{p}^{+}}-\frac{1}{\mu }\right)\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t.\end{array}Note that μ>p+\mu \gt {p}^{+}, by virtue of (4.30) and Proposition 2.4, we have {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}is bounded, so we assume that un⇀u{u}_{n}\rightharpoonup uin Wa1,p(t){W}_{a}^{1,p\left(t)}and un⇀u{u}_{n}\rightharpoonup uin Lp(t){L}^{p\left(t)}.Thanks to (4.29), thus (4.31)⟨ℒ(un),un−u⟩−∫Rwn1(un−u)dt+∫Rwn2(un−u)dt≤εn,∀n≥1.\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle -\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{1}\left({u}_{n}-u){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{2}\left({u}_{n}-u){\rm{d}}t\le {\varepsilon }_{n},\hspace{1em}\forall n\ge 1.Recall that wn1{w}_{n}^{1}, wn2∈Lp′(t)(R,RN){w}_{n}^{2}\in {L}^{p^{\prime} \left(t)}\left({\mathbb{R}},{{\mathbb{R}}}^{N}), so we have (4.32)∫Rwn1(un−u)dt→0,∫Rwn2(un−u)dt→0,\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{1}\left({u}_{n}-u){\rm{d}}t\to 0,\hspace{1em}\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{2}\left({u}_{n}-u){\rm{d}}t\to 0,as n→∞n\to \infty . Then, by (4.31) and (4.32), we obtain (4.33)limsupn→∞⟨ℒ(un),un−u⟩≤0.\mathop{\mathrm{limsup}}\limits_{n\to \infty }\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle \le 0.Combining with (4.33) and Proposition 2.9, we deduce that un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}. So φ\varphi satisfies PS condition.Step 2: ψ\psi satisfies nonsmooth mountain pass theorem.For any ε>0\varepsilon \gt 0, by H(f)2{}_{2}: (ii), (iv), one has (4.34)f1(t,u)≤ε∣u∣p++c(ε)∣u∣α(t),∀(t,u)∈R×RN.{f}_{1}\left(t,u)\le \varepsilon | u{| }^{{p}^{+}}+c\left(\varepsilon )| u{| }^{\alpha \left(t)},\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Choose ‖u‖=ρ\Vert u\Vert =\rho is small enough, from (4.27), (4.34) and Proposition 2.4, we obtain ψ(u)≥1p+∫R(∣u˙∣p(t)+a(t)∣u∣p(t))dt−ε∫R∣u∣p+dt−c(ε)∫R∣u∣α(t)dt≥1p+‖u‖p+−εcp+p+‖u‖p+−c(ε)cα−α−‖u‖α−\begin{array}{rcl}\psi \left(u)& \ge & \frac{1}{{p}^{+}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\varepsilon \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{{p}^{+}}{\rm{d}}t-c\left(\varepsilon )\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{\alpha \left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}\end{array}for ε>0\varepsilon \gt 0, let εcp+p+<12p+\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\lt \frac{1}{2{p}^{+}}, where cp+(cα−){c}_{{p}^{+}}({c}_{{\alpha }^{-}})is the embedding constant from Wa1,p(t){W}_{a}^{1,p\left(t)}to Lp+(Lα−){L}^{{p}^{+}}({L}^{{\alpha }^{-}}). Then, ψ(u)≥12p+‖u‖p+−c(ε)cα−α−‖u‖α−.\psi \left(u)\ge \frac{1}{2{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}.Note that p+<α−{p}^{+}\lt {\alpha }^{-}, there exists a constant r>0r\gt 0such that φ(u)≥r\varphi \left(u)\ge ras ‖u‖=ρ\Vert u\Vert =\rho , when ρ\rho is small enough.As in [41], it follows from H(f)2{}_{2}: (iii′^{\prime} ) and H(f)3{}_{3}: (iv) that (4.35)f1(t,σu)≥σμf1(t,u),∀(t,u)∈R×RN{f}_{1}\left(t,\sigma u)\ge {\sigma }^{\mu }{f}_{1}\left(t,u),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}and (4.36)f2(t,σu)≤σϱf2(t,u),∀(t,u)∈R×RN.{f}_{2}\left(t,\sigma u)\le {\sigma }^{\varrho }{f}_{2}\left(t,u),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Let ω∈Wa1,p(t)\omega \in {W}_{a}^{1,p\left(t)}be such that (4.37)∣ω(t)∣=1,as∣t∣≤1;0,as∣t∣≥2;≤1,as∣t∣∈(1,2].| \omega \left(t)| =\left\{\begin{array}{ll}1,\hspace{1.0em}& \hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \le 1;\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \ge 2;\\ \le 1,\hspace{1.0em}& \hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \in \left(1,2].\end{array}\right.In view of (4.36) and Proposition 2.8 (i), one has ∫−22f2(t,ω)dt=∫{t∈[−2,2]:∣ω∣>1}f2(t,ω)dt+∫{t∈[−2,2]:∣ω∣≤1}f2(t,ω)dt≤∫{t∈[−2,2]:∣ω∣>1}f2t,ω∣ω∣∣ω∣ϱdt+∫−22max∣ω∣≤1∣f2(t,ω)∣dt\begin{array}{rcl}\underset{-2}{\overset{2}{\displaystyle \int }}{f}_{2}\left(t,\omega ){\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{\left\{t\in \left[-2,2]:| \omega | \gt 1\right\}}{f}_{2}\left(t,\omega ){\rm{d}}t+\mathop{\displaystyle \int }\limits_{\left\{t\in \left[-2,2]:| \omega | \le 1\right\}}{f}_{2}\left(t,\omega ){\rm{d}}t\\ & \le & \mathop{\displaystyle \int }\limits_{\left\{t\in \left[-2,2]:| \omega | \gt 1\right\}}{f}_{2}\left(t,\frac{\omega }{| \omega | }\right)| \omega {| }^{\varrho }{\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t\end{array}(4.38)≤‖ω‖L∞(R)ϱ∫−22max∣ω∣=1f2(t,ω)dt+∫−22max∣ω∣≤1∣f2(t,ω)∣dt≤κϱ‖ω‖ϱ∫−22max∣ω∣=1f2(t,ω)dt+∫−22max∣ω∣≤1∣f2(t,ω)∣dt=M3‖ω‖ϱ+M4,\begin{array}{rcl}& \le & \Vert \omega {\Vert }_{{L}^{\infty }\left({\mathbb{R}})}^{\varrho }\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | =1}{f}_{2}\left(t,\omega ){\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t\\ & \le & {\kappa }^{\varrho }\Vert \omega {\Vert }^{\varrho }\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | =1}{f}_{2}\left(t,\omega ){\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t\\ & =& {M}_{3}\Vert \omega {\Vert }^{\varrho }+{M}_{4},\end{array}where M3=κϱ∫−22max∣ω∣=1f2(t,ω)dt,M4=∫−22max∣ω∣≤1∣f2(t,ω)∣dt.{M}_{3}={\kappa }^{\varrho }\underset{-2}{\overset{2}{\int }}\mathop{\max }\limits_{| \omega | =1}{f}_{2}\left(t,\omega ){\rm{d}}t,\hspace{1em}{M}_{4}=\underset{-2}{\overset{2}{\int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t.When σ>1\sigma \gt 1, by (4.35), we have (4.39)∫−22f1(t,σω(t))dt≥σμ∫−22f1(t,ω(t))dt=mσμ,\underset{-2}{\overset{2}{\int }}{f}_{1}\left(t,\sigma \omega \left(t)){\rm{d}}t\ge {\sigma }^{\mu }\underset{-2}{\overset{2}{\int }}{f}_{1}\left(t,\omega \left(t)){\rm{d}}t=m{\sigma }^{\mu },where m=∫−11f(t,ω)dt>0m={\int }_{-1}^{1}f\left(t,\omega ){\rm{d}}t\gt 0. Thus, it follows from (4.27), (4.36), (4.37), (4.38), (4.39) and Proposition 2.4 (ii), we derive (4.40)ψ(σω)=∫R1p(t)(∣σω˙∣p(t)+a(t)∣σω∣p(t))dt+∫R[f2(t,σω(t))−f1(t,σω(t))]dt≤σp+p−∫R(∣ω˙∣p(t)+a(t)∣ω∣p(t))dt+∫−22f2(t,σω)dt−∫−22f1(t,σω)dt≤σp+p−‖ω‖p++M3σϱ‖ω‖ϱ+M4σϱ−mσμ.\begin{array}{rcl}\psi \left(\sigma \omega )& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \sigma \dot{\omega }{| }^{p\left(t)}+a\left(t)| \sigma \omega {| }^{p\left(t)}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}{f}_{2}\left(t,\sigma \omega \left(t))-{f}_{1}\left(t,\sigma \omega \left(t))]{\rm{d}}t\\ & \le & \frac{{\sigma }^{{p}^{+}}}{{p}^{-}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| \dot{\omega }{| }^{p\left(t)}+a\left(t)| \omega {| }^{p\left(t)}){\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}{f}_{2}\left(t,\sigma \omega ){\rm{d}}t-\underset{-2}{\overset{2}{\displaystyle \int }}{f}_{1}\left(t,\sigma \omega ){\rm{d}}t\\ & \le & \frac{{\sigma }^{{p}^{+}}}{{p}^{-}}\Vert \omega {\Vert }^{{p}^{+}}+{M}_{3}{\sigma }^{\varrho }\Vert \omega {\Vert }^{\varrho }+{M}_{4}{\sigma }^{\varrho }-m{\sigma }^{\mu }.\end{array}Since μ>ϱ>p+\mu \gt \varrho \gt {p}^{+}, m>0m\gt 0, from (4.40), we can choose σ0>1{\sigma }_{0}\gt 1such that e=σ0ω∈Wa1,p(t)e={\sigma }_{0}\omega \in {W}_{a}^{1,p\left(t)}and φ(e)<0\varphi \left(e)\lt 0. Hence, from Lemma 2.19, there exists at least one nontrivial critical point, that is, system (4.2) has at least one homoclinic orbit.□Example 4.1Let p(t)=32+2π∣arctant∣p\left(t)=\frac{3}{2}+\frac{2}{\pi }| \arctan t| for t∈R,t\in {\mathbb{R}},and f(t,u)=a(t)(1+a(t))−1∣u∣5/2ln(1+∣u∣),∀(t,u)∈R×RN,f\left(t,u)=a\left(t){(1+a\left(t))}^{-1}| u{| }^{5\text{/}2}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where a(t)a\left(t)satisfies H(a),H(a)2\hspace{0.1em}\text{H(a)},{\text{H(a)}}_{2}. It is evident that ffis locally Lipschitz and ∂f(t,u)=a(t)(1+a(t))−152∣u∣1/2uln(1+∣u∣)+∣u∣3/2u1+∣u∣.\partial f\left(t,u)=a\left(t){\left(1+a\left(t))}^{-1}\left[\frac{5}{2}| u{| }^{1\text{/}2}u\mathrm{ln}(1+| u| )+\frac{| u{| }^{3\text{/}2}u}{1+| u| }\right].As −f0(t,u;−u)=52+∣u∣(1+∣u∣)ln(1+∣u∣)f(t,u)≥52+11+∣u∣f(t,u)>p++11+∣u∣f(t,u),\begin{array}{rcl}-{f}^{0}\left(t,u;-u)& =& \left[\frac{5}{2}+\frac{| u| }{(1+| u| )\mathrm{ln}(1+| u| )}\right]f\left(t,u)\\ & \ge & \left(\frac{5}{2}+\frac{1}{1+| u| }\right)f\left(t,u)\\ & \gt & \left({p}^{+}+\frac{1}{1+| u| }\right)f\left(t,u),\end{array}ffsatisfies H(f)2{\text{H(f)}}_{2}: (iii) with α=β=ν=1\alpha =\beta =\nu =1. Thus, we can show that ffsatisfies the hypothesis of Theorem 4.1.Moreover, it is similar to obtain that ffsatisfies the hypothesis of Theorem 4.2 with μ=52\mu =\frac{5}{2}.Example 4.2Let p(t)=5+11+t2p\left(t)=5+\frac{1}{1+{t}^{2}}for t∈R,t\in {\mathbb{R}},and f(t,u)=a(t)−1[∣u∣4∣sint∣+4+∣u∣2∣sint∣+2],∀(t,u)∈R×RN,f\left(t,u)=a{\left(t)}^{-1}{[}| u{| }^{4| \sin t| +4}+| u{| }^{2| \sin t| +2}],\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where a(t)=1+t2a\left(t)=1+{t}^{2}satisfies H(a),H(a)2\hspace{0.1em}\text{H(a)},{\text{H(a)}}_{2}. Obviously, ffis locally Lipschitz and ∂f(t,u)=a(t)−1(2∣sint∣+2)[2∣u∣4∣sint∣+2u+∣u∣2∣sint∣+1u].\partial f\left(t,u)=a{\left(t)}^{-1}(2| \sin t| +2){[}2| u{| }^{4| \sin t| +2}u+| u{| }^{2| \sin t| +1}u].Since ∣∂f(t,u)∣≤3(2∣sint∣+2)1+t2∣u∣2∣sint∣+1,∣u∣≤1;3(4∣sint∣+4)2(1+t2)∣u∣4∣sint∣+3,∣u∣≥1.| \partial f\left(t,u)| \le \left\{\begin{array}{ll}\frac{3(2| \sin t| +2)}{1+{t}^{2}}| u{| }^{2| \sin t| +1},\hspace{1.0em}& | u| \le 1;\\ \frac{3(4| \sin t| +4)}{2(1+{t}^{2})}| u{| }^{4| \sin t| +3},\hspace{1.0em}& | u| \ge 1.\end{array}\right.Then ffsatisfies H(f)2{\text{H(f)}}_{2}: (ii′)\left({\rm{ii}}^{\prime} )with α1(t)=2∣sint∣+2,α2(t)=4∣sint∣+4,a1(t)=31+t2,a2(t)=32(1+t2).{\alpha }_{1}\left(t)=2| \sin t| +2,\hspace{1em}{\alpha }_{2}\left(t)=4| \sin t| +4,\hspace{1em}{a}_{1}\left(t)=\frac{3}{1+{t}^{2}},\hspace{1em}{a}_{2}\left(t)=\frac{3}{2\left(1+{t}^{2})}.Let Ω=(−2,−2)\Omega =\left(-2,-2), γ(t)=2∣sint∣+2,\gamma \left(t)=2| \sin t| +2,one has f(t,u)≥15∣u∣2∣sint∣+2,∀∣u∣≤1.f\left(t,u)\ge \frac{1}{5}| u{| }^{2| \sin t| +2},\hspace{1em}\forall | u| \le 1.Hence, from Theorem 4.4, problem (4.1) has at least a nontrivial homoclinic solution.Example 4.3Let p(t)=32+2π∣arctant∣p\left(t)=\frac{3}{2}+\frac{2}{\pi }| \arctan t| for t∈Rt\in {\mathbb{R}}, f=f1−f2f={f}_{1}-{f}_{2}and f1(t,u)=a1(t)(1+a1(t))−1∣u∣7/2ln(1+∣u∣),∀(t,u)∈R×RN,{f}_{1}\left(t,u)={a}_{1}\left(t){(1+{a}_{1}\left(t))}^{-1}| u{| }^{7\text{/}2}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},f2(t,u)=a2(t)[sint∣u∣2+∣u∣3],∀(t,u)∈R×RN,{f}_{2}\left(t,u)={a}_{2}\left(t){[}\sin t| u{| }^{2}+| u{| }^{3}],\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where ai(t)(i=1,2.){a}_{i}\left(t)\left(i=1,2.)satisfies H(a),H(a)2\hspace{0.1em}\text{H(a)},{\text{H(a)}}_{2}and H(f)2{\text{H(f)}}_{2}: (ii′)\left({\rm{ii}}^{\prime} ), (v), respectively. It is visible that ffis locally Lipschitz, ∂f⊆∂f1−∂f2\partial f\subseteq \partial {f}_{1}-\partial {f}_{2}and f20(t,u;u)=a2(t)[2sint∣u∣2+3∣u∣3]≥3f2(t,u),{f}_{2}^{0}\left(t,u;\hspace{0.33em}u)={a}_{2}\left(t){[}2\sin t| u{| }^{2}+3| u{| }^{3}]\ge 3{f}_{2}\left(t,u),which implies, f2{f}_{2}satisfies H(f)3{\text{H(f)}}_{3}: (iv) with ϱ=3\varrho =3and μ=72\mu =\frac{7}{2}. It is easy to verify that ffsatisfies the hypothesis of Theorem 4.5. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Nonlinear Analysis de Gruyter

Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian

Advances in Nonlinear Analysis , Volume 12 (1): 1 – Jan 1, 2023

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de Gruyter
Copyright
© 2023 Jun Cheng et al., published by De Gruyter
eISSN
2191-950X
DOI
10.1515/anona-2022-0272
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Abstract

1IntroductionIn this article, we study the following nonlinear second-order p(t)p\left(t)-Laplacian system with nonsmooth potential (1.1)ddt(∣u˙(t)∣p(t)−2u˙(t))−a(t)∣u(t)∣p(t)−2u(t)∈∂f(t,u(t)),u(t)→0,as∣t∣→∞,\left\{\begin{array}{l}\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial f\left(t,u\left(t)),\\ u\left(t)\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \to \infty ,\end{array}\right.where p,a:R→R+p,a:{\mathbb{R}}\to {{\mathbb{R}}}^{+}, f:R×RN→Rf:{\mathbb{R}}\times {{\mathbb{R}}}^{N}\to {\mathbb{R}}, u↦f(t,u)u\mapsto f\left(t,u)is locally Lipschitz. Here ∂f(t,x)\partial f\left(t,x)denotes the subdifferential of the locally Lipschitz function u↦f(t,u)u\mapsto f\left(t,u).In recent years, the study on p(t)p\left(t)-Laplacian problems has attracted more and more attention. The p(t)p\left(t)-Laplacian possesses more complicated phenomena than the pp-Laplacian. For example, it is inhomogeneous, which causes many difficulties, and some classical theories and methods, such as the theory of Sobolev spaces, are not applicable. The study of various mathematical problems with variable exponent growth condition has received considerable attention in recent years; see [26,30,50,52]. One of the most studied models leading to problems of this type is the model of motion of electro-rheological fluids, which are characterized by their ability to drastically change the mechanical properties under the influence of an exterior electromagnetic field [59]. Problems with variable exponent growth conditions also appear in the mathematical modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the filtration processes of an ideal barotropic gas through a porous medium [2,3]. Another field of application of equations with variable exponent growth conditions is image processing [10]. We refer the reader to [23,55, 56,57,58,60,61] for an overview and references on this subject, and to [12,13, 14,15,28,29,41,45,53,54] for the study of the p(t)p\left(t)-Laplacian equations and the corresponding variational problems.Since many free boundary problems and obstacle problems may be reduced to partial differential equations (PDEs) with discontinuous nonlinearities, the existence of solutions for the problems with discontinuous nonlinearities has been widely investigated in recent years. Chang [4] extended the variational methods to a class of nondifferentiable functionals. In 2000, Kourogenis and Papageorgiou [35] obtained some nonsmooth critical point theorems. Subsequently, the nonsmooth version of the three critical points theorem and the nonsmooth Ricceri-type variational principle was established by Marano and Motreanu [36], who gave an application to elliptic problems involving the pp-Laplacian with discontinuous nonlinearities. Kandilakis et al. [34] obtained the local linking theorem