Xianhua Tang Periodic solutions for a differential inclusion problem involving the p ( t )-Laplacian

In recent years, the study on p(t)-Laplacian problems has attracted more and more attention. The p(t)Laplacian possesses more complicated nonlinearities than the p-Laplacian. For example, it is inhomogeneous, this causes many troubles, 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 been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. 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 [44]. 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 [1,2]. Another field of application of equations with variable exponent growth conditions is image processing [4]. We refer the reader to [35,40-44] for an overview of and references on this subject, and to [6-9,13,14,20,21,33,34, 37,39, 40-42] for the study of the p(t)-Laplacian equations and the corresponding variational problems. Recently, Wang and Yuan [37] obtained the existence of periodic solutionsfor p(t)-Laplacian system: { −(|u′(t)|p(t)−2u′(t))′ = ∇j(t, u(t)), a.e. t ∈ [0, T], u(0) = u(T), u′(0) = u′(T), (1.1)


Introduction
In recent years, the study on p(t)-Laplacian problems has attracted more and more attention. The p(t)-Laplacian possesses more complicated nonlinearities than the p-Laplacian. For example, it is inhomogeneous, this causes many troubles, 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 been received considerable attention in recent years. These problems are interesting in applications and raise many difficult mathematical problems. 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 [44]. 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 [1,2]. Another field of application of equations with variable exponent growth conditions is image processing [4]. We refer the reader to [35,[40][41][42][43][44] for an overview of and references on this subject, and to [6][7][8][9]13,14,20,21,33,34,37,39,[40][41][42] for the study of the p(t)-Laplacian equations and the corresponding variational problems.
Recently, Wang and Yuan [37] obtained the existence of periodic solutionsfor p(t)-Laplacian system: −(|u (t)| p(t)−2 u (t)) = ∇j(t, u(t)), a.e. t ∈ [0, T], u(0) = u(T), u (0) = u (T), (1.1) where j(t, u) is measurable in t ∈ [0, T], continuously diferentiable in u ∈ R N . More precisely, they were able to prove that, under suitable conditions, the system might have at least one solution, or have infinite number of solutions. Since many free boundary problems and obstacle problems may be reduced to partial differentia1 equations with discontinuous nonlinearities, now a question arises: whether there exist solutions for system (1.1) in the case where the potential function j(t, x) is nonsmooth in x ∈ R N . We require that j(t, ·) is only locally Lipschitz. That is the main problem which we want to solve in the present paper. The operator (|u (t)| p(t)−2 u (t)) is said to be p(t)-Laplacian, which becomes p-Laplacian when p(t) ≡ p (a constant) and the problem (1.1) reduces to the following where j(t, u) is locally Lipschitz in u ∈ R N . Periodic problems involving the scalar p-Laplacian were studied by many authors. We mention the works by Dang and Oppenheimer [10], Del Pino, Manasevich and Murua [11], Gasinski and Papageorgiou [16][17][18], Papageorgiou and Rǎdulescu [30], Yang [38] and the references [15,22,28,[30][31][32]38].
The goal of this paper is to discuss the existence of solutions of the following differential equation with p(t)-Laplacian and a nonsmooth potential where p(t) > 1, u ∈ R N , j(t, s) is locally Lipschitz function in the s-variable integrand (in general it can be nonsmooth), and ∂j(t, s) is the subdifferential with respect to the s-variable in the sense of Clarke [3,5]. This paper is organized as follows. In Section 2, we present some necessary preliminary knowledge on generalized gradient of the locally Lipschitz function and variable exponent Sobolev spaces. In Section 3, we give the main results and their proofs.

