A Non-Smooth Three Critical Points Theorem for General Hemivariational Inequality on Bounded Domains

In this paper we are concerned with the study of a hemivariational inequality with nonhomogeneous Neumann boundary condition. We establish the existence of at least three solutions of the problem by using the nonsmooth three critical points theorem and the principle of symmetric criticality for MotreanuPanagiotopoulos type functionals.


Introduction
In this paper, we study the following nonlinear elliptic differential inclusion with p(x)−Laplacian −∆ p(x) u + a(x)|u| p(x)−2 u = −µg (x, u) in Ω −|∇u| p(x)−2 ∂u ∂ν ∈ −λ∂F (x, u) on ∂Ω, where Ω ⊂ R N (N ≥ 2) is a bounded smooth domain, ∂u ∂ν is the outward unit normal derivative on ∂Ω, p : Ω → R is a continuous function satisfying and λ, µ ∈ [0, ∞).F : ∂Ω × R → R is a function in which F (•, u) is measurable for every u ∈ R and F (x, •) is locally Lipschitz for a.e.x ∈ ∂Ω.∂F (x, u) denotes the generalized Clarke gradient of F (x, u) at u ∈ R.Moreover, g : Ω × R → R is a Carathéodory function and G(x, u) = u 0 g(x, t)dt.In this paper, a class of problem for hemivariational inequality is studied which is defined on domains of the type B which are nonempty, closed, convex cone subsets of W 1,p(x) 0 (Ω).
The p(x)−Laplace operator ∆ p(x) u = div(|∇u| p(x)−2 ∇u) is a natural generalization of the p−Laplacian operator ∆ p u = div(|∇u| p−2 ∇u), where p > 1 is a real constant.The main difference between them is that p−Laplacian operator is (p − 1)−homogenous, but the p(x)−Laplacian operator, when p(x) is not constant is not homogeneous.For p(x)−Laplacian operator, we refer the readers to (cf.[13], [14], [15], [18], [23]) and references therein.In recent years, differential equations and variational problems have been studied in many papers, we refer to some interesting works (cf.[27], [28]).For a thorough treatment of the hemivariational inequality problems we refer to the monographs Naniewicz and Panagiotopoulos (cf.[26]) (based on pseudomonotonicity), Motreanu and Panagiotopoulos (cf.[24]), Motreanu and Rádulescu (cf.[25]) (based on compactness arguments).In these works (and in references therein) there are studied the elliptic problems on bounded domains.It is well known that many problems in mathematics and physics that comes from the real world by some authors have investigated (see cf. [1], [2], [29], [30], [31]).The applications to nonsmooth variational problems have been seen in (cf.[3]), Bonanno and Candito studied a class of variational-hemivariational inequalities; in (cf.[32]), Zhang and Liu studied an elliptic equation with discontinuous nonlinearities in R N .In recent years, the study of the three-critical-points theorem nonsmooth variational problems was investigated.The goal of this article is to apply a version for locally Lipschitz functionals (was established by Kristály, Marzantowicz and Varga A Non-Smooth Three Critical Points Theorem 97 in (cf.[21])).In the present article, we use a class of perturbed Motreanu-Panagiotopoulos functionals.We prove the existence of at least three solutions for a hemivariational inequality depending on two parameters.The paper is organized as follows.We prepare the basic definitions and properties in the framework of the generalized Lebesgue and Sobolev spaces.For this introductory part we refer to (cf.[6], [8], [9], [11], [12]).Moreover, some important properties of the p(x)−biharmonic operator, some basic notions about generalized directional derivative and hypotheses on F, the basic definitions and facts about the non-smooth three-critical-points theorem are given.Finally, we will give the proofs of our main results.

Preliminaries
We recall some basic facts about the variable exponent Lebesgue-Sobolev.The variable exponent Lebesgue space L p(•) (Ω) is defined by It is endowed with the Luxemburg norm For p ≡ const., the Luxemburg norm • p(•) coincides with the standard norm • p of the Lebesgue space L p (Ω).Then (L p(x) (Ω), • p(•) ) is a Banach space (cf.[22]).
The space W L,p(x) 0 (Ω) is a separable, uniformly convex and reflexive Banach space.