for locally Lipschitz functions. Dai [16] elaborated a nonsmooth version of the fountain theorem and gave an application to a Dirichlet-type differential inclusion. In 2019, Ge and Rădulescu [27] obtained infinitely many solutions for a nonhomogeneous differential inclusion with lack of compactness involving the p(x)p\left(x)-Laplacian.It is well known that homoclinic orbits play an important role in analyzing the chaos of dynamical systems. If a system has transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation, and its perturbed system probably produces chaotic phenomena. Therefore, it is of practical importance and mathematical significance to consider the existence of homoclinic orbits of problem (1.1). When p(t)≡pp\left(t)\equiv p, (1.1) reduces to pp-Laplacian system: (1.2)ddt(∣u˙(t)∣p−2u˙(t))−a(t)∣u(t)∣p−2u(t)∈∂f(t,u(t)),\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p-2}u\left(t)\in \partial f\left(t,u\left(t)),Hu and Papageorgious studied the existence of homoclinic solution using the theory of nonsmooth critical points and the idea of approximation [31,32] in the case of periodic nonsmooth potential with scalar equation. However, they did not prove the existence of homoclinic solutions and approached the problem differently from ours. Particularly, none of the works addressed the issues in the case of nonperiodic nonsmooth potential. With regard to the results of (1.1) in PDE, please refer to the literature [6,7,8, 9,17,19,22,33,42,47,48].To the best of our knowledge, there is few paper discussing the homoclinic solutions of problem (1.1) with nonsmooth potential via nonsmooth critical point theory can be found in the existing literature. In order to fill in this gap, inspired by [28,37,41,57], we study problem (1.1) from a more extensive viewpoint. More precisely, we would study the existence of nontrivial homoclinic solutions of problem (1.1) with the generalized subquadratic and superquadratic in two cases of the nonsmooth potential: periodic and nonperiodic, respectively. Moreover, our results generalize and improve the ones in (1.2). The resulting problem engages two major difficulties: first, due to the appearance of the variable exponent, which is not homogeneous, some special techniques and sharp estimation of inequality will be needed to study this type of problem (1.1). Another difficulty we must overcome is verifying the link geometry and certifying boundedness of the sequence of solutions {un}\left\{{u}_{n}\right\}associated with problem (1.1). It is worth to point out that commonly known methods and techniques for studying constant exponent equations fail in the setting of problems involving variable exponents. In these cases, we have to use techniques which are simpler and more direct in this article.Throughout this article, we formulate the hypotheses on p(t)p\left(t), a(t)a\left(t)and basic assumptions on f(t,u)f\left(t,u): H(p)p∈C(R,R+)p\in C\left({\mathbb{R}},{{\mathbb{R}}}^{+})and 1<p−≔inft∈Rp(t)≤supt∈Rp(t)≔p+<∞;1\lt {p}^{-}:= \mathop{\inf }\limits_{t\in {\mathbb{R}}}p\left(t)\le \mathop{\sup }\limits_{t\in {\mathbb{R}}}p\left(t):= {p}^{+}\lt \infty ;H(a)a∈C(R,R+)a\in C\left({\mathbb{R}},{{\mathbb{R}}}^{+})and there exists a0>0{a}_{0}\gt 0such that a(t)≥a0>0a\left(t)\ge {a}_{0}\gt 0for t∈Rt\in {\mathbb{R}};H(f)(i)the function f(t,⋅):R→Rf\left(t,\cdot ):{\mathbb{R}}\to {\mathbb{R}}is measurable for all u∈RNu\in {{\mathbb{R}}}^{N}and f(t,0)=0f\left(t,0)=0;(ii)the function f(⋅,u):RN→Rf\left(\cdot ,u):{{\mathbb{R}}}^{N}\to {\mathbb{R}}is locally Lipschitz for a.e. t∈Rt\in {\mathbb{R}}.Our approach is variationally based on the nonsmooth critical point theory (see Rădulescu and Repovš [51], Diening et al. [18] and the papers by Chang, Fan, Rădulescu, Papageorgiou, Papageorgiou and Zhao et al. [4,21,38,40,44]). For the convenience of the reader, in the next section we recall some basic definitions and facts from the theory, which we shall use in the sequel.This article is organized as follows. In Section 2, we present some necessary preliminary knowledge on the generalized gradient of the locally Lipschitz function and variable exponent Sobolev spaces. In Section 3, we establish and prove the existence of nontrivial homoclinic solution related to periodic problem (1.1). In Section 4, we establish and prove the existence of nontrivial homoclinic solution corresponding to nonperiodic problems (1.1) and (4.2), respectively.Throughout the article, we make use of the following notations: Ls(R)(1≤s<∞){L}^{s}\left({\mathbb{R}})\left(1\le s\lt \infty )denotes the Lebesgue space with the norm ‖u‖s=∫R∣u∣sdt1/s\Vert u{\Vert }_{s}={\left({\int }_{{\mathbb{R}}}| u{| }^{s}{\rm{d}}t\right)}^{1\text{/}s};For any x∈Rx\in {\mathbb{R}}and r>0r\gt 0, Br(x)≔{y∈R:∣y−x∣<r}{B}_{r}\left(x):= \{y\in {\mathbb{R}}:| y-x| \lt r\}and Br=Br(0){B}_{r}={B}_{r}\left(0);C1,C2,…{C}_{1},{C}_{2},\ldots denote positive constants possibly different in different places.2PreliminariesWe start with some preliminary basic results on variable exponent Sobolev spaces. For more details we refer the readers to the book of Rădulescu and Repovš [51], Diening et al. [18] and the papers by Chang, Fan, Rădulescu, Papageorgiou, Papageorgiou and Zhao et al. [4,21,38,40,44].2.1Weighted variable exponential Wa1,p(t){W}_{a}^{1,p\left(t)}spaceIn order to discuss problem (1.1), we recall some known results from critical point theory and the properties of space Wa1,p(t){W}_{a}^{1,p\left(t)}for the convenience of the readers.Let Ω\Omega be a subset of R{\mathbb{R}}, S(Ω,RN)≔{u:the functionu:Ω→RNis measurable}S(\Omega ,{{\mathbb{R}}}^{N}):= \{u:\hspace{0.33em}\hspace{0.1em}\text{the function}\hspace{0.1em}\hspace{0.33em}u:\Omega \to {{\mathbb{R}}}^{N}\hspace{0.1em}\text{is measurable}\hspace{0.1em}\}and any two elements in S(Ω,RN)S(\Omega ,{{\mathbb{R}}}^{N})which are almost equal are considered the same element. Let p,ap,asatisfy assumptions HH(p), HH(a), respectively.Define Lap(t)(Ω,RN){L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})(denoted by Lap(t){L}_{a}^{p\left(t)}) as follows: Lap(t)(Ω,RN)=u∈S(Ω,RN):∫Ωa(t)∣u(t)∣p(t)dt<∞{L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})=\left\{u\in S(\Omega ,{{\mathbb{R}}}^{N}):\mathop{\int }\limits_{\Omega }a\left(t)| u\left(t){| }^{p\left(t)}{\rm{d}}t\lt \infty \right\}endowed with the norm ∣u∣p(t),a=infλ>0:∫Ωa(t)uλp(t)dt≤1.| u{| }_{p\left(t),a}=\inf \left\{\lambda \gt 0:\mathop{\int }\limits_{\Omega }a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}{\rm{d}}t\le 1\right\}.If a(t)≡1a\left(t)\equiv 1, Lap(t){L}_{a}^{p\left(t)}and the corresponding norm ∣u∣p(t),a| u{| }_{p\left(t),a}are written simply by Lp(t){L}^{p\left(t)}, ∣u∣p(t)| u{| }_{p\left(t)}.Define Wa1,p(t)(Ω,RN){W}_{a}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})(denoted by Wa1,p(t){W}_{a}^{1,p\left(t)}) as follows: Wa1,p(t)(Ω,RN)={u∈Lap(t)(Ω,RN):u˙∈Lap(t)(Ω,RN)}{W}_{a}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}):\dot{u}\in {L}_{a}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})\}with the norm ‖u‖=infλ>0:∫Ωu˙λp(t)+a(t)uλp(t)dt≤1.\Vert u\Vert =\inf \left\{\lambda \gt 0:\mathop{\int }\limits_{\Omega }\left({\left|,\frac{\dot{u}}{\lambda },\right|}^{p\left(t)}+a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}\right){\rm{d}}t\le 1\right\}.In particular, if a(t)≡1a\left(t)\equiv 1, Wa1,p(t){W}_{a}^{1,p\left(t)}is reduced to W1,p(t)(Ω,RN)={u∈Lp(t)(Ω,RN):u˙∈Lp(t)(Ω,RN)}{W}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})=\{u\in {L}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}):\dot{u}\in {L}^{p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})\}and the norm ‖u‖=∣u∣p(t)+∣u˙∣p(t).\Vert u\Vert =| u{| }_{p\left(t)}+| \dot{u}{| }_{p\left(t)}.We use W01,p(t){W}_{0}^{1,p\left(t)}to represent the space of C0∞(Ω,RN){C}_{0}^{\infty }(\Omega ,{{\mathbb{R}}}^{N})consisting of infinitely continuous differentiable functions with compact supports on Ω\Omega completion in W1,p(t){W}^{1,p\left(t)}. We call the space Lp(t){L}^{p\left(t)}a generalized Lebesgue space, and it is a special kind of generalized Orlicz spaces. The space W1,p(t){W}^{1,p\left(t)}is called a generalized Sobolev space, it is a special kind of generalized Orlicz-Sobolev spaces. For more details on the general theory of generalized Orlicz spaces and generalized Orlicz-Sobolev spaces, see [18,20,51] and references therein.The following propositions summarize the main properties of this norm (see Alves and Liu [1], Rădulescu and Repovš [51] and Fan and Zhao [21]).Proposition 2.1Lap(t){L}_{a}^{p\left(t)}, Wa1,p(t){W}_{a}^{1,p\left(t)}, W01,p(t){W}_{0}^{1,p\left(t)}are reflexive Banach spaces with norms defined above when p−>1{p}^{-}\gt 1.Proposition 2.2Let ρ(u)=∫Ωa(t)∣u(t)∣p(t)dt\rho \left(u)={\int }_{\Omega }a\left(t)| u\left(t){| }^{p\left(t)}{\rm{d}}tfor any u,v∈Lap(t)u,v\in {L}_{a}^{p\left(t)}, then the following properties hold: (i)ρ(u)=0⇔u=0;\rho \left(u)=0\iff u=0;(ii)ρ(u)=ρ(−u);\rho \left(u)=\rho \left(-u);(iii)ρ(αu+βv)≤αρ(u)+βρ(v)\rho (\alpha u+\beta v)\le \alpha \rho \left(u)+\beta \rho \left(v)for any α,β≥0\alpha ,\beta \ge 0, α+β=1;\alpha +\beta =1;(iv)ρ(u+v)≤2p+(ρ(u)+ρ(v));\rho \left(u+v)\le {2}^{{p}^{+}}(\rho \left(u)+\rho \left(v));(v)If λ>1\lambda \gt 1, thenλp+ρ(u)≤ρ(λu)≤λp−ρ(u)≤λρ(u)≤ρ(u);{\lambda }^{{p}^{+}}\rho \left(u)\le \rho \left(\lambda u)\le {\lambda }^{{p}^{-}}\rho \left(u)\le \lambda \rho \left(u)\le \rho \left(u);(vi)‖u‖p(t),a=1\Vert u{\Vert }_{p\left(t),a}=1if and only if ρuλ=1\rho \left(\frac{u}{\lambda }\right)=1, for any u∈Lap(t)⧹{0}u\in {L}_{a}^{p\left(t)}\setminus \left\{0\right\}.Proposition 2.3For any u∈Lap(t)u\in {L}_{a}^{p\left(t)}, the following properties hold: (i)∣u∣p(t),a<1(=1;>1)⇔ρ(u)<1(=1;>1)| u{| }_{p\left(t),a}\lt 1\hspace{0.33em}\left(=1;\gt 1)\iff \rho \left(u)\lt 1\hspace{0.33em}\left(=1;\hspace{0.33em}\gt 1);(ii)If ∣u∣p(t),a>1| u{| }_{p\left(t),a}\gt 1, then ∣u∣p(t),ap−≤ρ(u)≤∣u∣p(t),ap+| u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{-}}\le \rho \left(u)\le | u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{+}};(iii)If ∣u∣p(t),a<1| u{| }_{p\left(t),a}\lt 1, then ∣u∣p(t),ap+≤ρ(u)≤∣u∣p(t),ap−| u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{+}}\le \rho \left(u)\le | u{\hspace{-0.25em}| }_{p\left(t),a}^{{p}^{-}};(iv)∣u∣p(t),a→0⇔ρ(u)→0| u{| }_{p\left(t),a}\to 0\iff \rho \left(u)\to 0;(v)∣u∣p(t),a→∞⇔ρ(u)→∞| u{| }_{p\left(t),a}\to \infty \iff \rho \left(u)\to \infty .Proposition 2.4Let ϕ(u)=∫Ω(∣u˙∣p(t)+a(t)∣u∣p(t))dt\phi \left(u)={\int }_{\Omega }(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}tfor any u∈Wa1,p(t)u\in {W}_{a}^{1,p\left(t)}, then the following properties hold: (i)‖u‖<1(=1;>1)⇔ϕ(u)<1(=1;>1)\Vert u\Vert \lt 1\hspace{0.25em}\left(=1;\gt 1)\iff \phi \left(u)\lt 1\hspace{0.25em}\left(=\hspace{0.25em}1;\gt 1);(ii)If ‖u‖>1\Vert u\Vert \gt 1, then ‖u‖p−≤ϕ(u)≤‖u‖p+\Vert u{\Vert }^{{p}^{-}}\le \phi \left(u)\le \Vert u{\Vert }^{{p}^{+}};(iii)If ‖u‖<1\Vert u\Vert \lt 1, then ‖u‖p+≤ϕ(u)≤‖u‖p−\Vert u{\Vert }^{{p}^{+}}\le \phi \left(u)\le \Vert u{\Vert }^{{p}^{-}};(iv)‖u‖→0⇔ϕ(u)→0\Vert u\Vert \to 0\iff \phi \left(u)\to 0;(v)‖u‖→∞⇔ϕ(u)→∞\Vert u\Vert \to \infty \iff \phi \left(u)\to \infty .Proposition 2.5Let ρ(u)=∫Ωa(t)∣u∣p(t)dt\rho \left(u)={\int }_{\Omega }a\left(t)| u{| }^{p\left(t)}{\rm{d}}tfor any u∈Lap(t)u\in {L}_{a}^{p\left(t)}, {un}⊂Lap(t)\{{u}_{n}\}\subset {L}_{a}^{p\left(t)}, then the following properties are equivalent: (i)limn→∞∣un−u∣p(t),a=0{\mathrm{lim}}_{n\to \infty }| {u}_{n}-u{| }_{p\left(t),a}=0;(ii)limn→∞ρ(un−u)=0{\mathrm{lim}}_{n\to \infty }\rho \left({u}_{n}-u)=0;(iii)un→u{u}_{n}\to ua.e. t∈Ωt\in \Omega and limn→∞ρ(un)=ρ(u){\mathrm{lim}}_{n\to \infty }\rho \left({u}_{n})=\rho \left(u).Proposition 2.6(Lp(t))∗=Lq(t){({L}^{p\left(t)})}^{\ast }={L}^{q\left(t)}with 1/p(t)+1/q(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}q\left(t)=1and∫Ωu(t)v(t)dt≤2∣u∣p(t)∣v∣q(t),∀u∈Lp(t),v∈Lq(t),\left|\mathop{\int }\limits_{\Omega }u\left(t)v\left(t){\rm{d}}t\right|\le 2| u{| }_{p\left(t)}| v{| }_{q\left(t)},\hspace{1em}\forall u\in {L}^{p\left(t)},\hspace{1em}v\in {L}^{q\left(t)},where (Lp(t))∗{\left({L}^{p\left(t)})}^{\ast }is the dual space of Lp(t){L}^{p\left(t)}.Proposition 2.7C0∞(R,RN){C}_{0}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})is dense in space Wa1,p(t){W}_{a}^{1,p\left(t)}.Proposition 2.8Let u∈Wa1,p(t)u\in {W}_{a}^{1,p\left(t)}, then(i)u∈C(R,RN)u\in C({\mathbb{R}},{{\mathbb{R}}}^{N})and u(t)→0u\left(t)\to 0as ∣t∣→∞| t| \to \infty . Moreover, the embedding Wa1,p(t)↪L∞(R,RN){W}_{a}^{1,p\left(t)}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})is continuous, and there exists a constant κ>0\kappa \gt 0such that‖u‖L∞≤κ‖u‖,∀u∈Wa1,p(t);\Vert u{\Vert }_{{L}^{\infty }}\le \kappa \Vert u\Vert ,\hspace{1em}\forall u\in {W}_{a}^{1,p\left(t)};(ii)If H(p),H(a)\hspace{0.1em}\text{H(p),H(a)}\hspace{0.1em}hold and a(t)→+∞a\left(t)\to +\infty as ∣t∣→∞| t| \to \infty , then the embedding Wa1,p(t)↪L∞(R,RN){W}_{a}^{1,p\left(t)}\hspace{0.33em}\hookrightarrow \hspace{0.33em}{L}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})is compact.Consider the following functional: I(u)=∫Ω1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt,∀u∈Wa1,p(t).I\left(u)=\mathop{\int }\limits_{\Omega }\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t,\hspace{1em}\forall u\in {W}_{a}^{1,p\left(t)}.We know that I∈C1(Wa1,p(t),R)I\in {C}^{1}\left({W}_{a}^{1,p\left(t)},{\mathbb{R}})under condition HH(a). Moreover, ⟨I′(u),v⟩=∫R(∣u˙∣p(t)−2u˙v˙+a(t)∣u∣p(t)−2uv)dt,∀u,v∈Wa1,p(t).