Preliminaries
The nonsmooth critical point theory for locally Lipschitz functionals is based on the subdifferential theory of Clarke [5].
Let X be a Banach space and let X * be its topological dual. By < · > we denote the duality brackets for the pair (X, X * ). A function ϕ : X → R is said to be locally Lipschitz, if for every x ∈ X, we can find a neighbourhood U of x and a constant k > 0 (depending on U), such that |ϕ(y) − ϕ(z)| ≤ k y − z , ∀ y, z ∈ U.
For a locally Lipschitz function ϕ : X → R we define It is obvious that the function h → ϕ 0 (x; h) is sublinear, continuous and so is the support function of a nonempty, convex and w * -compact set ∂ϕ(x) ⊆ X * , defined by The multifunction x → ∂ϕ(x) is known as the generalized (or Clarke) subdifferential of ϕ. If ϕ, ψ : X → R are locally Lipschitz functions, then Let ϕ : X → R be a locally Lipschitz function. A point x ∈ X is said to be a critical point of ϕ if 0 ∈ ∂ϕ(x). If x ∈ X is a critical point of ϕ, then c = ϕ(x) is a critical value of ϕ. It is easy to see that, if x ∈ X is a local extremum of ϕ, then 0 ∈ ∂ϕ(x). Moreover, the multifunction x → ∂ϕ(x) is upper semicontinuous from X into X * equipped with the w * topology, i.e., for any U ⊆ X * w * -open, the set {x ∈ X : ∂ϕ(x) ⊆ U} is open in X. For more details we refer to Clarke [5]. The critical point theory for smooth functions uses a compactness condition known as the Palais-Smale condition (PS). In the present nonsmooth setting this condition takes the following form: The locally Lipschitz function ϕ : X → R satisfies the nonsmooth PS condition if any sequence {xn} n≥1 ⊆ X such that {ϕ(xn)} n≥1 is bounded and m(xn) = min[ x * : x * ∈ ∂ϕ(xn)] → 0 as n → ∞, has a strongly convergent subsequence.
If ϕ ∈ C 1 (X, R), then as we already mentioned ∂ϕ(x) = {ϕ (x)} and so the above definition of the P.S. 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: The locally Lipschitz function ϕ : X → R satisfies the nonsmooth Cerami condition (nonsmooth C-condition for short), if any sequence {xn} n≥1 ⊆ X such that {ϕ(xn)} n≥1 is bounded and (1 + xn )m(xn) → 0 as n → ∞, has a strongly convergent subsequence. Lemma 2.1. [27] Assume that φ is a locally Lipschitz functional on a Banach space X and φ : X → R satisfies: (i) φ is weakly lower semicontinuous; (ii) φ is coercive. Then there exists x * ∈ X such that φ(x * ) = min x∈X φ(x). Lemma 2.2. [22] Let X be a Banach space and φ : X → R a locally Lipschitz functional satisfying the (C) condition. If X = Y ⊕ V with Y a finite-dimensional subspace of X. φ satisfies the nonsmooth C-condition, and there exists an r > 0 such that

Lemma 2.3. [22]
Let X be a reflexive Banach space, ϕ : X → R a locally Lipschitz functional satisfying the PS-condition. Assume that there exist x 0 , x 1 ∈ X, c 0 ∈ R and ϱ > 0 such that x 1 − x 0 > ϱ and Then, ϕ has a critical point x ∈ X with c = ϕ(x) ≥ c 0 given by In order to discuss (1.3), we recall some known results from critical point theory and the properties of space W 1,p(t) for the convenience of the readers. Let For u ∈ L 1 loc (0, T; R N ), let u denote the weak derivative of u, i.e., u ∈ L 1 loc (0, T; R N ) and satisfy We call the space L p(t) a generalized Lebesgue space, it is a special kind of generalized Orlicz spaces. The space W 1,p(t) is called a generalized Sobolev space, it is a special kind of generalized Orlicz-Sobolev spaces.
For the general theory of generalized Orlicz spaces and generalized Orlicz-Sobolev spaces, see [13,14].
Proposition 2.5. [13] L p(t) and W 1,p(t) are Banach spaces with the norms defined above. When p − > 1, they are reflexive.
then v is called a T-weak derivative of u and is denoted byu.
Consider the following functional: We know that (see [2]) J ∈ C 1 (W 1,p(t) per , R) and For every u ∈ W 1,p(t) per (0, T; R N ), set By virtue of [35], there exists a > 0 such that The corresponding functional φ :