Let p *
L denote the critical variable exponent related to p, defined for all x ∈ Ω by the pointwise relation is the critical exponent related to p.
Remark 2.6.(i) By the Proposition 2.5 there is a continuous and compact embedding of W

(Ω).
A Non-Smooth Three Critical Points Theorem

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In this paper, we denote by X = W 1,p(x) 0 (Ω) and X ⋆ the dual space.
The proof of this proposition is similar to the proof in (cf.[11]).
Let η : ∂Ω → R be a measurable.Define the weighted variable exponent Lebesgue space by where dσ is the measure on the boundary.
Here, we review the definitions and basic properties from the theory of generalized differentiation for locally Lipschitz functions.Let X be a Banach space and X ⋆ its topological dual.By • we will denote the norm in X and by < •, • > the duality brackets for the pair (X, X ⋆ ).A function h : X → R is said to be locally Lipschitz, if for every x ∈ X there exists a neighbourhood U of x and a constant K > 0 depending on U such that |h(y) − h(z)| ≤ K y − z for all y, z ∈ U.For a locally Lipschitz function h : X → R we define the generalized directional derivative of h at u ∈ X in the direction γ ∈ X is defined by The generalized gradient of h at u ∈ X is defined by It is nonempty, convex and w ⋆ −compact subset of X ⋆ , where < •, • > X is the duality pairing between X ⋆ and X, see (cf.[4]).
Proposition 2.11.(cf.[4])(Lebourg's mean value theorem) Let h : X → R be a locally Lipschitz functional.Then, for every u, v ∈ X there exists w ∈ Definition 2.12.(cf.[24]) Let X be a Banach space, I : X → (−∞, +∞] is called a Motreanu-Panagiotopoulos-type functional, if I = h + χ, where h : X → R is locally Lipschitz and χ : X → (−∞, +∞] is convex, proper and lower semicontinuous. Definition 2.13.(cf.[24]) An element u ∈ X is said to be a critical point of In most applications, the following special case is considered: Let h : X → R be a locally Lipschitz functional and we assume it is also given a functional χ : X → R ∪ {+∞} which is convex, lower semicontinuous and proper whose restriction to the set dom(χ) = {x ∈ X : (it is easily seen that χ B is proper, convex and lower semicontinuous), while its restriction to dom(χ B ) = B is the constant 0; clearly u ∈ X is a critical point for the Motreanu-Panagiotopoulos functional h + χ B iff u ∈ B and the following condition holds Definition 2.14.(cf.[17]) The functional I : X → X ⋆ verifies the (S + ) property if for any weakly convergence sequence {u n } n ⊂ X to u in X in which then {u n } n converges strongly to u in X.

Main Results
For the reader's convenience, we recall the non-smooth three critical points theorem.
Theorem 3.1.(cf.[19]) Let X be a separable and reflexive Banach space, Λ a real interval and B a nonempty, closed, convex subset of X. φ ∈ C 1 (X, R) a sequentially weakly l.s.c.functional and bounded on any bounded subset of X such that φ ′ is of type (S) + , suppose that F : X → R is a locally Lipschitz functional with compact gradient.Assume that: Then, there exist λ 1 , λ 2 ∈ Λ (λ 1 < λ 2 ) and σ > 0 such that for every λ ∈ [λ 1 , λ 2 ] and every locally Lipschitz functional G : X → R with compact derivative, there exists µ 1 > 0 such that for every µ ∈]0, µ 1 [ the functional φ − λF + µG has at least three critical points whose norms are less than σ.
Let us introduce the following conditions of our problem.We assume that F : ∂Ω × R → R is a Carathéodory function, which is locally Lipschitz in the second variable and satisfies the following properties: The next lemma displays some properties of φ (cf.[10]).
We need the following lemmas in the proof of our main result.Lemma 3.3.Let (F 1 ) be satisfied.Then F : X → R is locally Lipschitz functional with compact gradient.
Proof: First we prove that F is Lipschitz continuous on each bounded subset of X.Let u, v ∈ B(0, M ) (M > 0), and u , v ≥ 1. Utilizing Proposition 2.11 , from the Hölder inequality, and the embedding of X in L t(x) (∂Ω) and L z(x) (∂Ω) where c 1 , c 2 are positive constants.We prove that ∂F is compact.Let {u n } be a sequence in X such that u n ≤ M and choose u * n ∈ ∂F(u n ) for any n ∈ N. From (F 1 ) it follows that for any Consequently, ).The sequence (u * n ) is bounded and hence, up to a subsequence, u * n ⇀ u * .Suppose on the contrary; we assume there exists ǫ > 0 for which u * n − u * X * > ǫ (choose a subsequence if necessary).For every n ∈ N, we can find v n ∈ X with v n < 1 and Then, (v n ) is a bounded sequence and up to a subsequence,