\langle I^{\prime} \left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}(| \dot{u}{| }^{p\left(t)-2}\dot{u}\dot{v}+a\left(t)| u{| }^{p\left(t)-2}uv){\rm{d}}t,\hspace{1.0em}\forall u,v\in {W}_{a}^{1,p\left(t)}.Proposition 2.9I′I^{\prime} is a mapping of type (S)+{\left(S)}_{+}, i.e., ifun⇀uandlimn→∞(I′(un)−I′(u),un−u)≤0,{u}_{n}\rightharpoonup u\hspace{1em}and\hspace{1em}\mathop{\mathrm{lim}}\limits_{n\to \infty }(I^{\prime} \left({u}_{n})-I^{\prime} \left(u),{u}_{n}-u)\le 0,then un{u}_{n}has a convergent subsequence in Wa1,p(t){W}_{a}^{1,p\left(t)}.Denote (2.1)A=J′:Wa1,p(t)→(Wa1,p(t))∗,A=J^{\prime} :{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast },then we have ⟨A(u),v⟩=∫Ω(∣u′(t)∣p(t)−2u′(t)v′(t)+a(x)∣u(t)∣p(t)−2uv)dt\langle A\left(u),v\rangle =\mathop{\int }\limits_{\Omega }(| u^{\prime} \left(t){| }^{p\left(t)-2}u^{\prime} \left(t)v^{\prime} \left(t)+a\left(x)| u\left(t){| }^{p\left(t)-2}uv){\rm{d}}tfor all u,v∈Wa1,p(t).u,v\in {W}_{a}^{1,p\left(t)}.Proposition 2.10The mapping A is a strictly monotone, bounded homeomorphism and is of type (S)+{\left(S)}_{+}in Wa1,p(t){W}_{a}^{1,p\left(t)}.2.1.1Periodic variable exponential W2nb1,p(t){W}_{2nb}^{1,p\left(t)}spaceFor any b>0b\gt 0, n≥1n\ge 1, let Tn≐[−nb,nb]{T}_{n}\doteq \left[-nb,nb]. Define L2nbp(t)(Tn,RN){L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})(denoted by L2nbp(t){L}_{2nb}^{p\left(t)}) as follows: L2nbp(t)(Tn,RN)=u∈S(Tn,RN):∫−nbnba(t)∣u(t)∣p(t)dt<∞{L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})=\left\{u\in S({T}_{n},{{\mathbb{R}}}^{N}):\underset{-nb}{\overset{nb}{\int }}a\left(t)| u\left(t){| }^{p\left(t)}{\rm{d}}t\lt \infty \right\}endowed with the norm ∣u∣p(t)=infλ>0:∫−nbnba(t)uλp(t)dt≤1.| u{| }_{p\left(t)}=\inf \left\{\lambda \gt 0:\underset{-nb}{\overset{nb}{\int }}a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}{\rm{d}}t\le 1\right\}.Moreover, L2nb∞(Tn,RN){L}_{2nb}^{\infty }({T}_{n},{{\mathbb{R}}}^{N})(denoted by L2nb∞{L}_{2nb}^{\infty }) be a Banach space with the norm ‖u‖L2nb∞=esssup{∣u(t)∣:t∈[−nb,nb]}.\Vert u{\Vert }_{{L}_{2nb}^{\infty }}=\hspace{0.1em}\text{ess}\hspace{0.1em}\sup \{| u\left(t)| :t\in \left[-nb,nb]\}.Define W2nb1,p(t)(Tn,Rn){W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{n})(denoted by W2nb1,p(t){W}_{2nb}^{1,p\left(t)}) as follows: W2nb1,p(t)(Tn,Rn)={u∈L2nbp(t)(Tn,RN):u˙∈L2nbp(t)(Tn,RN)}{W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{n})=\{u\in {L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}):\dot{u}\in {L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})\}endowed with the norm ‖u‖1=infλ>0:∫−nbnbu˙λp(t)+a(t)uλp(t)dt≤1.\Vert u{\Vert }_{1}=\inf \left\{\lambda \gt 0:\underset{-nb}{\overset{nb}{\int }}\left({\left|,\frac{\dot{u}}{\lambda },\right|}^{p\left(t)}+a\left(t){\left|,\frac{u}{\lambda },\right|}^{p\left(t)}\right){\rm{d}}t\le 1\right\}.In particular, if b→+∞b\to +\infty , L2nbp(t)(Tn,RN){L}_{2nb}^{p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}), L2nb∞(Tn,RN){L}_{2nb}^{\infty }({T}_{n},{{\mathbb{R}}}^{N}), W2nb1,p(t)(Tn,Rn){W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{n})are written simply by Lap(t)(R,RN){L}_{a}^{p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), La∞(R,RN){L}_{a}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N}), Wa1,p(t)(R,Rn){W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{n}), respectively.2.2Nonsmooth analysis theoryThe nonsmooth critical point theory for locally Lipschitz functionals is based on the subdifferential theory of Clark [11], Rădulescu [49], Gasiński and Papageorgiou [25].Definition 2.11Let XXbe a Banach space and let X∗{X}^{\ast }be its topological dual. By ⟨⋅⟩\langle \cdot \rangle we denote the duality brackets for the pair (X,X∗)(X,{X}^{\ast }). A function ϕ:X→R\phi :X\to {\mathbb{R}}is said to be locally Lipschitz, if for every x∈Xx\in Xthere exist U∈N(x)U\in {\mathcal{N}}\left(x)and a constant kU>0{k}_{U}\gt 0, such that ∣ϕ(y)−ϕ(z)∣≤kU‖y−z‖X,∀y,z∈U.| \phi (y)-\phi \left(z)| \le {k}_{U}\Vert y-z{\Vert }_{X},\hspace{1em}\forall y,z\in U.Definition 2.12For a given locally Lipschitz function ϕ:X→R,\phi :X\to {\mathbb{R}},the generalized directional derivative of φ\varphi at x∈Xx\in Xin the direction h∈Xh\in Xis defined by ϕ0(x;h)≐limy→x;λ↓0supϕ(y+λh)−ϕ(y)λ=infε,δ>0sup‖x−y‖X<δ,0<λ<δϕ(y+λh)−ϕ(y)λ.\begin{array}{rcl}{\phi }^{0}\left(x;h)& \doteq & \mathop{\mathrm{lim}}\limits_{y\to x;\lambda \downarrow 0}\sup \frac{\phi (y+\lambda h)-\phi (y)}{\lambda }\\ & =& \mathop{\inf }\limits_{\varepsilon ,\delta \gt 0}\mathop{\sup }\limits_{\Vert x-y{\Vert }_{X}\lt \delta ,0\lt \lambda \lt \delta }\frac{\phi (y+\lambda h)-\phi (y)}{\lambda }.\end{array}Based on Definition 2.12, one can easily verify that the function h↦ϕ0(x;h)h\mapsto {\phi }^{0}\left(x;h)is sublinear, Lipschitz continuous (see [11, Proposition 2.1.1]).Definition 2.13Let ϕ:X→R\phi :X\to {\mathbb{R}}be a locally Lipschitz function. Then generalized subdifferential of ϕ\phi at x∈Xx\in Xis the nonempty set ∂ϕ(x)⊆X∗\partial \phi \left(x)\subseteq {X}^{\ast }defined by ∂ϕ(x)={x∗∈X∗:⟨x∗,h⟩≤ϕ0(x;h),∀h∈X}.\partial \phi \left(x)=\{{x}^{\ast }\in {X}^{\ast }:\langle {x}^{\ast },h\rangle \le {\phi }^{0}\left(x;h),\hspace{1em}\forall h\in X\}.The multifunction x→∂ϕ(x)x\to \partial \phi \left(x)is known as the generalized (or Clarke) subdifferential of ϕ\phi . If ϕ,ψ:X→R\phi ,\psi :X\to {\mathbb{R}}are locally Lipschitz functions, then ∂(ϕ+ψ)(x)⊆∂ϕ(x)+∂ψ(x)\partial \left(\phi +\psi )\left(x)\subseteq \partial \phi \left(x)+\partial \psi \left(x)and for every λ∈R\lambda \in {\mathbb{R}}, ∂(λϕ)(x)=λ∂ϕ(x)\partial \left(\lambda \phi )\left(x)=\lambda \partial \phi \left(x).Definition 2.14Let ϕ:X→R\phi :X\to {\mathbb{R}}be a locally Lipschitz function. A point x∈Xx\in Xis said to be a critical point of ϕ\phi if 0∈∂ϕ(x)0\in \partial \phi \left(x).If x∈Xx\in Xis a critical point of ϕ\phi , then c=ϕ(x)c=\phi \left(x)is a critical value of ϕ\phi . It is easy to see that, if x∈Xx\in Xis a local extremum of ϕ\phi , then 0∈∂ϕ(x)0\in \partial \phi \left(x). Moreover, the multifunction x→∂ϕ(x)x\to \partial \phi \left(x)is upper semicontinuous from XXinto X∗{X}^{\ast }equipped with the w∗{w}^{\ast }topology, i.e., for any U⊆X∗U\subseteq {X}^{\ast }w∗{w}^{\ast }-open, the set {x∈X:∂ϕ(x)⊆U}\left\{x\in X:\partial \phi \left(x)\subseteq U\right\}is open in XX. For more details we refer to Clarke [11, Proposition 2.1.2].Definition 2.15The locally Lipschitz function ϕ:X→R\phi :X\to {\mathbb{R}}satisfies the nonsmooth Palais-Smale (PS) condition, if any sequence {xn}n≥1⊆X{\left\{{x}_{n}\right\}}_{n\ge 1}\subseteq Xsuch that {ϕ(xn)}n≥1is boundedandm(xn)→0asn→∞,{\left\{\phi \left({x}_{n})\right\}}_{n\ge 1}\hspace{1em}\hspace{0.1em}\text{is bounded}\hspace{0.1em}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({x}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,has a strongly convergent subsequence, where m(xn)=min[‖x∗‖:x∗∈∂ϕ(xn)]m\left({x}_{n})=\min {[}\Vert {x}^{\ast }\Vert :{x}^{\ast }\in \partial \phi \left({x}_{n})].Lemma 2.16(Lebourg’s mean value Theorem [39]). Given the points x and y in X and a real-valued function ϕ\phi which is Lipschitz continuous on an open set containing the segment [x,y]={(1−t)x+ty:t∈[0,1]}\left[x,y]=\left\{\left(1-t)x+ty:t\in \left[0,1]\right\}, there exist z=x+t0(y−x)z=x+{t}_{0}(y-x), with 0<t0<10\lt {t}_{0}\lt 1, and x∗∈∂ϕ(z){x}^{\ast }\in \partial \phi \left(z)such thatϕ(y)−ϕ(x)=⟨x∗,y−x⟩.\phi (y)-\phi \left(x)=\langle {x}^{\ast },y-x\rangle .If ϕ∈C1(X,R)\phi \in {C}^{1}\left(X,{\mathbb{R}}), then as we already mentioned ∂ϕ(x)={ϕ′(x)}\partial \phi \left(x)=\left\{\phi ^{\prime} \left(x)\right\}and so the above definition of the PS condition coincides with the classical (smooth) one. In the context of the smooth theory, Cerami introduced a weaker compactness condition which in our nonsmooth setting has the following form:Definition 2.17The locally Lipschitz function ϕ:X→R\phi :X\to {\mathbb{R}}satisfies the nonsmooth Cerami condition (C-condition), if any sequence {xn}n≥1⊆X{\left\{{x}_{n}\right\}}_{n\ge 1}\subseteq Xsuch that {ϕ(xn)}n≥1is bounded and(1+‖xn‖)m(xn)→0asn→∞,{\left\{\phi \left({x}_{n})\right\}}_{n\ge 1}\hspace{0.33em}\hspace{0.1em}\text{is bounded and}\hspace{0.1em}\hspace{0.33em}(1+\Vert {x}_{n}\Vert )m\left({x}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,has a strongly convergent subsequence.Lemma 2.18(Weierstrass theorem [43]). Assume that φ\varphi is a locally Lipschitz functional on a Banach space X and φ:X→R\varphi :X\to {\mathbb{R}}satisfies: (i)φ\varphi is weakly lower semicontinuous;(ii)φ\varphi is coercive.Then there exists x∗∈X{x}^{\ast }\in Xsuch that φ(x∗)=minx∈Xφ(x)\varphi \left({x}^{\ast })={\min }_{x\in X}\varphi \left(x).Lemma 2.19(Nonsmooth mountain pass theorem [35]). Let X be a reflexive Banach space, ϕ:X→R\phi :X\to {\mathbb{R}}a locally Lipschitz functional satisfying the PS-condition. Assume that there exist x0,x1∈X,{x}_{0},{x}_{1}\in X,c0∈R{c}_{0}\in {\mathbb{R}}and ϱ>0\varrho \gt 0such that ‖x1−x0‖>ϱ\Vert {x}_{1}-{x}_{0}\Vert \gt \varrho andmax{ϕ(x0),ϕ(x1)}<c0=inf[ϕ(y):‖y−x0‖=ϱ].\max \left\{\phi \left({x}_{0}),\phi \left({x}_{1})\right\}\lt {c}_{0}=\inf {[}\phi (y):\Vert y-{x}_{0}\Vert =\varrho ].Then ϕ\phi has a critical point x∈Xx\in Xwith c=ϕ(x)≥c0c=\phi \left(x)\ge {c}_{0}, where c is given byc=infγ∈Γ1maxt∈Tϕ(γ(t)),c=\mathop{\inf }\limits_{\gamma \in {\Gamma }_{1}}\mathop{\max }\limits_{t\in T}\phi \left(\gamma \left(t)),Γ1={γ∈C([0,1],x):γ(0)=x0,γ(1)=x1}.{\Gamma }_{1}=\left\{\gamma \in C\left(\left[0,1],x):\gamma \left(0)={x}_{0},\gamma \left(1)={x}_{1}\right\}.3Periodic p(t)p\left(t)-Laplacian inclusion systemIn this section, we establish the existence of homoclinic solutions with periodic assumption for problem (1.1). In this situation, our hypotheses on p,ap,aand ffare the following: H(p)1p(t)p\left(t)is 2b2b-periodic;H(a)1a(t)a\left(t)is 2b2b-periodic;H(f)1(i)the function f(t,⋅):R→Rf\left(t,\cdot ):{\mathbb{R}}\to {\mathbb{R}}is 2b2b-periodic;(ii)for almost all t∈T=[−b,b]t\in T=\left[-b,b], there exists a function α(t)∈C(R)∩Lγ(t)γ(t)−α(t)(R)\alpha \left(t)\in C\left({\mathbb{R}})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}\left({\mathbb{R}})such that ∣ω∣≤a(t)(1+∣u∣α(t)−1),∀u∈RN,ω∈∂f(t,u(t)),| \omega | \le a\left(t)\left(1+| u{| }^{\alpha \left(t)-1}),\hspace{1em}\forall u\in {{\mathbb{R}}}^{N},\hspace{0.33em}\omega \in \partial f\left(t,u\left(t)),where a∈L∞(R)a\in {L}^{\infty }\left({\mathbb{R}}), α+<γ−<γ(t)<γ+<p−{\alpha }^{+}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(iii)there exist constants M,α,β>0M,\alpha ,\beta \gt 0such that 0≤p++1α+β∣u∣νf(t,u)≤−f0(t,u;−u)∀t∈T,∣u∣≥M,0\le \left({p}^{+}+\frac{1}{\alpha +\beta | u{| }^{\nu }}\right)f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\forall t\in T,\hspace{1em}\hspace{1em}| u| \ge M,where ν<p−\nu \lt {p}^{-};(iii′)there exist constants μ>p+,M>0\mu \gt {p}^{+},M\gt 0such that μf(t,u)≤−f0(t,u;−u)∀t∈T,∣u∣≥M;\mu f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\forall t\in T,\hspace{0.33em}| u| \ge M;(iv)there exists a function q(t)>0q\left(t)\gt 0such that lim∣u∣→0(w,u)∣u∣p(t)≤0,lim∣u∣→+∞inff(t,u)∣u∣q(t)>0,∀t∈T,w∈∂f(t,u),\mathop{\mathrm{lim}}\limits_{| u| \to 0}\frac{\left(w,u)}{| u{| }^{p\left(t)}}\le 0,\hspace{1em}\mathop{\mathrm{lim}}\limits_{| u| \to +\infty }\inf \frac{f\left(t,u)}{| u{| }^{q\left(t)}}\gt 0,\hspace{1em}\forall t\in T,\hspace{0.33em}w\in \partial f\left(t,u),where p+<q−{p}^{+}\lt {q}^{-}.Our main results can be stated as follows.Theorem 3.1If hypotheses H(p), H(p)1{\text{H(p)}}_{1}, H(a), H(a)1{\text{H(a)}}_{1}, H(f), and H(f)1{\text{H(f)}}_{1}: (i), (ii), (iii), (iv) hold, then problem (1.1) has a nontrivial homoclinic solution.Theorem 3.2If hypotheses H(p), H(p)1{\text{H(p)}}_{1}, H(a), H(a)1{\text{H(a)}}_{1}, H(f), and H(f)1{\text{H(f)}}_{1}: (i), (ii), (iii′^{\prime} ), (iv) hold, then problem (1.1) has a nontrivial homoclinic solution.Proof of Theorem 3.1We consider the following auxiliary periodic problem: (3.1)−ddt(∣u˙(t)∣p(t)−2u˙(t))+a(t)∣u(t)∣p(t)−2u(t)∈∂f(t,u(t)),a.e.t∈Tn,u(−nb)=u(nb),u˙(−nb)=u˙(nb).\left\{\begin{array}{l}-\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))+a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial f\left(t,u\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e.}\hspace{0.1em}\hspace{1em}t\in {T}_{n},\hspace{1.0em}\\ u\left(-nb)=u\left(nb),\dot{u}\left(-nb)=\dot{u}\left(nb).\hspace{1.0em}\end{array}\right.From [5], we know that problem (3.1) has a nontrivial solution un∈C2nb1(Tn,RN){u}_{n}\in {C}_{2nb}^{1}({T}_{n},{{\mathbb{R}}}^{N}). Let φn:W2nb1,p(t)(Tn,RN)→R{\varphi }_{n}:{W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})\to {\mathbb{R}}be defined by (3.2)φn(u)=∫−nbnb1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−∫−nbnbf(t,u(t))dt=φ˜n(u)−∫−nbnbf(t,u(t))dt.{\varphi }_{n}\left(u)=\underset{-nb}{\overset{nb}{\int }}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}f\left(t,u\left(t)){\rm{d}}t={\widetilde{\varphi }}_{n}\left(u)-\underset{-nb}{\overset{nb}{\int }}f\left(t,u\left(t)){\rm{d}}t.We claim that φn{\varphi }_{n}be the locally Lipschitz functional. In fact, for all u1,u2∈W2nb1,p(t)(Tn,RN){u}_{1},{u}_{2}\in {W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}), one has (3.3)∣φ˜n(u1)−φ˜n(u2)∣=∣φ˜n′(u˜)⋅(u1−u2)∣,| {\widetilde{\varphi }}_{n}({u}_{1})-{\widetilde{\varphi }}_{n}({u}_{2})| =| {\widetilde{\varphi }}_{n}^{^{\prime} }\left(\tilde{u})\cdot ({u}_{1}-{u}_{2})| ,where u˜=su1+(1−s)u2,s∈(0,1)\tilde{u}=s{u}_{1}+\left(1-s){u}_{2},s\in \left(0,1). Let Ω⊂Tn\Omega \subset {T}_{n}, fix u0∈W2nb1,p(t)(Ω,RN){u}_{0}\in {W}_{2nb}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N})and Br={u∈W2nb1,p(t)(Tn,RN):∥u−u0∥1≤r}.{B}_{r}=\{u\in {W}_{2nb}^{1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N}):{\parallel u-{u}_{0}\parallel }_{1}\le r\}.Note that Br{B}_{r}is compact, which yields that there exists C1>0{C}_{1}\gt 0such that (3.4)∥φ˜n′(u˜)∥W2nb−1,p(t)(Tn,RN)≤C1,{\parallel {\widetilde{\varphi }}_{n}^{^{\prime} }\left(\tilde{u})\parallel }_{{W}_{2nb}^{-1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})}\le {C}_{1},as r→0r\to 0. Then, it follows from (3.3) and (3.4), we obtain (3.5)φ˜n(u1)−φ˜n(u2)∣=∣φ˜n(u˜)⋅(u1−u2)∣≤∥φ˜n′(u˜)∥W2nb−1,p(t)(Tn,RN)∥u1−u2∥1≤C1∥u1−u2∥1,{\widetilde{\varphi }}_{n}({u}_{1})-{\widetilde{\varphi }}_{n}({u}_{2})| =| {\widetilde{\varphi }}_{n}\left(\tilde{u})\cdot ({u}_{1}-{u}_{2})| \le {\parallel {\widetilde{\varphi }}_{n}^{^{\prime} }\left(\tilde{u})\parallel }_{{W}_{2nb}^{-1,p\left(t)}({T}_{n},{{\mathbb{R}}}^{N})}{\parallel {u}_{1}-{u}_{2}\parallel }_{1}\le {C}_{1}{\parallel {u}_{1}-{u}_{2}\parallel }_{1},for all u1,u2∈W2nb1,p(t)(Ω,RN){u}_{1},{u}_{2}\in {W}_{2nb}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}).On the other hand, it follows from H(f)1{}_{1}: (ii) and Lemma 2.16, for all u1,u2∈W2nb1,p(t)(Ω,RN){u}_{1},{u}_{2}\in {W}_{2nb}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}), we have (3.6)∣f(t,u1)−f(t,u2)∣≤a(t)(1+∣u˜∣α(t)−1)∣u1−u2∣| f(t,{u}_{1})-f(t,{u}_{2})| \le a\left(t)(1+| \tilde{u}{| }^{\alpha \left(t)-1})| {u}_{1}-{u}_{2}| and a(t)∣u˜∣α(t)−1≤(γ(t)−α(t))∣a(t)∣γ(t)−1γ(t)−α(t)γ(t)−1+α(t)−1γ(t)−1∣u˜∣γ(t)−1,a\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}\le \frac{\left(\gamma \left(t)-\alpha \left(t))| a\left(t){| }^{\tfrac{\gamma \left(t)-1}{\gamma \left(t)-\alpha \left(t)}}}{\gamma \left(t)-1}+\frac{\alpha \left(t)-1}{\gamma \left(t)-1}| \tilde{u}{| }^{\gamma \left(t)-1},which imply that there exist some constants C2{C}_{2}, C3>0{C}_{3}\gt 0such that (3.7)(a(t)∣u˜∣α(t)−1)γ(t)γ(t)−1≤C2∣a(t)∣γ(t)γ(t)−α(t)+C3∣u˜∣γ(t).