Homoclinic solutions for a di ﬀ erential inclusion system involving the p ( t )- Laplacian

: The aim of this article is to study nonlinear problem driven by the ( ) p t - Laplacian with nonsmooth potential. We establish the existence of homoclinic solutions by using variational principle for locally Lipschitz functions and the properties of the generalized Lebesgue - Sobolev space under two cases of the nonsmooth potential: periodic and nonperiodic, respectively. The resulting problem engages two major di ﬃ culties: ﬁ rst, due to the appearance of the variable exponent, commonly known methods and techni ques for studying constant exponent equations fail in the setting of problems involving variable exponents. Another di ﬃ culty we must overcome is verifying the link geometry and certifying boundedness of the Palais - Smale sequence. To our best knowledge, our theorems appear to be the ﬁ rst such result about homoclinic solution for di ﬀ erential inclusion system involving the ( ) p t - Laplacian.


Introduction
In this article, we study the following nonlinear second-order ( ) p t -Laplacian system with nonsmooth potential (1.1) where → + p a , : , ( ) ↦ u f t u , is locally Lipschitz. Here ( ) ∂f t x , denotes the subdifferential of the locally Lipschitz function ( ) ↦ u f t u , . In recent years, the study on ( ) p t -Laplacian problems has attracted more and more attention. The ( ) p t -Laplacian possesses more complicated phenomena than the p-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 -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 p-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 -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 p, (1.1) reduces to p-Laplacian system: 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 { } u n 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 , ( ) a t and basic assumptions on ( ) f t u , : 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: denote positive constants possibly different in different places.

Preliminaries
We start with some preliminary basic results on variable exponent Sobolev spaces. For more details we refer the readers to the book of Rădulescu p t a p t , Ω If ( ) ≡ a t 1, ( ) L a p t and the corresponding norm | | ( ) u p t a , are written simply by ( ) ) as follows: We use ( ) to represent the space of ( ) consisting of infinitely continuous differentiable functions with compact supports on Ω completion in ( ) . We call the space ( ) L p t a generalized Lebesgue space, and it is a special kind of generalized Orlicz spaces. The space 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]).
a p t , then the following properties hold: if and only if ( ) = ∈ u L a p t , the following properties hold: .
, then the following properties hold: p t , then the following properties are equivalent: Consider the following functional: under condition H (a). Moreover, then u n has a convergent subsequence in Proposition 2.10. The mapping A is a strictly monotone, bounded homeomorphism and is of type ) as follows: ) as follows: , respectively.
Definition 2.11. Let X be a Banach space and let * X be its topological dual. By ⟨⋅⟩ we denote the duality brackets for the pair ( , . U X Definition 2.12. For a given locally Lipschitz function : be a locally Lipschitz function. Then generalized subdifferential of ϕ at ∈ x X is the nonempty set ( )  1 , then as we already mentioned ( ) { ( )} ∂ = ′ ϕ x ϕ x 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: such that x m x n is bounded and 1 0 as , n n n n 1 has a strongly convergent subsequence.
Lemma 2.18. (Weierstrass theorem [43]). Assume that φ is a locally Lipschitz functional on a Banach space X and → φ X : satisfies: Lemma 2.19. (Nonsmooth mountain pass theorem [35]). Let X be a reflexive Banach space, → ϕ X : a locally Lipschitz functional satisfying the PS-condition. Assume that there exist ∈ x x X , , where c is given by In this section, we establish the existence of homoclinic solutions with periodic assumption for problem (1.1). In this situation, our hypotheses on p a , and f are the following: Our main results can be stated as follows.  Proof of Theorem 3.1. We consider the following auxiliary periodic problem: From [5], we know that problem (3.1) has a nontrivial solution We claim that φ n be the locally Lipschitz functional. In fact, for all ( ) , one has where On the other hand, it follows from H(f)1: (ii) and Lemma 2.16, for all ( ) and Then, in virtue of (3.6), (3.7) and Hölder inequality, one has .  which yields that φ 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: (iii), (iv), for ≥ σ 1, there exist > C C , 0 6 7 such that be defined as follows: if . Note that ( ) = f t, 0 0, then for all ≥ σ σ 0 , we deduce that ( ) ( ) = φ σu φ σûn 1 . As in [5], we see that the solution ( ) ∈ u C T , n nb n N 2 1 of problem (3.1) is obtained via the nonsmooth mountain pass theorem. One will immediately obtain the fact that there exists > ρ 0 such that  which implies that Then, it follows from (3.10), we obtain Using (2.1), one has , a.e. , n n n nb n n n 2 which yields that n n ν n n n n 0 Hence, from H(f): (ii), (3.13) and Proposition 2.8 (i), there exists a constant ξ 0 which is independent of n such that for ‖ ‖ ≥ u 1 n . Since < − ν p , there exists a constant ξ 1 which is independent of n such that where ξ 2 is a constant which is independent of n. Thus, by (3.14), we obtain is independent of n. Moreover, by an argument as in the proof of [46, (2.19)], there exists a constant ξ 4 which is independent of n such that In what follows, we extend by periodicity u n and w n to all of . From (3.15) and the fact that the embedding is compact, we may assume that where > ξ 0 5 is a constant which is independent of n. Passing to a subsequence if needed, we may assume that in T n for all ≥ n 1 and ( ) ( ( )) ∈ ∂ w t f t u t , on . For any > τ 0, we have Thus, from (3.18) and (3.19), one has By the arbitrariness of > τ 0, from (3.20), we deduce that , then ⊆ θ T supp n for large ≥ n 1, which together with (3.15), performs integration by parts and Proposition 2.6, we have ,˙d˙, d˙. Note that ( ( ) ( )) a.e. on for all n. Therefore, by generalized Lebesgue-dominated convergence theorem, we see that , which leads to Furthermore, by (3.17), one has . So it follows that for all ≥ n 1. By (3.22) and Proposition 2.1, let Define the continuous functional → λ : by Then, from (3.23) that Hence, it follows from (3.24), we obtain For any ≥ n 1, we have From the arbitrary of ( ) , we can deduce that a.e. and ( )  Then, it follows from (3.28) and (3.29) that It is evident that f is locally Lipschitz and which implies that f satisfies ( ) H f 1 :(iii) with = = = α β ν 1. Hence, from Theorem 3.1, problem (1.1) has a nontrivial homoclinic solution.

Nonperiodic p t ( )-Laplacian inclusion system
In 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: and another problem where a satisfy the following hypothesis: H(a)2 ( ) = +∞ | |→+∞ a t lim t .
y r For the nonlinearity f , we suppose the following hypotheses: Then problem (4.1) has at least a nontrivial homoclinic solution.
With regard to problem (4.2), in this situation, assume that p, a and f 1 satisfy all the conditions in Theorem 4.2 and f 2 satisfies the following conditions: is locally Lipschitz for a.e. ∈ t ; (iii) for almost all ∈ t , there exists a function ( ) ( ) ( ) as follows: Using the same type of reasoning as the proof of Theorem 3.1, it is easy to verify that φ is the nonsmooth Lipschitz energy functional corresponding to problem (1.1). as follows: According to the literature [21], is monotonic and semicontinuous, so it is maximal monotone (see also [24]), therefore, , we obtain , 0 , n n n n (4.5) which yields that