Main results and their Proofs
Our hypotheses on the function p(t) and j(t, u) are the following: In the previous existence theorems the energy functional φ defined in (2.2) was coercive and so the solution was obtained by an application of the least action principle. In the next existence theorem the energy functional φ is bounded below but not necessarily coercive. In this case the hypotheses on the nonsmooth potential j(t, u) are the following: H(j) 3 j : [0, T] × R N is a function such that j(·, 0) ∈ L 1 [0, T] and: Next, we will establish some existence results by using nonsmooth Mountain Pass theorem. Our hypotheses on the nonsmooth potential function j(t, u) are the following:
Owing to the fact the compact embedding of W 1,p(t) per (0, T; R N ) → C(0, T; R N ) and the weak lower semicontinuity of the norm functional in a Banach space, we infer that φ is weakly lower semicontinuous. So by hypothesis H(j) 1 (v) and the Weierstrass theorem (Lemma 2.1) we can find u 0 such that Next, we will show that u is the solution of (1.3). In fact, let u ∈ W 1,p(t) per (0, T; R N ) be such that 0 ∈ ∂φ(u). Define the nonlinear operator A : W 1,p(t) per (0, T; R N ) → (W 1,p(t) per (0, T; R N )) * as follows Thus, where w ∈ L q(t) (0, T; R N ) and w ∈ ∂j(t, u(t)). Here 1 p(t) + 1 q(t) = 1. For every x, y ∈ R N , the following inequalities hold [23]: [1,2]. (3.5) and An argument similar to the one used in [37] shows that for every v ∈ W 1,p(t) per (0, T; R N ), thus which implies that (|u (t)| p(t)−2 u (t)) has T-weak derivative and satisfy −(|u (t)| p(t)−2 u (t)) = w.

Proof of Theorem 3.3.
Consider the locally Lipschitz energy functional φ : W 1,p(t) per (0, T; R N ) → R defined by Step 1: φ satisfies the nonsmooth C-condition.
Since ∂φ(un) ⊆ W 1,p(t) per (0, T; R N ) * is weakly compact and the norm functional in a Banach space is weakly compact, from the Weierstrass theorem (Lemma 2.1), we can find u * n ∈ ∂φ(un) such that m(un) = u * n . Define the nonlinear operator A : Then where w n ∈ ∂j(t, un). From the choice of the sequence {un} n≥1 ⊆ W 1,p(t) per (0, T; R N ) we have  where c 5 > 0 is independent of n. Therefore, from (3.10) we have |u n | p(t) ≤ ϵn + μM 1 + c 6 , n ≥ 1.
In fact, from the choice of the sequence {un} n≥1 we have Recall that u * n = A(un) − wn, then we have Then limn→∞ < A(un), un − u >≤ 0.
Step 2: Similar to the proof in [30,31], we have j(t, sx) ≤ s μ j(t, x) for every s ≥ 1, we omit its proof course.
Let v ∈ V be such that |{t ∈ T : v(t) > M }| 1 > 0, we have Because of hypothesis H(j) 3 (iii) and the mean value theorem for locally Lipschitz functions, we see that there exists c 7 > 0 such that Also using Step 2, we have Noting that for u = M, from the subdifferential chain rule, for almost all t ∈ T we have (3.14) By (3.13) and (3.14), we have Using (3.15) and Poincare-Wirtinger inequality (2.1) we have which yields that φ| V is coercive since μ < p − .
Step 1, 3 and 4 permit the application of the nonsmooth saddle point theorem, so we can find u ∈ W 1,p(t) per (0, T; R N ) such that 0 ∈ ∂φ(u).

Proof of Theorem 3.4.
Consider the locally Lipschitz energy functional φ : W 1,p(t) per (0, T; R N ) → R defined by We divide our proof into two steps.
As before, we can show that u ∈ W 1,p(t) per (0, T; R N ) solves (1.3), we omit its proof process.