{(a\left(t)| \tilde{u}{| }^{\alpha \left(t)-1})}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}\le {C}_{2}| a\left(t){| }^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}+{C}_{3}| \tilde{u}{| }^{\gamma \left(t)}.Then, in virtue of (3.6), (3.7) and Hölder inequality, one has (3.8)∫−nbnbf(t,u1)dt−∫−nbnbf(t,u2)dt≤∫−nbnba(x)(1+∣u˜∣α(t)−1)∣u1−u2∣dt≤∣a(t)∣γ(t)γ(t)−1+∣a(t)∣u˜∣α(t)−1∣γ(t)γ(t)−1∣u1−u2∣γ(t)≤C4∣∣u1−u2∣∣1.\begin{array}{rcl}\left|\underset{-nb}{\overset{nb}{\displaystyle \int }}f(t,{u}_{1}){\rm{d}}t-\underset{-nb}{\overset{nb}{\displaystyle \int }}f(t,{u}_{2}){\rm{d}}t\right|& \le & \underset{-nb}{\overset{nb}{\displaystyle \int }}a\left(x)(1+| \tilde{u}{| }^{\alpha \left(t)-1})| {u}_{1}-{u}_{2}| {\rm{d}}t\\ & \le & \left[{| a\left(t)| }_{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}+{| a\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| }_{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}\right]{| {u}_{1}-{u}_{2}| }_{\gamma \left(t)}\\ & \le & {C}_{4}| | {u}_{1}-{u}_{2}| {| }_{1}.\end{array}Hence, from (3.2), (3.5) and (3.8), we obtain φn(u)=φ˜n(u)−∫−nbnbf(t,u(t))dt≤C1∣∣u1−u2∣∣1+C4∣∣u1−u2∣∣1≤C5∣∣u1−u2∣∣1,{\varphi }_{n}\left(u)={\widetilde{\varphi }}_{n}\left(u)-\underset{-nb}{\overset{nb}{\int }}f\left(t,u\left(t)){\rm{d}}t\le {C}_{1}| | {u}_{1}-{u}_{2}| {| }_{1}+{C}_{4}| | {u}_{1}-{u}_{2}| {| }_{1}\le {C}_{5}| | {u}_{1}-{u}_{2}| {| }_{1},which yields that φn{\varphi }_{n}be the nonsmooth locally Lipschitz energy functional corresponding to problem (3.1). Therefore, it follows from (3.2), H(f): (ii) and H(f)1{}_{1}: (iii), (iv), for σ≥1\sigma \ge 1, there exist C6,C7>0{C}_{6},{C}_{7}\gt 0such that φ1(σu)=∫−bb1p(t)(∣σu˙∣p(t)+a(t)∣σu∣p(t))dt−∫−bbf(t,σu)dt≤σp+∫−bb1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−σq−∫−bb∣u∣q(t)dt−C6σp+∫−bb∣u∣p+dt+2bC7.\begin{array}{rcl}{\varphi }_{1}\left(\sigma u)& =& \underset{-b}{\overset{b}{\displaystyle \int }}\frac{1}{p\left(t)}(| \sigma \dot{u}{| }^{p\left(t)}+a\left(t)| \sigma u{| }^{p\left(t)}){\rm{d}}t-\underset{-b}{\overset{b}{\displaystyle \int }}f\left(t,\sigma u){\rm{d}}t\\ & \le & {\sigma }^{{p}^{+}}\underset{-b}{\overset{b}{\displaystyle \int }}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-{\sigma }^{{q}^{-}}\underset{-b}{\overset{b}{\displaystyle \int }}| u{| }^{q\left(t)}{\rm{d}}t-{C}_{6}{\sigma }^{{p}^{+}}\underset{-b}{\overset{b}{\displaystyle \int }}| u{| }^{{p}^{+}}{\rm{d}}t+2b{C}_{7}.\end{array}Since p+<q−{p}^{+}\lt {q}^{-}, there exists a constant σ0>0{\sigma }_{0}\gt 0such that σ>σ0\sigma \gt {\sigma }_{0}and u¯∈W2b1,p(t)(T1,RN)\bar{u}\in {W}_{2b}^{1,p\left(t)}({T}_{1},{{\mathbb{R}}}^{N}), we have φ1(σu¯)<0{\varphi }_{1}(\sigma \bar{u})\lt 0.Let uˆ∈W2b1,p(t)(T1,RN)\hat{u}\in {W}_{2b}^{1,p\left(t)}({T}_{1},{{\mathbb{R}}}^{N})be defined as follows: uˆ(t)=u¯(t),ift∈T1;0,ift∈Tn⧹T1.\hat{u}\left(t)=\left\{\begin{array}{ll}\bar{u}\left(t),\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\in {T}_{1};\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\in {T}_{n}\setminus {T}_{1}.\end{array}\right.Note that f(t,0)=0f\left(t,0)=0, then for all σ≥σ0\sigma \ge {\sigma }_{0}, we deduce that φn(σuˆ)=φ1(σu¯){\varphi }_{n}\left(\sigma \hat{u})={\varphi }_{1}\left(\sigma \bar{u}).As in [5], we see that the solution un∈C2nb1(Tn,RN){u}_{n}\in {C}_{2nb}^{1}\left({T}_{n},{{\mathbb{R}}}^{N})of problem (3.1) is obtained via the nonsmooth mountain pass theorem. One will immediately obtain the fact that there exists ρ>0\rho \gt 0such that cn≔infγ∈Γnsupt∈[0,1]φn(γ(t))=φn(un)≥inf[φn(u):‖u‖=ρ]>0,{c}_{n}:= {\inf }_{\gamma \in {\Gamma }_{n}}\mathop{\sup }\limits_{t\in \left[0,1]}{\varphi }_{n}\left(\gamma \left(t))={\varphi }_{n}\left({u}_{n})\ge \inf {[}{\varphi }_{n}\left(u):\Vert u\Vert =\rho ]\gt 0,where Γn={γ∈C([0,1],W2nb1,p(t)):γ(0)=0,γ(1)=σuˆ}{\Gamma }_{n}=\{\gamma \in C(\left[0,1],{W}_{2nb}^{1,p\left(t)}):\gamma \left(0)=0,\gamma \left(1)=\sigma \hat{u}\}for σ≥σ0\sigma \ge {\sigma }_{0}and 0∈∂φn(un)0\in \partial {\varphi }_{n}\left({u}_{n})for all n≥1n\ge 1. Extending by constant, as n1≤n2{n}_{1}\le {n}_{2}we see that Wn11,p(t)⊆Wn21,p(t)andΓn1⊆Γn2{W}_{{n}_{1}}^{1,p\left(t)}\subseteq {W}_{{n}_{2}}^{1,p\left(t)}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}{\Gamma }_{{n}_{1}}\subseteq {\Gamma }_{{n}_{2}}and consequently cn2≤cn1,∀n1<n2.{c}_{{n}_{2}}\le {c}_{{n}_{1}},\hspace{1em}\forall {n}_{1}\lt {n}_{2}.This way we have produced a decreasing sequence {cn}n≥1{\left\{{c}_{n}\right\}}_{n\ge 1}of critical values. For every n≥1n\ge 1, from (3.2), we have (3.9)cn=φn(un)=∫−nbnb1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫−nbnbf(t,un(t))dt≤c1,{c}_{n}={\varphi }_{n}\left({u}_{n})=\underset{-nb}{\overset{nb}{\int }}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}f\left(t,{u}_{n}\left(t)){\rm{d}}t\le {c}_{1},which implies that (3.10)∫−nbnbp+p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫−nbnbp+f(t,un(t))dt≤p+c1.\underset{-nb}{\overset{nb}{\int }}\frac{{p}^{+}}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}{p}^{+}f\left(t,{u}_{n}\left(t)){\rm{d}}t\le {p}^{+}{c}_{1}.Then, it follows from (3.10), we obtain (3.11)∫−nbnb(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫−nbnbp+f(t,un(t))dt≤p+c1.\underset{-nb}{\overset{nb}{\int }}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\underset{-nb}{\overset{nb}{\int }}{p}^{+}f\left(t,{u}_{n}\left(t)){\rm{d}}t\le {p}^{+}{c}_{1}.Using (2.1), one has A(un)=wn,wn∈L2nb∞,wn(t)∈∂f(t,un(t)),a.e.t∈Tn,\begin{array}{l}A\left({u}_{n})={w}_{n},\hspace{1em}{w}_{n}\in {L}_{2nb}^{\infty },\\ {w}_{n}\left(t)\in \partial f\left(t,{u}_{n}\left(t)),\hspace{1em}\hspace{0.1em}\text{a.e.}\hspace{0.1em}\hspace{0.33em}t\in {T}_{n},\end{array}which yields that (3.12)−∫−nbnb(∣u˙n∣p(t)+a(t)∣un∣p(t))dt=∫−nbnb⟨wn(t),−un(t)⟩dt≤∫−nbnbf0(t,un(t);−un(t))dt.-\underset{-nb}{\overset{nb}{\int }}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t=\underset{-nb}{\overset{nb}{\int }}\langle {w}_{n}\left(t),-{u}_{n}\left(t)\rangle {\rm{d}}t\le \underset{-nb}{\overset{nb}{\int }}{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t)){\rm{d}}t.Combining (3.11) with (3.12), we obtain (3.13)∫−nbnb(−p+f(t,un)−f0(t,un;−un))dt≤p+c1.\underset{-nb}{\overset{nb}{\int }}(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t\le {p}^{+}{c}_{1}.By virtue of H(f)1{}_{1}: (iii), one has f(t,un)≤(α+β∣un∣ν)(−p+f(t,un)−f0(t,un;−un)),∀∣un∣>M.f\left(t,{u}_{n})\le (\alpha +\beta | {u}_{n}{| }^{\nu })(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})),\hspace{1em}\forall | {u}_{n}| \gt M.Hence, from H(f): (ii), (3.13) and Proposition 2.8 (i), there exists a constant ξ0{\xi }_{0}which is independent of nnsuch that ∫−nbnbf(t,un)dt=∫Tn∩{∣un∣>M}f(t,un)dt+∫Tn∩{∣un∣≤M}f(t,un)dt≤∫Tn∩{∣un∣>M}(α+β∣un∣ν)(−p+f(t,un)−f0(t,un;−un))dt+ξ0≤(α+β‖un‖∞ν)∫Tn∩{∣un∣>M}(−p+f(t,un)−f0(t,un;−un))dt+ξ0≤(α+β‖un‖∞ν)∫−nbnb(−p+f(t,un)−f0(t,un;−un))dt+ξ0≤p+c1(α+βκν‖un‖ν)+ξ0,\begin{array}{rcl}\underset{-nb}{\overset{nb}{\displaystyle \int }}f\left(t,{u}_{n}){\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \gt M\right\}}f\left(t,{u}_{n}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \le M\right\}}f\left(t,{u}_{n}){\rm{d}}t\\ & \le & \mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \gt M\right\}}(\alpha +\beta | {u}_{n}{| }^{\nu })(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t+{\xi }_{0}\\ & \le & (\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\mathop{\displaystyle \int }\limits_{{T}_{n}\cap \left\{| {u}_{n}| \gt M\right\}}(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t+{\xi }_{0}\\ & \le & (\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\underset{-nb}{\overset{nb}{\displaystyle \int }}(-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t+{\xi }_{0}\\ & \le & {p}^{+}{c}_{1}(\alpha +\beta {\kappa }^{\nu }\Vert {u}_{n}{\Vert }^{\nu })+{\xi }_{0},\end{array}which, together with (3.2), (3.9) and Proposition 2.4 (ii), imply that 1p+‖un‖p−≤1p+∫−nbnb(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤∫−nbnb1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt=φn(un)+∫−nbnbf(t,un)dt≤c1+p+c1(α+βκν‖un‖ν)+ξ0,\begin{array}{rcl}\frac{1}{{p}^{+}}\Vert {u}_{n}{\Vert }^{{p}^{-}}& \le & \frac{1}{{p}^{+}}\underset{-nb}{\overset{nb}{\displaystyle \int }}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & \le & \underset{-nb}{\overset{nb}{\displaystyle \int }}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & =& {\varphi }_{n}\left({u}_{n})+\underset{-nb}{\overset{nb}{\displaystyle \int }}f\left(t,{u}_{n}){\rm{d}}t\\ & \le & {c}_{1}+{p}^{+}{c}_{1}(\alpha +\beta {\kappa }^{\nu }\Vert {u}_{n}{\Vert }^{\nu })+{\xi }_{0},\end{array}for ‖un‖≥1\Vert {u}_{n}\Vert \ge 1. Since ν<p−\nu \lt {p}^{-}, there exists a constant ξ1{\xi }_{1}which is independent of nnsuch that (3.14)∫−nbnb1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤c1+p+c1(α+βκνξ1ν)+ξ0≔ξ2,\underset{-nb}{\overset{nb}{\int }}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\le {c}_{1}+{p}^{+}{c}_{1}(\alpha +\beta {\kappa }^{\nu }{\xi }_{1}^{\nu })+{\xi }_{0}:= {\xi }_{2},where ξ2{\xi }_{2}is a constant which is independent of nn. Thus, by (3.14), we obtain (3.15)‖un‖W2nb1,p(t)≤ξ3,\Vert {u}_{n}{\Vert }_{{W}_{2nb}^{1,p\left(t)}}\le {\xi }_{3},where ξ3>0{\xi }_{3}\gt 0is independent of nn. Moreover, by an argument as in the proof of [46, (2.19)], there exists a constant ξ4{\xi }_{4}which is independent of nnsuch that (3.16)‖un‖L2nb∞≤ξ4.\Vert {u}_{n}{\Vert }_{{L}_{2nb}^{\infty }}\le {\xi }_{4}.In what follows, we extend by periodicity un{u}_{n}and wn{w}_{n}to all of R{\mathbb{R}}. From (3.15) and the fact that the embedding W2nb1,p(t)↪C(Tn,RN){W}_{2nb}^{1,p\left(t)}\hspace{0.33em}\hookrightarrow \hspace{0.33em}C\left({T}_{n},{{\mathbb{R}}}^{N})is compact, we may assume that (3.17)un→uinCloc(R,RN),{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N}),hence u∈C(R,RN)u\in C\left({\mathbb{R}},{{\mathbb{R}}}^{N}). In view of hypothesis H(f)1{}_{1}: (ii), one has ‖wn(t)‖≤‖a‖∞(1+‖un(t)‖α(t)−1)=ξ5,\Vert {w}_{n}\left(t)\Vert \le \Vert a{\Vert }_{\infty }(1+\Vert {u}_{n}\left(t){\Vert }^{\alpha \left(t)-1})={\xi }_{5},where ξ5>0{\xi }_{5}\gt 0is a constant which is independent of nn. Passing to a subsequence if needed, we may assume that wn⇀w{w}_{n}\rightharpoonup win La∞(R,RN){L}_{a}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})and wn⇀w{w}_{n}\rightharpoonup win L2nbq(t)(Tn,RN){L}_{2nb}^{q\left(t)}({T}_{n},{{\mathbb{R}}}^{N}), where 1/p(t)+1/q(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}q\left(t)=1. It is obvious that w∈La∞(R,RN)∩Llocq(t)(R,RN)w\in {L}_{a}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N})\cap {L}_{{\rm{loc}}}^{q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), thus w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t))in Tn{T}_{n}for all n≥1n\ge 1and w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t))on R{\mathbb{R}}.For any τ>0\tau \gt 0, we have ∫−ττ∣un(t)−u(t)∣p(t)dt→0,asn→∞,\underset{-\tau }{\overset{\tau }{\int }}| {u}_{n}\left(t)-u\left(t){| }^{p\left(t)}{\rm{d}}t\to 0,\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to \infty ,which shows that (3.18)limn→∞∫−ττ∣un(t)∣p(t)dt=∫−ττ∣u(t)∣p(t)dt.\mathop{\mathrm{lim}}\limits_{n\to \infty }\underset{-\tau }{\overset{\tau }{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t=\underset{-\tau }{\overset{\tau }{\int }}| u\left(t){| }^{p\left(t)}{\rm{d}}t.Choose n0≥1{n}_{0}\ge 1such that [−τ,τ]⊆Tn0=[−n0b,n0b]\left[-\tau ,\tau ]\subseteq {T}_{{n}_{0}}=\left[-{n}_{0}b,{n}_{0}b]and for n≥n0n\ge {n}_{0}, from (3.15), we derive (3.19)∫−ττ∣un(t)∣p(t)dt≤∫−n0bn0b∣un(t)∣p(t)dt≤max{ξ3p+,ξ3p−}=ξ30.\underset{-\tau }{\overset{\tau }{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t\le \underset{-{n}_{0}b}{\overset{{n}_{0}b}{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t\le \max \{{\xi }_{3}^{{p}^{+}},{\xi }_{3}^{{p}^{-}}\}={\xi }_{3}^{0}.Thus, from (3.18) and (3.19), one has (3.20)∫−ττ∣u(t)∣p(t)dt≤ξ30.\underset{-\tau }{\overset{\tau }{\int }}| u\left(t){| }^{p\left(t)}{\rm{d}}t\le {\xi }_{3}^{0}.By the arbitrariness of τ>0\tau \gt 0, from (3.20), we deduce that u∈Lp(t)u\in {L}^{p\left(t)}.Let θ∈C0∞(R,RN)\theta \in {C}_{0}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N}), then suppθ⊆Tn\hspace{0.1em}\text{supp}\hspace{0.1em}\theta \subseteq {T}_{n}for large n≥1n\ge 1, which together with (3.15), performs integration by parts and Proposition 2.6, we have ∫R(un(t),θ˙(t))dt=∫R(θ(t),u˙n(t))dt=∫−nbnb(u˙n(t),θ(t))dt≤‖u˙n‖L2nbp(t)‖θ‖L2nbq(t)≤ξ3‖θ‖L2nbq(t).\left|\mathop{\int }\limits_{{\mathbb{R}}}({u}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t\right|=\left|\mathop{\int }\limits_{{\mathbb{R}}}(\theta \left(t),{\dot{u}}_{n}\left(t)){\rm{d}}t\right|=\left|\underset{-nb}{\overset{nb}{\int }}({\dot{u}}_{n}\left(t),\theta \left(t)){\rm{d}}t\right|\le \Vert {\dot{u}}_{n}{\Vert }_{{L}_{2nb}^{p\left(t)}}\Vert \theta {\Vert }_{{L}_{2nb}^{q\left(t)}}\le {\xi }_{3}\Vert \theta {\Vert }_{{L}_{2nb}^{q\left(t)}}.Note that (un(t),θ˙(t))→(u(t),θ˙(t))({u}_{n}\left(t),\dot{\theta }\left(t))\to (u\left(t),\dot{\theta }\left(t))in Cloc(R,RN){C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N}). By (3.16), we have ∣(un(t),θ˙(t))∣≤‖un‖L2nb∞∣θ˙(t)∣≤ξ4∣θ˙(t)∣a.e. onTn=[−nb,nb].| \left({u}_{n}\left(t),\dot{\theta }\left(t))| \le \Vert {u}_{n}{\Vert }_{{L}_{2nb}^{\infty }}| \dot{\theta }\left(t)| \le {\xi }_{4}| \dot{\theta }\left(t)| \hspace{1em}\hspace{0.1em}\text{a.e. on}\hspace{0.1em}\hspace{0.33em}{T}_{n}=\left[-nb,nb].Let η(t)=ξ4∣θ˙(t)∣,ift∈suppθ;0,ift∉suppθ.