1).
For u , v ≤ 1 the proof is similar.✷ Lemma 3.4.Let G be satisfied.Then G is a locally Lipschitz functional with compact derivative.
Proof: G(u) = Ω G(x, u)dx is locally continuous on each bounded subset of X.Indeed, let u, v ∈ B(0, M )(M > 0) and apply Theorem 2.5, the Hölder inequality and mean value Theorem there is a functional ω(x) in which where c 5 is positive constant.Hence, G is locally Lipschitz.It remains to show that G ′ is compact.Let (u n ) ⊂ X be a sequence such that u n ⇀ u.From compact embedding of X into L q(x) (Ω), we can assume up to subsequence u n → u in L q(x) (Ω).According to the Krasnoselki's theorem, the Nemytskii Fariba Fattahi and Mohsen Alimohammady operator N g : u(x) → g(x, u(x)) is a continuous bounded operator from L q(x) (Ω) to L q(x) q(x)−1 (Ω).Using Hölder's inequality and the continuous embedding of X to L q(x) (Ω), it follows that This inequality shows that the operator A : L . Then F is well-defined and The next lemma points out the relationship between the critical points of I(u) and solutions of Problem (1.2).
Using Proposition 3.5 and the property (ii) of Proposition 2.10, we obtain the desired inequality.✷ Lemma 3.7.If (F 2 ) holds, then for any λ ∈ (0, +∞) Proof: For u ∈ X such that u ≥ 1 and using (F 2 ), By the embedding theorem for suitable positive constant c 7 , c 8 it implies that ).On the other hand from Proposition 2.7, A Non-Smooth Three Critical Points Theorem
Proof: According to Lemma 3.6, it is sufficient to prove the existence of a critical point of functional I.For this, we check if I satisfies the conditions of the nonsmooth three critical points Theorem 3.1.First, we note that Lemma 3.2 guarantees that φ satisfies the weakly sequentially lower semicontinuous property and φ ′ is of type (S + ).Besides, Due to Lemma (3.3), the functional F is weakly sequentially continuous.Lemma 3.7, implies that φ − λF is coercive on X for all λ ∈ Λ =]0, +∞[; the assumption (i) of Theorem 3.1, verified.
For assumption (ii), let us consider two cases.Case 1.Let us assume that u < 1.
Put for every r > 0, we prove that In view of (F 1 ), it is follows that for every ǫ > 0, there exists c(ǫ) > 0 such that for every x ∈ ∂Ω, u ∈ R and ξ ∈ ∂F (x, u) It implies that for every u ∈ X by the Sobolev embedding theorem, there exist suitable positive constants c 9 and c 10 From (F 3 ) û = 0. Hence, due to (3.3), there is r ∈ R in which especially, ρ 0 < F(û).
We claim that It is obvious that the mapping is upper semicontinuous on Λ and Therefore, there exists λ ∈ Λ in which We consider two cases: A Non-Smooth Three Critical Points Theorem

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We claim that Infact, for every u ∈ B there are two cases: From (3.6), (3.7) and the assumption (ii) of Theorem 3.1, this case verified.Case 2. Assume that u > 1.In a similar way like the case 1: Put for every r > 0 We claim that In order to (3.4), for every u ∈ X for continuous and compact embedding, it implies the existence of c 11 and c 12 such that It follows from min{t + , z + } > p + that lim r→0 + θ 2 (r) r = 0.

A
Non-Smooth Three Critical Points Theorem 103 where c 3 , c 4 are positive constants.