\eta \left(t)=\left\{\begin{array}{ll}{\xi }_{4}| \dot{\theta }\left(t)| ,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\in \hspace{0.1em}\text{supp}\hspace{0.1em}\theta ;\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}t\notin \hspace{0.1em}\text{supp}\hspace{0.1em}\theta .\end{array}\right.Then, η∈L1(R)\eta \in {L}^{1}\left({\mathbb{R}})and ∣(un(t),θ˙(t))∣≤η(t)| ({u}_{n}\left(t),\dot{\theta }\left(t))| \le \eta \left(t)a.e. on R{\mathbb{R}}for all nn. Therefore, by generalized Lebesgue-dominated convergence theorem, we see that ∫R(un(t),θ˙(t))dt→∫R(u(t),θ˙(t))dt\mathop{\int }\limits_{{\mathbb{R}}}({u}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t\to \mathop{\int }\limits_{{\mathbb{R}}}(u\left(t),\dot{\theta }\left(t)){\rm{d}}tand ∫R(u(t),θ˙(t))dt≤ξ3‖θ‖L2nbq(t).\left|\mathop{\int }\limits_{{\mathbb{R}}}(u\left(t),\dot{\theta }\left(t)){\rm{d}}t\right|\le {\xi }_{3}\Vert \theta {\Vert }_{{L}_{2nb}^{q\left(t)}}.It follows from [5] that u∈Wa1,p(t)(R,RN)u\in {W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}).Recall that wn⇀w∈Llocq(t)(R,RN){w}_{n}\rightharpoonup w\in {L}_{{\rm{loc}}}^{q\left(t)}\left({\mathbb{R}},{{\mathbb{R}}}^{N}), which leads to (3.21)∫R(wn(t),θ(t))dt→∫R(w(t),θ(t))dt.\mathop{\int }\limits_{{\mathbb{R}}}({w}_{n}\left(t),\theta \left(t)){\rm{d}}t\to \mathop{\int }\limits_{{\mathbb{R}}}(w\left(t),\theta \left(t)){\rm{d}}t.Furthermore, by (3.17), one has (3.22)∫R(a(t)∣un(t)∣p(t)−2un(t),θ(t))dt→∫R(a(t)∣u(t)∣p(t)−2u(t),θ(t))dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(a\left(t)| {u}_{n}\left(t){| }^{p\left(t)-2}{u}_{n}\left(t),\theta \left(t)){\rm{d}}t\to \mathop{\int }\limits_{{\mathbb{R}}}\left(a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t),\theta \left(t)){\rm{d}}t.Since un{u}_{n}is a solution of problem (3.1), then we obtain ddt(∣u˙n(t)∣p(t)−2u˙n(t))∈L2nbp(t)\frac{{\rm{d}}}{{\rm{d}}t}(| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t))\in {L}_{2nb}^{p\left(t)}. So it follows that ∣u˙n(t)∣p(t)−2u˙n(t)∈W2nb1,p(t)| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t)\in {W}_{2nb}^{1,p\left(t)}for all n≥1n\ge 1. By (3.22) and Proposition 2.1, let ∣u˙n∣p(t)−2u˙n⇀vinWloc1,p(t)(R,RN),| {\dot{u}}_{n}{| }^{p\left(t)-2}{\dot{u}}_{n}\hspace{0.33em}\rightharpoonup \hspace{0.33em}v\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{W}_{{\rm{loc}}}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}),then we have (3.23)∣u˙n∣p(t)−2u˙n→vinCloc(R,RN).| {\dot{u}}_{n}{| }^{p\left(t)-2}{\dot{u}}_{n}\to v\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{C}_{{\rm{loc}}}\left({\mathbb{R}},{{\mathbb{R}}}^{N}).Define the continuous functional λ:R→R\lambda :{\mathbb{R}}\to {\mathbb{R}}by λ(u)=∣u∣p(t)−2u,∀u∈Cloc(R,RN).\lambda \left(u)=| u{| }^{p\left(t)-2}u,\hspace{1em}\forall u\in {C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N}).Then, from (3.23) that u˙n=λ−1(∣u˙n∣p(t)−2u˙n)→λ−1(v)inLloc1(R,RN),{\dot{u}}_{n}={\lambda }^{-1}(| {\dot{u}}_{n}{| }^{p\left(t)-2}{\dot{u}}_{n})\to {\lambda }^{-1}(v)\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{L}_{{\rm{loc}}}^{1}\left({\mathbb{R}},{{\mathbb{R}}}^{N}),which shows that λ−1(v)=u˙{\lambda }^{-1}\left(v)=\dot{u}, so we have v=∣u˙∣p(t)−2u˙v=| \dot{u}{| }^{p\left(t)-2}\dot{u}. Also, integration by parts yields that (3.24)∫R(∣u˙(t)∣p(t)−2u˙(t),θ˙(t))dt=−∫Rddt(∣u˙(t)∣p(t)−2u˙(t)),θ(t)dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t),\dot{\theta }\left(t)){\rm{d}}t=-\mathop{\int }\limits_{{\mathbb{R}}}\left(\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t)),\theta \left(t)\right){\rm{d}}t.Hence, it follows from (3.24), we obtain (3.25)∫R(∣u˙n(t)∣p(t)−2u˙n(t),θ˙(t))dt→−∫Rddt(∣u˙(t)∣p(t)−2u˙(t)),θ(t)dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t\to -\mathop{\int }\limits_{{\mathbb{R}}}\left(\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t)),\theta \left(t)\right){\rm{d}}t.For any n≥1n\ge 1, we have (3.26)∫R(∣u˙n(t)∣p(t)−2u˙n(t),θ˙(t))dt+∫R(a(t)∣un(t)∣p(t)−2un(t),θ(t))dt=∫R(wn(t),θ(t))dt.\mathop{\int }\limits_{{\mathbb{R}}}\left(| {\dot{u}}_{n}\left(t){| }^{p\left(t)-2}{\dot{u}}_{n}\left(t),\dot{\theta }\left(t)){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}\left(a\left(t)| {u}_{n}\left(t){| }^{p\left(t)-2}{u}_{n}\left(t),\theta \left(t)){\rm{d}}t=\mathop{\int }\limits_{{\mathbb{R}}}\left({w}_{n}\left(t),\theta \left(t)){\rm{d}}t.Letting n→∞n\to \infty , then from (3.21), (3.22), (3.25), and (3.26), we obtain −∫Rddt(∣u˙(t)∣p(t)−2u˙(t))θ(t)dt+∫Ra(t)∣u(t)∣p(t)−2u(t)θ(t)dt=∫R(w(t),θ(t))dt.-\mathop{\int }\limits_{{\mathbb{R}}}\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))\theta \left(t){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\theta \left(t){\rm{d}}t=\mathop{\int }\limits_{{\mathbb{R}}}\left(w\left(t),\theta \left(t)){\rm{d}}t.From the arbitrary of θ∈C0∞(R,RN)\theta \in {C}_{0}^{\infty }({\mathbb{R}},{{\mathbb{R}}}^{N}), we can deduce that −ddt(∣u˙(t)∣p(t)−2u˙(t))+a(t)∣u(t)∣p(t)−2u(t)=w(t),a.e.t∈R-\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))+a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)=w\left(t),\hspace{1em}\hspace{0.1em}\text{a.e.}\hspace{0.1em}\hspace{0.33em}t\in {\mathbb{R}}and w∈Llocq(t)(R,RN)w\in {L}_{{\rm{loc}}}^{q\left(t)}\left({\mathbb{R}},{{\mathbb{R}}}^{N}), w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t)).Next, we show that u(±∞)=u˙(±∞)=0u\left(\pm \infty )=\dot{u}\left(\pm \infty )=0.Recall that Proposition 2.8 (i), since u∈Wa1,p(t)(R,RN)u\in {W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), we have u(t)→0u\left(t)\to 0as ∣t∣→∞| t| \to \infty . Because w(t)∈∂f(t,u(t))w\left(t)\in \partial f\left(t,u\left(t)), by virtue of H(f)1{}_{1}: (ii), one has (3.27)‖w(t)‖≤a1(t)(1+‖u(t)‖α(t)−1).\Vert w\left(t)\Vert \le {a}_{1}\left(t)(1+\Vert u\left(t){\Vert }^{\alpha \left(t)-1}).Note that u∈Wa1,p(t)(R,RN)u\in {W}_{a}^{1,p\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), then ∣u∣p(t)−2u∈Lq(t)(R,RN)| u{| }^{p\left(t)-2}u\in {L}^{q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}), so w∈Lq(t)(R,RN)w\in {L}^{q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}). Therefore, ∣u˙∣p(t)−2u˙∈Wa1,q(t)(R,RN)| \dot{u}{| }^{p\left(t)-2}\dot{u}\in {W}_{a}^{1,q\left(t)}({\mathbb{R}},{{\mathbb{R}}}^{N}). Again from Proposition 2.8 (i) that ∣u˙(t)∣p(t)−1→0| \dot{u}\left(t){| }^{p\left(t)-1}\to 0as ∣t∣→∞| t| \to \infty . Thus, we obtain u˙(t)→0\dot{u}\left(t)\to 0, i.e., u˙(±t)=0\dot{u}\left(\pm t)=0as ∣t∣→∞| t| \to \infty .Finally, we show that u(t)≠0u\left(t)\ne 0. For all n≥1n\ge 1, from (3), we have (3.28)a0∫−nbnb∣un∣p(t)dt≤∫−nbnb(wn(t),un(t))dt.{a}_{0}\underset{-nb}{\overset{nb}{\int }}| {u}_{n}{| }^{p\left(t)}{\rm{d}}t\le \underset{-nb}{\overset{nb}{\int }}\left({w}_{n}\left(t),{u}_{n}\left(t)){\rm{d}}t.Let (3.29)hn(t)=(wn(t),un(t))∣un(t)∣p(t),ifun(t)≠0;0,ifun(t)=0.{h}_{n}\left(t)=\left\{\begin{array}{ll}\frac{\left({w}_{n}\left(t),{u}_{n}\left(t))}{| {u}_{n}\left(t){| }^{p\left(t)}},\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{u}_{n}\left(t)\ne 0;\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{if}\hspace{0.1em}\hspace{0.33em}{u}_{n}\left(t)=0.\end{array}\right.Then, it follows from (3.28) and (3.29) that a0∫−nbnb∣un∣p(t)dt≤∫−nbnbhn(t)∣un(t)∣p(t)dt≤esssupTnhn∫−nbnb∣un(t)∣p(t)dt.{a}_{0}\underset{-nb}{\overset{nb}{\int }}| {u}_{n}{| }^{p\left(t)}{\rm{d}}t\le \underset{-nb}{\overset{nb}{\int }}{h}_{n}\left(t)| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t\le \hspace{0.1em}\text{ess}\hspace{0.1em}{\sup }_{{T}_{n}}{h}_{n}\underset{-nb}{\overset{nb}{\int }}| {u}_{n}\left(t){| }^{p\left(t)}{\rm{d}}t.By virtue of hypothesis H(f)1{}_{1}: (iv), for a given ε>0\varepsilon \gt 0, we can find δ>0\delta \gt 0such that (3.30)(wn(t),un(t))∣un(t)∣p(t)≤ε,∣un(t)∣p(t)≤δ,∀t∈Tn,ω∈∂f(t,un).\frac{\left({w}_{n}\left(t),{u}_{n}\left(t))}{| {u}_{n}\left(t){| }^{p\left(t)}}\le \varepsilon ,\hspace{1em}| {u}_{n}\left(t){| }^{p\left(t)}\le \delta ,\hspace{1em}\forall t\in {T}_{n},\hspace{0.33em}\omega \in \partial f\left(t,{u}_{n}).If u=0u=0, then it follows from (3.17) that (3.31)un(t)⟶0inCloc(R,RN){u}_{n}\left(t)\hspace{0.33em}\longrightarrow \hspace{0.33em}0\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{1em}{C}_{{\rm{loc}}}({\mathbb{R}},{{\mathbb{R}}}^{N})and so we can find n0≥1{n}_{0}\ge 1such that ∣un(t)∣p(t)≤δ,∀t∈Tn,n≥n0.| {u}_{n}\left(t){| }^{p\left(t)}\le \delta ,\hspace{1.0em}\forall t\in {T}_{n},\hspace{0.33em}n\ge {n}_{0}.Thus for all n≥n0n\ge {n}_{0}and almost all t∈Tnt\in {T}_{n}, we have hn(t)≤ε{h}_{n}\left(t)\le \varepsilon and so a0≤esssupTnhn=esssupRhn≤ε,∀n≥n0.{a}_{0}\le \mathop{{\rm{esssup}}}\limits_{{T}_{n}}\hspace{1em}{h}_{n}=\mathop{{\rm{esssup}}}\limits_{{\mathbb{R}}}{h}_{n}\le \varepsilon ,\hspace{1.0em}\forall n\ge {n}_{0}.In the above derivation process, we use the fact that un{u}_{n}and ωn{\omega }_{n}are extended by periodicity to all of R{\mathbb{R}}. Let ε↘0\varepsilon \searrow 0reach a contradiction since 0<a00\lt {a}_{0}. This proves that u≠0u\ne 0.□Proof of Theorem 3.2Similar to the proof of Theorem 3.1, we can prove Theorem 3.2, so we omit its course.□Example 3.1Let p(t)=52+12sintp\left(t)=\frac{5}{2}+\frac{1}{2}\sin tfor t∈Rt\in {\mathbb{R}}, and f(t,u)=a(t)∣u∣3ln(1+∣u∣),∀(t,u)∈[−π,π]×RN,f\left(t,u)=a\left(t)| u{| }^{3}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in \left[-\pi ,\pi ]\times {{\mathbb{R}}}^{N},where a(t)a\left(t)satisfies H(a), H(a)1{}_{1}with b=πb=\pi . It is evident that ffis locally Lipschitz and ∂f(t,u)=a(t)3∣u∣uln(1+∣u∣)+∣u∣2u1+∣u∣.\partial f\left(t,u)=a\left(t)\left[3| u| u\mathrm{ln}(1+| u| )+\frac{| u{| }^{2}u}{1+| u| }\right].Then −f0(t,u;−u)=3+∣u∣(1+∣u∣)ln(1+∣u∣)f(t,u)≥3+11+∣u∣f(t,u)≥p++11+∣u∣f(t,u),\begin{array}{rcl}-{f}^{0}\left(t,u;-u)& =& \left[3+\frac{| u| }{(1+| u| )\mathrm{ln}(1+| u| )}\right]f\left(t,u)\\ & \ge & \left(3+\frac{1}{1+| u| }\right)f\left(t,u)\\ & \ge & \left({p}^{+}+\frac{1}{1+| u| }\right)f\left(t,u),\end{array}which implies that ffsatisfies H(f)1{\text{H(f)}}_{1}:(iii) with α=β=ν=1\alpha =\beta =\nu =1. Hence, from Theorem 3.1, problem (1.1) has a nontrivial homoclinic solution.Example 3.2Let p(t)=52+12sintp\left(t)=\frac{5}{2}+\frac{1}{2}\sin tfor t∈Rt\in {\mathbb{R}}, and f(t,u)=a(t)∣u∣7/2ln(1+∣u∣),∀(t,u)∈[−π,π]×RN.f\left(t,u)=a\left(t)| u{| }^{7\text{/}2}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in \left[-\pi ,\pi ]\times {{\mathbb{R}}}^{N}.Similar to Example 3.1, problem (1.1) has a nontrivial homoclinic solution by Theorem 3.2.4Nonperiodic p(t)p\left(t)-Laplacian inclusion systemIn this section, we investigate the question of existence of homoclinic (to zero) solutions without periodic assumptions. Namely, we examine the following two types of problems: (4.1)ddt(∣u˙(t)∣p(t)−2u˙(t))−a(t)∣u(t)∣p(t)−2u(t)∈∂f(t,u(t)),\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial f\left(t,u\left(t)),and another problem (4.2)ddt(∣u˙(t)∣p(t)−2u˙(t))−a(t)∣u(t)∣p(t)−2u(t)∈∂f1(t,u(t))−∂f2(t,u(t)),\frac{{\rm{d}}}{{\rm{d}}t}(| \dot{u}\left(t){| }^{p\left(t)-2}\dot{u}\left(t))-a\left(t)| u\left(t){| }^{p\left(t)-2}u\left(t)\in \partial {f}_{1}\left(t,u\left(t))-\partial {f}_{2}\left(t,u\left(t)),where aasatisfy the following hypothesis: H(a)2lim∣t∣→+∞a(t)=+∞{\mathrm{lim}}_{| t| \to +\infty }a\left(t)=+\infty .Note that a∈C(R,R+)a\in C({\mathbb{R}},{{\mathbb{R}}}^{+})is coercive, then H′(a)2̲\underline{\hspace{0.1em}\text{H}^{\prime} {\text{(a)}}_{2}}is satisfied, namely, H′(a)2there exists r>0r\gt 0such that lim∣y∣→∞meas({t∈Br(y):a(t)≤b})=0for anyb>0.\mathop{\mathrm{lim}}\limits_{| y| \to \infty }\hspace{0.1em}\text{meas}\hspace{0.1em}(\{t\in {B}_{r}(y):a\left(t)\le b\})=0\hspace{1em}\hspace{0.1em}\text{for any}\hspace{0.1em}\hspace{0.33em}b\gt 0.For the nonlinearity ff, we suppose the following hypotheses: H(f)2(ii)for almost all t∈Rt\in {\mathbb{R}}, there exist a function α(t)∈C(R)∩Lγ(t)γ(t)−α(t)(R)\alpha \left(t)\in C\left({\mathbb{R}})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}\left({\mathbb{R}})such that ∣ω∣≤a(t)(1+∣u∣α(t)−1),∀u∈RN,ω∈∂f(t,u(t)),| \omega | \le a\left(t)\left(1+| u{| }^{\alpha \left(t)-1}),\hspace{1em}\forall u\in {{\mathbb{R}}}^{N},\hspace{1em}\omega \in \partial f\left(t,u\left(t)),where a∈L∞(R)a\in {L}^{\infty }\left({\mathbb{R}}), α+<γ−<γ(t)<γ+<p−{\alpha }^{+}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(ii′)there exist two functions ai(t)<a(t)(i=1,2){a}_{i}\left(t)\lt a\left(t)\left(i=1,2)such that ∣ω∣≤a1(t)α1(t)∣u∣α1(t)−1,∀t∈R,∣u∣≤1,| \omega | \le {a}_{1}\left(t){\alpha }_{1}\left(t)| u{| }^{{\alpha }_{1}\left(t)-1},\hspace{1em}\forall t\in {\mathbb{R}},\hspace{0.33em}| u| \le 1,and ∣ω∣≤a2(t)α2(t)∣u∣α2(t)−1,∀t∈R,∣u∣≥1,| \omega | \le {a}_{2}\left(t){\alpha }_{2}\left(t)| u{| }^{{\alpha }_{2}\left(t)-1},\hspace{1em}\forall t\in {\mathbb{R}},\hspace{0.33em}| u| \ge 1,where ω∈∂f(t,u(t))\omega \hspace{-0.08em}\in \hspace{-0.08em}\partial f\left(t,u\left(t)), ai(t)∈C(R,R+){a}_{i}\left(t)\hspace{-0.08em}\in \hspace{-0.08em}C\left({\mathbb{R}},{{\mathbb{R}}}^{+})and αi(t)∈C(R,R+)∩Lγ(t)γ(t)−αi(t)(R){\alpha }_{i}\left(t)\hspace{-0.08em}\in \hspace{-0.08em}C\left({\mathbb{R}},{{\mathbb{R}}}^{+})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-{\alpha }_{i}\left(t)}}\left({\mathbb{R}}), αi+<γ−<γ(t)<γ+<p−{\alpha }_{i}^{+}\hspace{-0.08em}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(iii)there exist constants M,α,β>0M,\alpha ,\beta \gt 0such that 0≤p++1α+β∣u∣νf(t,u)≤−f0(t,u;−u)∀(t,u)∈R×RN,0\le \left({p}^{+}+\frac{1}{\alpha +\beta | u{| }^{\nu }}\right)f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where ν<p−\nu \lt {p}^{-};(iii′)there exist constants μ>p+,M>0\mu \gt {p}^{+},M\gt 0such that μf(t,u)≤−f0(t,u;−u)∀(t,u)∈R×RN;\mu f\left(t,u)\le -{f}^{0}\left(t,u;-u)\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N};(iv)there exists a function q(t)>0q\left(t)\gt 0such that lim∣u∣→0w∣u∣p+−1=0,lim∣u∣→+∞inff(t,u)∣u∣q(t)>0,∀t∈R,w∈∂f(t,u),\mathop{\mathrm{lim}}\limits_{| u| \to 0}\frac{w}{| u{| }^{{p}^{+}-1}}=0,\hspace{1em}\mathop{\mathrm{lim}}\limits_{| u| \to +\infty }\inf \frac{f\left(t,u)}{| u{| }^{q\left(t)}}\gt 0,\hspace{1em}\forall t\in {\mathbb{R}},w\in \partial f\left(t,u),where p+<q−{p}^{+}\lt {q}^{-}.Theorem 4.1If hypotheses H(p), H(a), H(f) and H(f)2{\text{H(f)}}_{2}: (ii), (iii), (iv) hold, then problem (4.1) has at least a nontrivial homoclinic solution.Theorem 4.2If hypotheses H(p), H(a), H(f)\hspace{0.1em}\text{H(p), H(a), H(f)}\hspace{0.1em}and H(f)2{\text{H(f)}}_{2}: (ii),(iii′),(iv)\left(ii),\hspace{0.33em}\left(iii^{\prime} ),\hspace{0.33em}\left(iv)hold, then problem (4.1) has at least a nontrivial homoclinic solution.Remark 4.3Note that H(f)2̲\underline{{\text{H(f)}}_{2}}: (ii) is weaker than [27, f1{}_{1}], by virtue of hypothesis H′(a)2̲,\underline{\hspace{0.1em}\text{H}^{\prime} {\text{(a)}}_{2}},we cannot immediately obtain [27, Theorem 1.2] for problem (4.1) with symmetrical condition f(t,−u)=f(t,u)f\left(t,-u)=f\left(t,u)for all (t,u)∈R×RN\left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Theorem 4.4If hypotheses H(p), H(a), H(a)2{\text{H(a)}}_{2}, H(f), H(f)2{\text{H(f)}}_{2}: (ii′)\left(ii^{\prime} )and the following condition hold:(v)there exist an open subset Ω⊂R\Omega \subset {\mathbb{R}}and function γ(t)\gamma \left(t)such thatf(t,u)≥η∣u∣γ(t),∀(t,u)∈Ω×RN,∣u∣≤1,f\left(t,u)\ge \eta | u{| }^{\gamma \left(t)},\hspace{1em}\forall \left(t,u)\in \Omega \times {{\mathbb{R}}}^{N},\hspace{1em}| u| \le 1,where γ(t)\gamma \left(t)satisfies H(p)\hspace{0.1em}\text{H(p)}\hspace{0.1em}and γ+<p−{\gamma }^{+}\lt {p}^{-}, η>0\eta \gt 0is a constant.Then problem (4.1) has at least a nontrivial homoclinic solution.With regard to problem (4.2), in this situation, assume that pp, aaand f1{f}_{1}satisfy all the conditions in Theorem 4.2 and f2{f}_{2}satisfies the following conditions: H(f)3(i)the function f2(t,⋅):R→R{f}_{2}\left(t,\cdot ):{\mathbb{R}}\to {\mathbb{R}}is measurable for all u∈RNu\in {{\mathbb{R}}}^{N}and f2(t,0)=0{f}_{2}\left(t,0)=0;(ii)the function f2(⋅,u):RN→R{f}_{2}\left(\cdot ,u):{{\mathbb{R}}}^{N}\to {\mathbb{R}}is locally Lipschitz for a.e. t∈Rt\in {\mathbb{R}};(iii)for almost all t∈Rt\in {\mathbb{R}}, there exists a function α(t)∈C(R)∩Lγ(t)γ(t)−α(t)(R)\alpha \left(t)\in C\left({\mathbb{R}})\cap {L}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-\alpha \left(t)}}\left({\mathbb{R}})such that ∣ω∣≤a(t)(1+∣u∣α(t)−1),∀u∈RN,ω∈∂f2(t,u(t)),| \omega | \le a\left(t)\left(1+| u{| }^{\alpha \left(t)-1}),\hspace{1em}\forall u\in {{\mathbb{R}}}^{N},\hspace{1em}\omega \in \partial {f}_{2}\left(t,u\left(t)),where a∈L∞(R)a\in {L}^{\infty }\left({\mathbb{R}}), α+<γ−<γ(t)<γ+<p−{\alpha }^{+}\lt {\gamma }^{-}\lt \gamma \left(t)\lt {\gamma }^{+}\lt {p}^{-};(iv)there exists a constant ϱ∈[p+,μ)\varrho \in {[}{p}^{+},\mu )such that f20(t,u;u)≤ϱf2(t,u),∀(t,u)∈R×RN.{f}_{2}^{0}\left(t,u;\hspace{0.33em}u)\le \varrho {f}_{2}\left(t,u),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Theorem 4.5If hypotheses H(p), H(a)\hspace{0.1em}\text{H(p), H(a)}\hspace{0.1em}, H(f)3{\text{H(f)}}_{3}hold, then problem (4.2) has at least a nontrivial homoclinic solution.Proof of Theorem 4.1In order to prove Theorems 4.1, 4.2 and 4.4, we first define a functional φ:Wa1,p(t)→R\varphi :{W}_{a}^{1,p\left(t)}\to {\mathbb{R}}as follows: (4.3)φ(u)=∫R1p(t)(∣u˙(t)∣p(t)+a(t)∣u(t)∣p(t))dt−∫Rf(t,u(t))dt.\varphi \left(u)=\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}\left(t){| }^{p\left(t)}+a\left(t)| u\left(t){| }^{p\left(t)}){\rm{d}}t-\mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t.Using the same type of reasoning as the proof of Theorem 3.1, it is easy to verify that φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1).Let {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}be such that (4.4)∣φ(un)∣≤M1andm(un)→0asn→+∞,| \varphi \left({u}_{n})| \le {M}_{1}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({u}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to +\infty ,where M1>0{M}_{1}\gt 0is a constant. Our proofs are divided into two steps.Step 1: un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}.Since ∂φ(un)⊆W−1,p(t)\partial \varphi \left({u}_{n})\subseteq {W}^{-1,p\left(t)}is weakly compact and norm is weakly lower semicontinuous, according to Weierstrass theorem (Lemma 2.18). We can find un∗∈∂φ(un){u}_{n}^{\ast }\in \partial \varphi \left({u}_{n})such that m(un)=‖un∗‖m\left({u}_{n})=\Vert {u}_{n}^{\ast }\Vert for n≥1n\ge 1.Define nonlinear operator ℒ:Wa1,p(t)→(Wa1,p(t))∗{\mathcal{ {\mathcal L} }}:{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast }as follows: ⟨ℒ(u),v⟩=∫R∣u˙(t)∣p(t)−2(u˙(t),v˙(t))dt,∀u,v∈Wa1,p(t).\langle {\mathcal{ {\mathcal L} }}\left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}| \dot{u}\left(t){| }^{p\left(t)-2}(\dot{u}\left(t),\dot{v}\left(t)){\rm{d}}t,\hspace{1em}\forall u,v\in {W}_{a}^{1,p\left(t)}.According to the literature [21], ℒ{\mathcal{ {\mathcal L} }}is monotonic and semicontinuous, so it is maximal monotone (see also [24]), therefore, un∗=ℒ(un)−wn{u}_{n}^{\ast }={\mathcal{ {\mathcal L} }}\left({u}_{n})-{w}_{n}for n≥1n\ge 1, and wn∈∂f(t,un){w}_{n}\in \partial f\left(t,{u}_{n}), wn∈Lp′(t){w}_{n}\in {L}^{p^{\prime} \left(t)}, where 1/p′(t)+1/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} \left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1.In another way, by the selection of sequence{un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}, we obtain (4.5)∣⟨un∗,un⟩∣≤εn,εn↓0,| \langle {u}_{n}^{\ast },{u}_{n}\rangle | \le {\varepsilon }_{n},\hspace{1em}{\varepsilon }_{n}\downarrow 0,which yields that (4.6)−∫R(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rωnundt≤εn.-\mathop{\int }\limits_{{\mathbb{R}}}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{\omega }_{n}{u}_{n}{\rm{d}}t\le {\varepsilon }_{n}.Note that ⟨wn,−un⟩≤f0(t,un;−un),\langle {w}_{n},-{u}_{n}\rangle \le {f}^{0}\left(t,{u}_{n};-{u}_{n}),using this fact and by (4.3), (4.4) and (4.6), one has (4.7)p+M1≥p+φ(un)−⟨un∗,un⟩≥∫Rp+p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−p+∫Rf(t,un)dt−∫R(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rwnundt≥∫R[−p+f(t,un)−f0(t,un;−un)]dt.\begin{array}{rcl}{p}^{+}{M}_{1}& \ge & {p}^{+}\varphi \left({u}_{n})-\langle {u}_{n}^{\ast },{u}_{n}\rangle \\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{{p}^{+}}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-{p}^{+}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}){\rm{d}}t\\ & & -\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{w}_{n}{u}_{n}{\rm{d}}t\\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})]{\rm{d}}t.\end{array}For any (t,un)∈R×(RN⧹{0})\left(t,{u}_{n})\in {\mathbb{R}}\times ({{\mathbb{R}}}^{N}\setminus \left\{0\right\}), by H(f)2{}_{2}: (iii), we have (4.8)f(t,un(t))≤(α+β∣un(t)∣ν)[−p+f(t,un)−f0(t,un;−un)].f\left(t,{u}_{n}\left(t))\le (\alpha +\beta | {u}_{n}\left(t){| }^{\nu }){[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})].Hence, without loss of generality, we assume that ‖un‖≥1\Vert {u}_{n}\Vert \ge 1. Otherwise, ‖un‖\Vert {u}_{n}\Vert is bounded. It follows from (4.3), (4.4), (4.7), (4.8), Propositions 2.4 (ii) and 2.8 (i) that (4.9)1p+‖un‖p−≤∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt=φ(un)+∫Rf(t,un(t))dt≤φ(un)+∫R(α+β∣un(t)∣ν)[−p+f(t,un)−f0(t,un;−un)]dt≤M1+(α+β‖un‖∞ν)∫R[−p+f(t,un)−f0(t,un;−un)]dt≤M1+p+M1(α+β‖un‖∞ν)≤M1+p+M1(α+βκν‖un‖ν).\begin{array}{rcl}\frac{1}{{p}^{+}}\Vert {u}_{n}{\Vert }^{{p}^{-}}& \le & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & =& \varphi \left({u}_{n})+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}\left(t)){\rm{d}}t\\ & \le & \varphi \left({u}_{n})+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\alpha +\beta | {u}_{n}\left(t){| }^{\nu }){[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})]{\rm{d}}t\\ & \le & {M}_{1}+\left(\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}-{p}^{+}f\left(t,{u}_{n})-{f}^{0}\left(t,{u}_{n};-{u}_{n})]{\rm{d}}t\\ & \le & {M}_{1}+{p}^{+}{M}_{1}(\alpha +\beta \Vert {u}_{n}{\Vert }_{\infty }^{\nu })\\ & \le & {M}_{1}+{p}^{+}{M}_{1}(\alpha +\beta {\kappa }^{\nu }\Vert {u}_{n}{\Vert }^{\nu }).\end{array}Note that ν<p−\nu \lt {p}^{-}, in light of (4.9), it is easy to show that {‖un‖}\left\{\Vert {u}_{n}\Vert \right\}is bounded. So passing to a subsequence if necessary, it can be assumed that un⇀u{u}_{n}\hspace{0.33em}\rightharpoonup \hspace{0.33em}uin Wa1,p(t){W}_{a}^{1,p\left(t)}, un⇀u{u}_{n}\hspace{0.33em}\rightharpoonup \hspace{0.33em}uin Lp(t){L}^{p\left(t)}. Because (4.5), then ⟨ℒ(un),un−u⟩−∫Rwn(un−u)dt≤εn,∀n≥1.\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle -\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}\left({u}_{n}-u){\rm{d}}t\le {\varepsilon }_{n},\hspace{1em}\forall n\ge 1.By virtue of the fact that {wn}n≥1{\left\{{w}_{n}\right\}}_{n\ge 1}is bounded in Lp′(t){L}^{p^{\prime} \left(t)}, then limsupn→∞⟨ℒ(un),un−u⟩≤0{\mathrm{limsup}}_{n\to \infty }\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle \le 0. By Proposition 2.9, we deduce that un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}.Step 2: φ\varphi satisfies nonsmooth mountain pass theorem.Let ε>0\varepsilon \gt 0be small enough, in view of hypotheses H(f)2{}_{2}: (ii), (iv), one has f(t,u)≤ε∣u∣p++c(ε)∣u∣α(t),∀(t,u)∈R×RN.f\left(t,u)\le \varepsilon | u{| }^{{p}^{+}}+c\left(\varepsilon )| u{| }^{\alpha \left(t)},\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Let ‖u‖=ρ\Vert u\Vert =\rho be small enough, then by Proposition 2.4, we have (4.10)φ(u)≥1p+∫R(∣u˙∣p(t)+a(t)∣u∣p(t))dt−ε∫R∣u∣p+dt−c(ε)∫R∣u∣α(t)dt≥1p+‖u‖p+−εcp+p+‖u‖p+−c(ε)cα−α−‖u‖α−\begin{array}{rcl}\varphi \left(u)& \ge & \frac{1}{{p}^{+}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\varepsilon \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{{p}^{+}}{\rm{d}}t-c\left(\varepsilon )\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{\alpha \left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}\end{array}for the above ε\varepsilon , let εcp+p+<12p+\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\lt \frac{1}{2{p}^{+}}, where cp+(cα−){c}_{{p}^{+}}({c}_{{\alpha }^{-}})is the embedding constant from Wa1,p(t){W}_{a}^{1,p\left(t)}to Lp+(Lα−){L}^{{p}^{+}}({L}^{{\alpha }^{-}}). Then, from (4.10), we obtain φ(u)≥12p+‖u‖p+−c(ε)cα−α−‖u‖α−.\varphi \left(u)\ge \frac{1}{2{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}.Note that p+<α−{p}^{+}\lt {\alpha }^{-}, then there exists a constant r>0r\gt 0such that φ(u)≥r\varphi \left(u)\ge rwhen ‖u‖=ρ\Vert u\Vert =\rho for ρ\rho small enough.By H(f)2{}_{2}: (ii), (iv), we see that (4.11)f(t,u)≥∣u∣q(t),∀t∈R,∣u∣≥Mf\left(t,u)\ge | u{| }^{q\left(t)},\hspace{1em}\forall t\in {\mathbb{R}},\hspace{1em}| u| \ge Mand (4.12)∣f(t,u)∣≤c0a(t),∀t∈R,∣u∣<M,| f\left(t,u)| \le {c}_{0}a\left(t),\hspace{1em}\forall t\in {\mathbb{R}},\hspace{1em}| u| \lt M,respectively. Thus, for any u∈W1,p(t)⧹{0}u\in {W}^{1,p\left(t)}\setminus \left\{0\right\}and σ>1\sigma \gt 1, from (4.3), (4.11) and (4.12), we obtain φ(σu)=∫R1p(t)(∣σu˙∣p(t)+a(t)∣σu∣p(t))dt−∫Rf(t,σu)dt≤σp+∫R1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−σq−∫R∣u∣q(t)dt+c0∫Ra(t)dt.\begin{array}{rcl}\varphi \left(\sigma u)& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \sigma \dot{u}{| }^{p\left(t)}+a\left(t)| \sigma u{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,\sigma u){\rm{d}}t\\ & \le & {\sigma }^{{p}^{+}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-{\sigma }^{{q}^{-}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{q\left(t)}{\rm{d}}t+{c}_{0}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}a\left(t){\rm{d}}t.\end{array}Since p+<q−{p}^{+}\lt {q}^{-}, it is easy to show that φ(σu)→−∞\varphi \left(\sigma u)\to -\infty as σ→+∞\sigma \to +\infty . Because φ(0)=0\varphi \left(0)=0and φ\varphi satisfies Lemma 2.19, hence there exists at least one nontrivial critical point, that is, system (4.1) has at least one homoclinic orbit.□Proof of Theorem 4.2Applying the proof of Theorem 4.1, we deduce that φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1).Let {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}be such that (4.13)∣φ(un)∣≤M1andm(un)→0asn→+∞,| \varphi \left({u}_{n})| \le {M}_{1}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({u}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to +\infty ,where M1>0{M}_{1}\gt 0as given in (4.4). As ∂φ(un)⊆W−1,p(t)\partial \varphi \left({u}_{n})\subseteq {W}^{-1,p\left(t)}is weakly compact and the norm is weakly lower semicontinuous, by Lemma 2.18, we can choose un∗∈∂φ(un){u}_{n}^{\ast }\in \partial \varphi \left({u}_{n})such that m(un)=‖un∗‖m\left({u}_{n})=\Vert {u}_{n}^{\ast }\Vert for n≥1n\ge 1.Define nonlinear operator ℒ:Wa1,p(t)→(Wa1,p(t))∗{\mathcal{ {\mathcal L} }}:{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast }as follows: ⟨ℒ(u),v⟩=∫R∣u˙(t)∣p(t)−2(u˙(t),v˙(t))dt,∀u,v∈Wa1,p(t).\langle {\mathcal{ {\mathcal L} }}\left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}| \dot{u}\left(t){| }^{p\left(t)-2}(\dot{u}\left(t),\dot{v}\left(t)){\rm{d}}t,\hspace{1em}\forall u,v\in {W}_{a}^{1,p\left(t)}.According to the literature [21], ℒ{\mathcal{ {\mathcal L} }}is monotonic and semicontinuous, so it is maximal monotone (see also [24]), therefore un∗=ℒ(un)−wn{u}_{n}^{\ast }={\mathcal{ {\mathcal L} }}\left({u}_{n})-{w}_{n}for n≥1n\ge 1, and wn∈∂f(t,un){w}_{n}\in \partial f\left(t,{u}_{n}), wn∈Lp′(t){w}_{n}\in {L}^{p^{\prime} \left(t)}, where 1/p′(t)+1/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} \left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1.In another way, by the selection of sequence {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}, we obtain (4.14)∣⟨un∗,un⟩∣≤εn,εn↓0,| \langle {u}_{n}^{\ast },{u}_{n}\rangle | \le {\varepsilon }_{n},\hspace{1em}{\varepsilon }_{n}\downarrow 0,which yields (4.15)−p+∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rωnundt≤εn.-{p}^{+}\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{\omega }_{n}{u}_{n}{\rm{d}}t\le {\varepsilon }_{n}.Note that ⟨wn,−un⟩≤f0(t,un;−un),\langle {w}_{n},-{u}_{n}\rangle \le {f}^{0}\left(t,{u}_{n};-{u}_{n}),using this fact and by (4.3), (4.13) and (4.15), one has μM1+εn≥μφ(un)+⟨un∗,−un⟩=μ∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫Rf(t,un)dt+⟨un∗,−un⟩=μ∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫Rf(t,un)dt−p+∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+∫Rωnundt≥(μ−p+)∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫R(μf(t,un)+f0(t,un;−un))dt,\begin{array}{rcl}\mu {M}_{1}+{\varepsilon }_{n}& \ge & \mu \varphi \left({u}_{n})+\langle {u}_{n}^{\ast },-{u}_{n}\rangle \\ & =& \mu \left[\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}){\rm{d}}t\right]+\langle {u}_{n}^{\ast },-{u}_{n}\rangle \\ & =& \mu \left[\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,{u}_{n}){\rm{d}}t\right]-{p}^{+}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{\omega }_{n}{u}_{n}{\rm{d}}t\\ & \ge & (\mu -{p}^{+})\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\mu f\left(t,{u}_{n})+{f}^{0}\left(t,{u}_{n};-{u}_{n})){\rm{d}}t,\end{array}which leads to (4.16)(μ−p+)∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤μM1+εn+∫R(μf(t,un(t))+f0(t,un(t);−un(t)))dt.(\mu -{p}^{+})\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\le \mu {M}_{1}+{\varepsilon }_{n}+\mathop{\int }\limits_{{\mathbb{R}}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t.By H(f)2{}_{2}: (ii), there exist two functions a1(t),b1(t)∈L∞(R)+{a}_{1}\left(t),{b}_{1}\left(t)\in {L}^{\infty }{\left({\mathbb{R}})}_{+}such that ∣f(t,un(t))∣≤a1(t)+b1(t)∣un(t)∣α(t).| f\left(t,{u}_{n}\left(t))| \le {a}_{1}\left(t)+{b}_{1}\left(t)| {u}_{n}\left(t){| }^{\alpha \left(t)}.Recall that u↦f(t,u)u\mapsto f\left(t,u)is local Lipschitz, there exists c(t)∈L∞(R)+c\left(t)\in {L}^{\infty }{\left({\mathbb{R}})}_{+}such that f0(t,un(t);−un(t))≤c(t)∣un(t)∣,∀u∈RN.{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))\le c\left(t)| {u}_{n}\left(t)| ,\hspace{1em}\forall u\in {{\mathbb{R}}}^{N}.Thus, there exist a2>0,b2∈L∞(R)+{a}_{2}\gt 0,{b}_{2}\in {L}^{\infty }{\left({\mathbb{R}})}_{+}such that μf(t,un(t))+f0(t,un(t);−un(t))≤a2b2(t),∀t∈R,∣un∣<M,\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))\le {a}_{2}{b}_{2}\left(t),\hspace{1em}\forall t\in {\mathbb{R}},\hspace{0.33em}| {u}_{n}| \lt M,which implies that there exists a constant C3>0{C}_{3}\gt 0such that (4.17)∫{∣un∣<M}(μf(t,un(t))+f0(t,un(t);−un(t)))dt≤C3.\mathop{\int }\limits_{\left\{| {u}_{n}| \lt M\right\}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t\le {C}_{3}.Combining with (4.17) and H(f)2{\text{H(f)}}_{2}: (iii′)\left({\rm{iii}}^{\prime} ), we can deduce (4.18)∫R(μf(t,un(t))+f0(t,un(t);−un(t)))dt=∫{∣un∣<M}(μf(t,un(t))+f0(t,un(t);−un(t)))dt+∫{∣un∣≥M}(μf(t,un(t))+f0(t,un(t);−un(t)))dt≤C3.\begin{array}{l}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t\\ \hspace{1.0em}=\mathop{\displaystyle \int }\limits_{\left\{| {u}_{n}| \lt M\right\}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t+\mathop{\displaystyle \int }\limits_{\left\{| {u}_{n}| \ge M\right\}}(\mu f\left(t,{u}_{n}\left(t))+{f}^{0}\left(t,{u}_{n}\left(t);-{u}_{n}\left(t))){\rm{d}}t\le {C}_{3}.\end{array}Hence, from (4.16) and (4.18), we obtain (4.19)(μ−p+)∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt≤C4.(\mu -{p}^{+})\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\le {C}_{4}.Note that μ>p+\mu \gt {p}^{+}, it follows from (4.19) that {un}n≥1⊆W1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}^{1,p\left(t)}is bounded. So passing to a subsequence if necessary, it can be assumed that un⇀u{u}_{n}\rightharpoonup uin Wa1,p(t){W}_{a}^{1,p\left(t)}, un⇀u{u}_{n}\rightharpoonup uin Lp(t){L}^{p\left(t)}. Because (4.14), then ⟨ℒ(un),un−u⟩−∫Rwn(un−u)dt≤εn,∀n≥1.\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle -\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}\left({u}_{n}-u){\rm{d}}t\le {\varepsilon }_{n},\hspace{1em}\forall n\ge 1.By virtue of the fact that {wn}n≥1{\left\{{w}_{n}\right\}}_{n\ge 1}is bounded in Lp′(t){L}^{p^{\prime} \left(t)}, then limsupn→∞⟨ℒ(un),un−u⟩≤0.\mathop{\mathrm{limsup}}\limits_{n\to \infty }\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle \le 0.By Proposition 2.9, we obtain un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}.Next, we need only to verify that φ\varphi satisfies nonsmooth mountain pass theorem, i.e., Lemma 2.19, the proof is similar to Theorem 4.1, so we omitted its course.□Proof of Theorem 4.4Consider the functional φ:Wa1,p(t)→R\varphi :{W}_{a}^{1,p\left(t)}\to {\mathbb{R}}be defined as (4.3), i.e., (4.20)φ(u)=∫R1p(t)(∣u˙(t)∣p(t)+a(t)∣u(t)∣p(t))dt−∫Rf(t,u(t))dt≔φ˜(u)−∫Rf(t,u(t))dt.\varphi \left(u)=\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}\left(t){| }^{p\left(t)}+a\left(t)| u\left(t){| }^{p\left(t)}){\rm{d}}t-\mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t:= \widetilde{\varphi }\left(u)-\mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t.First, we prove that φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1). However, similar arguments as Theorem 4.1, it is easy to see that φ˜\widetilde{\varphi }is the locally Lipschitz functional. So we only need to show ∫Rf(t,u(t))dt{\int }_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}tis the locally Lipschitz functional.Let Ω⊂R\Omega \subset {\mathbb{R}}, from H(f)2{}_{2}: (ii)′^{\prime} and Lemma 2.16, for all u1,u2∈Wa1,p(t)(Ω,RN){u}_{1},{u}_{2}\in {W}_{a}^{1,p\left(t)}(\Omega ,{{\mathbb{R}}}^{N}), one has (4.21)∣f(t,u1)−f(t,u2)∣≤ai(t)αi(t)∣u˜∣α(t)−1∣u1−u2∣,i=1,2| f(t,{u}_{1})-f(t,{u}_{2})| \le {a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| {u}_{1}-{u}_{2}| ,\hspace{1em}i=1,2and ai(t)αi(t)∣u˜∣αi(t)−1≤(γ(t)−αi(t))∣ai(t)αi(t)∣γ(t)−1γ(t)−αi(t)γ(t)−1+αi(t)−1γ(t)−1∣u˜∣γ(t)−1,i=1,2,{a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{{\alpha }_{i}\left(t)-1}\le \frac{\left(\gamma \left(t)-{\alpha }_{i}\left(t))| {a}_{i}\left(t){\alpha }_{i}\left(t){| }^{\tfrac{\gamma \left(t)-1}{\gamma \left(t)-{\alpha }_{i}\left(t)}}}{\gamma \left(t)-1}+\frac{{\alpha }_{i}\left(t)-1}{\gamma \left(t)-1}| \tilde{u}{| }^{\gamma \left(t)-1},\hspace{1em}i=1,2,which yields that (4.22)(ai(t)αi(t)∣u˜∣αi(t)−1)γ(t)γ(t)−1≤C8∣a(t)∣γ(t)γ(t)−αi(t)+C9∣u˜∣γ(t),i=1,2,{({a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{{\alpha }_{i}\left(t)-1})}^{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}\le {C}_{8}| a\left(t){| }^{\tfrac{\gamma \left(t)}{\gamma \left(t)-{\alpha }_{i}\left(t)}}+{C}_{9}| \tilde{u}{| }^{\gamma \left(t)},\hspace{1em}i=1,2,where u˜=su1+(1−s)u2,s∈(0,1)\tilde{u}=s{u}_{1}+\left(1-s){u}_{2},s\in \left(0,1), C8,C9>0{C}_{8},{C}_{9}\gt 0. Then, from (4.21), (4.22) and Hölder inequality, we obtain ∫Rf(t,u1)dt−∫Rf(t,u2)dt≤∫Rai(t)αi(t)∣u˜∣α(t)−1∣u1−u2∣dt≤∣ai(t)αi(t)∣u˜∣α(t)−1∣γ(t)γ(t)−1∣u1−u2∣γ(t)≤C10∣∣u1−u2∣∣,i=1,2.\begin{array}{rcl}\left|\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f(t,{u}_{1}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f(t,{u}_{2}){\rm{d}}t\right|& \le & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| {u}_{1}-{u}_{2}| {\rm{d}}t\\ & \le & {| {a}_{i}\left(t){\alpha }_{i}\left(t)| \tilde{u}{| }^{\alpha \left(t)-1}| }_{\tfrac{\gamma \left(t)}{\gamma \left(t)-1}}{| {u}_{1}-{u}_{2}| }_{\gamma \left(t)}\\ & \le & {C}_{10}| | {u}_{1}-{u}_{2}| | ,\hspace{1em}i=1,2.\end{array}Hence, φ\varphi is the nonsmooth Lipschitz energy functional corresponding to problem (1.1).Next, our proofs are divided into three steps.Step 1: φ\varphi is coercive.It follows from H(f)2{\text{H(f)}}_{2}: (ii′)\left({\rm{ii}}^{\prime} )that (4.23)f(t,u)≤a1(t)∣u∣α1(t),∣u∣≤1;a2(t)∣u∣α2(t),∣u∣>1.f\left(t,u)\le \left\{\begin{array}{ll}{a}_{1}\left(t)| u{| }^{{\alpha }_{1}\left(t)},\hspace{1.0em}& | u| \le 1;\\ {a}_{2}\left(t)| u{| }^{{\alpha }_{2}\left(t)},\hspace{1.0em}& | u| \gt 1.\end{array}\right.Let ‖u‖≥1\Vert u\Vert \ge 1, it follows from (4.3), Propositions 2.2, 2.4, and 2.6 that (4.24)φ(u)=∫R1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−∫Rf(t,u)dt≥1p+‖u‖p−−∫{t:∣u∣≤1}f(t,u)dt−∫{t:∣u∣>1}f(t,u)dt≥1p+‖u‖p−−∫{t:∣u∣≤1}a1(t)∣u∣α1(t)dt−∫{t:∣u∣>1}a2(t)∣u∣α2(t)dt≥1p+‖u‖p−−C11∫{t:∣u∣≤1}bα1(t)p(t)aα1(t)p(t)∣u∣α1(t)dt−C12∫{t:∣u∣>1}bα2(t)p(t)aα2(t)p(t)∣u∣α2(t)dt≥1p+‖u‖p−−2C11∣bα1(t)p(t)∣Lr1(t)∣u∣p(t),aα˜1−2C12∣bα2(t)p(t)∣Lr2(t)∣u∣p(t),aα˜2≥1p+‖u‖p−−2C11∣bα1(t)p(t)∣Lr1(t)‖u‖α˜1−2C12∣bα2(t)p(t)∣Lr2(t)‖u‖α˜2,\begin{array}{rcl}\varphi \left(u)& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,u){\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \le 1\right\}}f\left(t,u){\rm{d}}t-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \gt 1\right\}}f\left(t,u){\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \le 1\right\}}{a}_{1}\left(t)| u{| }^{{\alpha }_{1}\left(t)}{\rm{d}}t-\mathop{\displaystyle \int }\limits_{\left\{t:| u| \gt 1\right\}}{a}_{2}\left(t)| u{| }^{{\alpha }_{2}\left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-{C}_{11}\mathop{\displaystyle \int }\limits_{\left\{t:| u| \le 1\right\}}{b}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}{a}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}| u{| }^{{\alpha }_{1}\left(t)}{\rm{d}}t-{C}_{12}\mathop{\displaystyle \int }\limits_{\left\{t:| u| \gt 1\right\}}{b}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}{a}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}| u{| }^{{\alpha }_{2}\left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-2{C}_{11}| {b}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{1}\left(t)}}| u{| }_{p\left(t),a}^{{\widetilde{\alpha }}_{1}}-2{C}_{12}| {b}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{2}\left(t)}}| u{| }_{p\left(t),a}^{{\widetilde{\alpha }}_{2}}\hspace{1em}\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{-}}-2{C}_{11}| {b}^{\tfrac{{\alpha }_{1}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{1}\left(t)}}\Vert u{\Vert }^{{\widetilde{\alpha }}_{1}}-2{C}_{12}| {b}^{\tfrac{{\alpha }_{2}\left(t)}{p\left(t)}}{| }_{{L}^{{r}_{2}\left(t)}}\Vert u{\Vert }^{{\widetilde{\alpha }}_{2}},\end{array}where Ci+10=supt∈Rai(t)(i=1,2.){C}_{i+10}={\sup }_{t\in {\mathbb{R}}}{a}_{i}\left(t)\left(i=1,2.), b(t)=a(t)−1b\left(t)=a{\left(t)}^{-1}, 1/ri(t)+αi(t)/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}{r}_{i}\left(t)+{\alpha }_{i}\left(t)\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1, and α˜i∈[αi−,αi+]{\widetilde{\alpha }}_{i}\in {[}{\alpha }_{i}^{-},{\alpha }_{i}^{+}]is a constant. From H(f)2{}_{2}: (ii′)\left({\rm{ii}}^{\prime} ), we know αi−<αi+<p−{\alpha }_{i}^{-}\lt {\alpha }_{i}^{+}\lt {p}^{-}, so αi˜<p−\widetilde{{\alpha }_{i}}\lt {p}^{-}. Hence, by H(a)2{}_{2}, we have φ(u)→+∞\varphi \left(u)\to +\infty as ‖u‖→+∞\Vert u\Vert \to +\infty . Thus, φ\varphi is bounded below.Step 2: φ\varphi is sequence weakly lower semicontinuous.Without loss of generality, we assume that un⇀u{u}_{n}\rightharpoonup uin Wa1,p(t){W}_{a}^{1,p\left(t)}, so from Proposition 2.8 (ii), we have un→u{u}_{n}\to uin L∞(R){L}^{\infty }\left({\mathbb{R}}), which implies that un→uandf(t,un(t))→f(t,u(t)),∀t∈R.{u}_{n}\to u\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}f(t,{u}_{n}\left(t))\to f\left(t,u\left(t)),\hspace{1em}\forall t\in {\mathbb{R}}.By Fatou lemma, we have (4.25)limn→∞sup∫Rf(t,un(t))dt≤∫Rf(t,u(t))dt.\mathop{\mathrm{lim}}\limits_{n\to \infty }\sup \mathop{\int }\limits_{{\mathbb{R}}}f(t,{u}_{n}\left(t)){\rm{d}}t\le \mathop{\int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t.Hence, from (4.3) and (4.25), we obtain (4.26)liminfn→∞φ(un)≥liminfn→∞∫R1p(t)(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−limsupn→∞∫Rf(t,un(t))dt≥∫R1p(t)(∣u˙∣p(t)+a(t)∣u∣p(t))dt−∫Rf(t,u(t))dt=φ(u),\begin{array}{rcl}\mathop{\mathrm{liminf}}\limits_{n\to \infty }\varphi ({u}_{n})& \ge & \mathop{\mathrm{liminf}}\limits_{n\to \infty }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}({| {\dot{u}}_{n}| }^{p\left(t)}+a\left(t){| {u}_{n}| }^{p\left(t)}){\rm{d}}t-\mathop{\mathrm{limsup}}\limits_{n\to \infty }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f(t,{u}_{n}\left(t)){\rm{d}}t\\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,u\left(t)){\rm{d}}t\\ & =& \varphi \left(u),\end{array}which shows that φ\varphi is sequence weakly lower semicontinuous.Using Lemma 2.18, there is a global minimum point u0∈Wa1,p(t){u}_{0}\in {W}_{a}^{1,p\left(t)}such that φ(u0)=minu∈Wa1,p(t)φ(u).\varphi \left({u}_{0})=\mathop{\min }\limits_{u\in {W}_{a}^{1,p\left(t)}}\varphi \left(u).Step 3: φ(u0)<0\varphi \left({u}_{0})\lt 0.Let u0∈(W01,p(t)⋂Wa1,p(t))⧹{0}{u}_{0}\in ({W}_{0}^{1,p\left(t)}\hspace{0.33em}\bigcap \hspace{0.33em}{W}_{a}^{1,p\left(t)})\setminus \left\{0\right\}with ‖u0‖=1\Vert {u}_{0}\Vert =1, by (4.3) and condition (v), for 0<s<10\lt s\lt 1, we obtain φ(su0)=∫R1p(t)(∣su˙0∣p(t)+a(t)∣su0∣p(t))dt−∫Rf(t,su0(t))dt≤sp−p−−∫Ωf(t,su0(t))dt≤sp−p−−ηsγ+∫Ω∣u0(t)∣γ(t)dt.\begin{array}{rcl}\varphi \left(s{u}_{0})& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| s{\dot{u}}_{0}{| }^{p\left(t)}+a\left(t)| s{u}_{0}{| }^{p\left(t)}){\rm{d}}t-\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}f\left(t,s{u}_{0}\left(t)){\rm{d}}t\\ & \le & \frac{{s}^{{p}^{-}}}{{p}^{-}}-\mathop{\displaystyle \int }\limits_{\Omega }f\left(t,s{u}_{0}\left(t)){\rm{d}}t\\ & \le & \frac{{s}^{{p}^{-}}}{{p}^{-}}-\eta {s}^{{\gamma }^{+}}\mathop{\displaystyle \int }\limits_{\Omega }| {u}_{0}\left(t){| }^{\gamma \left(t)}{\rm{d}}t.\end{array}Note that 1<γ+<p−1\lt {\gamma }^{+}\lt {p}^{-}, it is easy to show that φ(su0)<0\varphi \left(s{u}_{0})\lt 0as s>0s\gt 0small enough.□Proof of Theorem 4.5Define a functional ψ:Wa1,p(t)→R\psi :{W}_{a}^{1,p\left(t)}\to {\mathbb{R}}as follows: (4.27)ψ(u)=∫R1p(t)(∣u˙(t)∣p(t)+a(t)∣u(t)∣p(t))dt−∫Rf1(t,u(t))dt+∫Rf2(t,u(t))dt.\psi \left(u)=\mathop{\int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \dot{u}\left(t){| }^{p\left(t)}+a\left(t)| u\left(t){| }^{p\left(t)}){\rm{d}}t-\mathop{\int }\limits_{{\mathbb{R}}}{f}_{1}\left(t,u\left(t)){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{f}_{2}\left(t,u\left(t)){\rm{d}}t.Arguments as in proof of Theorems 4.1 and 4.5, we can easily obtain ψ\psi as the nonsmooth Lipschitz energy functional corresponding to problem (4.2).Let {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}be such that (4.28)∣ψ(un)∣≤M2andm(un)→0asn→+∞,| \psi \left({u}_{n})| \le {M}_{2}\hspace{1em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{1em}m\left({u}_{n})\to 0\hspace{1em}\hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}n\to +\infty ,where M2>0{M}_{2}\gt 0is a constant. Because ∂ψ(un)⊆W−1,p(t)\partial \psi \left({u}_{n})\subseteq {W}^{-1,p\left(t)}is weakly compact and the norm is weakly lower semicontinuous. By Lemma 2.18, we can choose un∗∈∂ψ(un){u}_{n}^{\ast }\in \partial \psi \left({u}_{n})such that m(un)=‖un∗‖m\left({u}_{n})=\Vert {u}_{n}^{\ast }\Vert for n≥1n\ge 1.Define nonlinear operator ℒ:Wa1,p(t)→(Wa1,p(t))∗{\mathcal{ {\mathcal L} }}:{W}_{a}^{1,p\left(t)}\to {({W}_{a}^{1,p\left(t)})}^{\ast }as follows: ⟨ℒ(u),v⟩=∫R∣u˙(t)∣p(t)−2(u˙(t),v˙(t))dt,∀u,v∈Wa1,p(t).\langle {\mathcal{ {\mathcal L} }}\left(u),v\rangle =\mathop{\int }\limits_{{\mathbb{R}}}| \dot{u}\left(t){| }^{p\left(t)-2}(\dot{u}\left(t),\dot{v}\left(t)){\rm{d}}t,\hspace{1em}\forall u,v\in {W}_{a}^{1,p\left(t)}.According to the literature [21], ℒ{\mathcal{ {\mathcal L} }}is monotonic and semicontinuous, so it is maximal monotone (see also [24]), therefore, un∗=ℒ(un)−wn1+wn2{u}_{n}^{\ast }={\mathcal{ {\mathcal L} }}\left({u}_{n})-{w}_{n}^{1}+{w}_{n}^{2}for n≥1n\ge 1, and wn1∈∂f1(t,un){w}_{n}^{1}\in \partial {f}_{1}\left(t,{u}_{n}), wn2∈∂f2(t,un){w}_{n}^{2}\in \partial {f}_{2}\left(t,{u}_{n}), wn1{w}_{n}^{1}, wn2∈Lp′(t){w}_{n}^{2}\in {L}^{p^{\prime} \left(t)}, where 1/p′(t)+1/p(t)=11\hspace{0.1em}\text{/}\hspace{0.1em}p^{\prime} \left(t)+1\hspace{0.1em}\text{/}\hspace{0.1em}p\left(t)=1.In another way, by the selection of sequence {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}, we obtain (4.29)∣⟨un∗,un⟩∣≤εn,εn↓0.| \langle {u}_{n}^{\ast },{u}_{n}\rangle | \le {\varepsilon }_{n},\hspace{1em}{\varepsilon }_{n}\downarrow 0.Then, it follows from (4.27), (4.28), (4.29), H(f)2{}_{2}: (iii′^{\prime} ) and H(f)3{}_{3}: (iv), we can show that (4.30)M2+εnμ≥ψ(un)−1μ⟨un∗,un⟩=∫R1p(t)−1μ(∣u˙n∣p(t)+a(t)∣un∣p(t))dt−∫Rf1(t,un(t))dt+∫Rf2(t,un(t))dt+1μ∫R(⟨wn1,un⟩−⟨wn2,un⟩)dt≥∫R1p(t)−1μ(∣u˙n∣p(t)+a(t)∣un∣p(t))dt+1μ∫R[−f10(t,un;−un)−μf1(t,un)]dt+1μ∫R[μf2(t,un)−f20(t,un;un)]dt≥1p+−1μ∫R(∣u˙n∣p(t)+a(t)∣un∣p(t))dt.\begin{array}{rcl}{M}_{2}+\frac{{\varepsilon }_{n}}{\mu }& \ge & \psi \left({u}_{n})-\frac{1}{\mu }\langle {u}_{n}^{\ast },{u}_{n}\rangle \\ & =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\left(\frac{1}{p\left(t)}-\frac{1}{\mu }\right)(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & & -\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{f}_{1}\left(t,{u}_{n}\left(t)){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{f}_{2}\left(t,{u}_{n}\left(t)){\rm{d}}t+\frac{1}{\mu }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(\langle {w}_{n}^{1},{u}_{n}\rangle -\langle {w}_{n}^{2},{u}_{n}\rangle ){\rm{d}}t\\ & \ge & \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\left(\frac{1}{p\left(t)}-\frac{1}{\mu }\right)(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t\\ & & +\frac{1}{\mu }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}-{f}_{1}^{0}\left(t,{u}_{n};-{u}_{n})-\mu {f}_{1}\left(t,{u}_{n})]{\rm{d}}t+\frac{1}{\mu }\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}\mu {f}_{2}\left(t,{u}_{n})-{f}_{2}^{0}\left(t,{u}_{n};{u}_{n})]{\rm{d}}t\\ & \ge & \left(\frac{1}{{p}^{+}}-\frac{1}{\mu }\right)\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| {\dot{u}}_{n}{| }^{p\left(t)}+a\left(t)| {u}_{n}{| }^{p\left(t)}){\rm{d}}t.\end{array}Note that μ>p+\mu \gt {p}^{+}, by virtue of (4.30) and Proposition 2.4, we have {un}n≥1⊆Wa1,p(t){\left\{{u}_{n}\right\}}_{n\ge 1}\subseteq {W}_{a}^{1,p\left(t)}is bounded, so we assume that un⇀u{u}_{n}\rightharpoonup uin Wa1,p(t){W}_{a}^{1,p\left(t)}and un⇀u{u}_{n}\rightharpoonup uin Lp(t){L}^{p\left(t)}.Thanks to (4.29), thus (4.31)⟨ℒ(un),un−u⟩−∫Rwn1(un−u)dt+∫Rwn2(un−u)dt≤εn,∀n≥1.\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle -\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{1}\left({u}_{n}-u){\rm{d}}t+\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{2}\left({u}_{n}-u){\rm{d}}t\le {\varepsilon }_{n},\hspace{1em}\forall n\ge 1.Recall that wn1{w}_{n}^{1}, wn2∈Lp′(t)(R,RN){w}_{n}^{2}\in {L}^{p^{\prime} \left(t)}\left({\mathbb{R}},{{\mathbb{R}}}^{N}), so we have (4.32)∫Rwn1(un−u)dt→0,∫Rwn2(un−u)dt→0,\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{1}\left({u}_{n}-u){\rm{d}}t\to 0,\hspace{1em}\mathop{\int }\limits_{{\mathbb{R}}}{w}_{n}^{2}\left({u}_{n}-u){\rm{d}}t\to 0,as n→∞n\to \infty . Then, by (4.31) and (4.32), we obtain (4.33)limsupn→∞⟨ℒ(un),un−u⟩≤0.\mathop{\mathrm{limsup}}\limits_{n\to \infty }\langle {\mathcal{ {\mathcal L} }}\left({u}_{n}),{u}_{n}-u\rangle \le 0.Combining with (4.33) and Proposition 2.9, we deduce that un→u{u}_{n}\to uin Wa1,p(t){W}_{a}^{1,p\left(t)}. So φ\varphi satisfies PS condition.Step 2: ψ\psi satisfies nonsmooth mountain pass theorem.For any ε>0\varepsilon \gt 0, by H(f)2{}_{2}: (ii), (iv), one has (4.34)f1(t,u)≤ε∣u∣p++c(ε)∣u∣α(t),∀(t,u)∈R×RN.{f}_{1}\left(t,u)\le \varepsilon | u{| }^{{p}^{+}}+c\left(\varepsilon )| u{| }^{\alpha \left(t)},\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Choose ‖u‖=ρ\Vert u\Vert =\rho is small enough, from (4.27), (4.34) and Proposition 2.4, we obtain ψ(u)≥1p+∫R(∣u˙∣p(t)+a(t)∣u∣p(t))dt−ε∫R∣u∣p+dt−c(ε)∫R∣u∣α(t)dt≥1p+‖u‖p+−εcp+p+‖u‖p+−c(ε)cα−α−‖u‖α−\begin{array}{rcl}\psi \left(u)& \ge & \frac{1}{{p}^{+}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| \dot{u}{| }^{p\left(t)}+a\left(t)| u{| }^{p\left(t)}){\rm{d}}t-\varepsilon \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{{p}^{+}}{\rm{d}}t-c\left(\varepsilon )\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}| u{| }^{\alpha \left(t)}{\rm{d}}t\\ & \ge & \frac{1}{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}\end{array}for ε>0\varepsilon \gt 0, let εcp+p+<12p+\varepsilon {c}_{{p}^{+}}^{{p}^{+}}\lt \frac{1}{2{p}^{+}}, where cp+(cα−){c}_{{p}^{+}}({c}_{{\alpha }^{-}})is the embedding constant from Wa1,p(t){W}_{a}^{1,p\left(t)}to Lp+(Lα−){L}^{{p}^{+}}({L}^{{\alpha }^{-}}). Then, ψ(u)≥12p+‖u‖p+−c(ε)cα−α−‖u‖α−.\psi \left(u)\ge \frac{1}{2{p}^{+}}\Vert u{\Vert }^{{p}^{+}}-c\left(\varepsilon ){c}_{{\alpha }^{-}}^{{\alpha }^{-}}\Vert u{\Vert }^{{\alpha }^{-}}.Note that p+<α−{p}^{+}\lt {\alpha }^{-}, there exists a constant r>0r\gt 0such that φ(u)≥r\varphi \left(u)\ge ras ‖u‖=ρ\Vert u\Vert =\rho , when ρ\rho is small enough.As in [41], it follows from H(f)2{}_{2}: (iii′^{\prime} ) and H(f)3{}_{3}: (iv) that (4.35)f1(t,σu)≥σμf1(t,u),∀(t,u)∈R×RN{f}_{1}\left(t,\sigma u)\ge {\sigma }^{\mu }{f}_{1}\left(t,u),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}and (4.36)f2(t,σu)≤σϱf2(t,u),∀(t,u)∈R×RN.{f}_{2}\left(t,\sigma u)\le {\sigma }^{\varrho }{f}_{2}\left(t,u),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N}.Let ω∈Wa1,p(t)\omega \in {W}_{a}^{1,p\left(t)}be such that (4.37)∣ω(t)∣=1,as∣t∣≤1;0,as∣t∣≥2;≤1,as∣t∣∈(1,2].| \omega \left(t)| =\left\{\begin{array}{ll}1,\hspace{1.0em}& \hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \le 1;\\ 0,\hspace{1.0em}& \hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \ge 2;\\ \le 1,\hspace{1.0em}& \hspace{0.1em}\text{as}\hspace{0.1em}\hspace{0.33em}| t| \in \left(1,2].\end{array}\right.In view of (4.36) and Proposition 2.8 (i), one has ∫−22f2(t,ω)dt=∫{t∈[−2,2]:∣ω∣>1}f2(t,ω)dt+∫{t∈[−2,2]:∣ω∣≤1}f2(t,ω)dt≤∫{t∈[−2,2]:∣ω∣>1}f2t,ω∣ω∣∣ω∣ϱdt+∫−22max∣ω∣≤1∣f2(t,ω)∣dt\begin{array}{rcl}\underset{-2}{\overset{2}{\displaystyle \int }}{f}_{2}\left(t,\omega ){\rm{d}}t& =& \mathop{\displaystyle \int }\limits_{\left\{t\in \left[-2,2]:| \omega | \gt 1\right\}}{f}_{2}\left(t,\omega ){\rm{d}}t+\mathop{\displaystyle \int }\limits_{\left\{t\in \left[-2,2]:| \omega | \le 1\right\}}{f}_{2}\left(t,\omega ){\rm{d}}t\\ & \le & \mathop{\displaystyle \int }\limits_{\left\{t\in \left[-2,2]:| \omega | \gt 1\right\}}{f}_{2}\left(t,\frac{\omega }{| \omega | }\right)| \omega {| }^{\varrho }{\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t\end{array}(4.38)≤‖ω‖L∞(R)ϱ∫−22max∣ω∣=1f2(t,ω)dt+∫−22max∣ω∣≤1∣f2(t,ω)∣dt≤κϱ‖ω‖ϱ∫−22max∣ω∣=1f2(t,ω)dt+∫−22max∣ω∣≤1∣f2(t,ω)∣dt=M3‖ω‖ϱ+M4,\begin{array}{rcl}& \le & \Vert \omega {\Vert }_{{L}^{\infty }\left({\mathbb{R}})}^{\varrho }\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | =1}{f}_{2}\left(t,\omega ){\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t\\ & \le & {\kappa }^{\varrho }\Vert \omega {\Vert }^{\varrho }\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | =1}{f}_{2}\left(t,\omega ){\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t\\ & =& {M}_{3}\Vert \omega {\Vert }^{\varrho }+{M}_{4},\end{array}where M3=κϱ∫−22max∣ω∣=1f2(t,ω)dt,M4=∫−22max∣ω∣≤1∣f2(t,ω)∣dt.{M}_{3}={\kappa }^{\varrho }\underset{-2}{\overset{2}{\int }}\mathop{\max }\limits_{| \omega | =1}{f}_{2}\left(t,\omega ){\rm{d}}t,\hspace{1em}{M}_{4}=\underset{-2}{\overset{2}{\int }}\mathop{\max }\limits_{| \omega | \le 1}| {f}_{2}\left(t,\omega )| {\rm{d}}t.When σ>1\sigma \gt 1, by (4.35), we have (4.39)∫−22f1(t,σω(t))dt≥σμ∫−22f1(t,ω(t))dt=mσμ,\underset{-2}{\overset{2}{\int }}{f}_{1}\left(t,\sigma \omega \left(t)){\rm{d}}t\ge {\sigma }^{\mu }\underset{-2}{\overset{2}{\int }}{f}_{1}\left(t,\omega \left(t)){\rm{d}}t=m{\sigma }^{\mu },where m=∫−11f(t,ω)dt>0m={\int }_{-1}^{1}f\left(t,\omega ){\rm{d}}t\gt 0. Thus, it follows from (4.27), (4.36), (4.37), (4.38), (4.39) and Proposition 2.4 (ii), we derive (4.40)ψ(σω)=∫R1p(t)(∣σω˙∣p(t)+a(t)∣σω∣p(t))dt+∫R[f2(t,σω(t))−f1(t,σω(t))]dt≤σp+p−∫R(∣ω˙∣p(t)+a(t)∣ω∣p(t))dt+∫−22f2(t,σω)dt−∫−22f1(t,σω)dt≤σp+p−‖ω‖p++M3σϱ‖ω‖ϱ+M4σϱ−mσμ.\begin{array}{rcl}\psi \left(\sigma \omega )& =& \mathop{\displaystyle \int }\limits_{{\mathbb{R}}}\frac{1}{p\left(t)}(| \sigma \dot{\omega }{| }^{p\left(t)}+a\left(t)| \sigma \omega {| }^{p\left(t)}){\rm{d}}t+\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}{[}{f}_{2}\left(t,\sigma \omega \left(t))-{f}_{1}\left(t,\sigma \omega \left(t))]{\rm{d}}t\\ & \le & \frac{{\sigma }^{{p}^{+}}}{{p}^{-}}\mathop{\displaystyle \int }\limits_{{\mathbb{R}}}(| \dot{\omega }{| }^{p\left(t)}+a\left(t)| \omega {| }^{p\left(t)}){\rm{d}}t+\underset{-2}{\overset{2}{\displaystyle \int }}{f}_{2}\left(t,\sigma \omega ){\rm{d}}t-\underset{-2}{\overset{2}{\displaystyle \int }}{f}_{1}\left(t,\sigma \omega ){\rm{d}}t\\ & \le & \frac{{\sigma }^{{p}^{+}}}{{p}^{-}}\Vert \omega {\Vert }^{{p}^{+}}+{M}_{3}{\sigma }^{\varrho }\Vert \omega {\Vert }^{\varrho }+{M}_{4}{\sigma }^{\varrho }-m{\sigma }^{\mu }.\end{array}Since μ>ϱ>p+\mu \gt \varrho \gt {p}^{+}, m>0m\gt 0, from (4.40), we can choose σ0>1{\sigma }_{0}\gt 1such that e=σ0ω∈Wa1,p(t)e={\sigma }_{0}\omega \in {W}_{a}^{1,p\left(t)}and φ(e)<0\varphi \left(e)\lt 0. Hence, from Lemma 2.19, there exists at least one nontrivial critical point, that is, system (4.2) has at least one homoclinic orbit.□Example 4.1Let p(t)=32+2π∣arctant∣p\left(t)=\frac{3}{2}+\frac{2}{\pi }| \arctan t| for t∈R,t\in {\mathbb{R}},and f(t,u)=a(t)(1+a(t))−1∣u∣5/2ln(1+∣u∣),∀(t,u)∈R×RN,f\left(t,u)=a\left(t){(1+a\left(t))}^{-1}| u{| }^{5\text{/}2}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where a(t)a\left(t)satisfies H(a),H(a)2\hspace{0.1em}\text{H(a)},{\text{H(a)}}_{2}. It is evident that ffis locally Lipschitz and ∂f(t,u)=a(t)(1+a(t))−152∣u∣1/2uln(1+∣u∣)+∣u∣3/2u1+∣u∣.\partial f\left(t,u)=a\left(t){\left(1+a\left(t))}^{-1}\left[\frac{5}{2}| u{| }^{1\text{/}2}u\mathrm{ln}(1+| u| )+\frac{| u{| }^{3\text{/}2}u}{1+| u| }\right].As −f0(t,u;−u)=52+∣u∣(1+∣u∣)ln(1+∣u∣)f(t,u)≥52+11+∣u∣f(t,u)>p++11+∣u∣f(t,u),\begin{array}{rcl}-{f}^{0}\left(t,u;-u)& =& \left[\frac{5}{2}+\frac{| u| }{(1+| u| )\mathrm{ln}(1+| u| )}\right]f\left(t,u)\\ & \ge & \left(\frac{5}{2}+\frac{1}{1+| u| }\right)f\left(t,u)\\ & \gt & \left({p}^{+}+\frac{1}{1+| u| }\right)f\left(t,u),\end{array}ffsatisfies H(f)2{\text{H(f)}}_{2}: (iii) with α=β=ν=1\alpha =\beta =\nu =1. Thus, we can show that ffsatisfies the hypothesis of Theorem 4.1.Moreover, it is similar to obtain that ffsatisfies the hypothesis of Theorem 4.2 with μ=52\mu =\frac{5}{2}.Example 4.2Let p(t)=5+11+t2p\left(t)=5+\frac{1}{1+{t}^{2}}for t∈R,t\in {\mathbb{R}},and f(t,u)=a(t)−1[∣u∣4∣sint∣+4+∣u∣2∣sint∣+2],∀(t,u)∈R×RN,f\left(t,u)=a{\left(t)}^{-1}{[}| u{| }^{4| \sin t| +4}+| u{| }^{2| \sin t| +2}],\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where a(t)=1+t2a\left(t)=1+{t}^{2}satisfies H(a),H(a)2\hspace{0.1em}\text{H(a)},{\text{H(a)}}_{2}. Obviously, ffis locally Lipschitz and ∂f(t,u)=a(t)−1(2∣sint∣+2)[2∣u∣4∣sint∣+2u+∣u∣2∣sint∣+1u].\partial f\left(t,u)=a{\left(t)}^{-1}(2| \sin t| +2){[}2| u{| }^{4| \sin t| +2}u+| u{| }^{2| \sin t| +1}u].Since ∣∂f(t,u)∣≤3(2∣sint∣+2)1+t2∣u∣2∣sint∣+1,∣u∣≤1;3(4∣sint∣+4)2(1+t2)∣u∣4∣sint∣+3,∣u∣≥1.| \partial f\left(t,u)| \le \left\{\begin{array}{ll}\frac{3(2| \sin t| +2)}{1+{t}^{2}}| u{| }^{2| \sin t| +1},\hspace{1.0em}& | u| \le 1;\\ \frac{3(4| \sin t| +4)}{2(1+{t}^{2})}| u{| }^{4| \sin t| +3},\hspace{1.0em}& | u| \ge 1.\end{array}\right.Then ffsatisfies H(f)2{\text{H(f)}}_{2}: (ii′)\left({\rm{ii}}^{\prime} )with α1(t)=2∣sint∣+2,α2(t)=4∣sint∣+4,a1(t)=31+t2,a2(t)=32(1+t2).{\alpha }_{1}\left(t)=2| \sin t| +2,\hspace{1em}{\alpha }_{2}\left(t)=4| \sin t| +4,\hspace{1em}{a}_{1}\left(t)=\frac{3}{1+{t}^{2}},\hspace{1em}{a}_{2}\left(t)=\frac{3}{2\left(1+{t}^{2})}.Let Ω=(−2,−2)\Omega =\left(-2,-2), γ(t)=2∣sint∣+2,\gamma \left(t)=2| \sin t| +2,one has f(t,u)≥15∣u∣2∣sint∣+2,∀∣u∣≤1.f\left(t,u)\ge \frac{1}{5}| u{| }^{2| \sin t| +2},\hspace{1em}\forall | u| \le 1.Hence, from Theorem 4.4, problem (4.1) has at least a nontrivial homoclinic solution.Example 4.3Let p(t)=32+2π∣arctant∣p\left(t)=\frac{3}{2}+\frac{2}{\pi }| \arctan t| for t∈Rt\in {\mathbb{R}}, f=f1−f2f={f}_{1}-{f}_{2}and f1(t,u)=a1(t)(1+a1(t))−1∣u∣7/2ln(1+∣u∣),∀(t,u)∈R×RN,{f}_{1}\left(t,u)={a}_{1}\left(t){(1+{a}_{1}\left(t))}^{-1}| u{| }^{7\text{/}2}\mathrm{ln}(1+| u| ),\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},f2(t,u)=a2(t)[sint∣u∣2+∣u∣3],∀(t,u)∈R×RN,{f}_{2}\left(t,u)={a}_{2}\left(t){[}\sin t| u{| }^{2}+| u{| }^{3}],\hspace{1em}\forall \left(t,u)\in {\mathbb{R}}\times {{\mathbb{R}}}^{N},where ai(t)(i=1,2.){a}_{i}\left(t)\left(i=1,2.)satisfies H(a),H(a)2\hspace{0.1em}\text{H(a)},{\text{H(a)}}_{2}and H(f)2{\text{H(f)}}_{2}: (ii′)\left({\rm{ii}}^{\prime} ), (v), respectively. It is visible that ffis locally Lipschitz, ∂f⊆∂f1−∂f2\partial f\subseteq \partial {f}_{1}-\partial {f}_{2}and f20(t,u;u)=a2(t)[2sint∣u∣2+3∣u∣3]≥3f2(t,u),{f}_{2}^{0}\left(t,u;\hspace{0.33em}u)={a}_{2}\left(t){[}2\sin t| u{| }^{2}+3| u{| }^{3}]\ge 3{f}_{2}\left(t,u),which implies, f2{f}_{2}satisfies H(f)3{\text{H(f)}}_{3}: (iv) with ϱ=3\varrho =3and μ=72\mu =\frac{7}{2}. It is easy to verify that ffsatisfies the hypothesis of Theorem 4.5.

Journal

Advances in Nonlinear Analysisde Gruyter

Published: Jan 1, 2023

Keywords: p ( t )-Laplacian; homoclinic solution; locally Lipschitz; nonsmooth critical point theory; 34C25; 58E30; 47H04

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