EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR P (X)-LAPLACIAN DIFFERENTIAL INCLUSIONS INVOLVING CRITICAL GROWTH∗

This paper concernes with the existence and multiplicity of solutions for p(x)-Laplacian differential inclusions involving critical growth. The main tools are the nonsmooth analysis and variational methods. Our main results generalize some recent results in the literature into nonsmooth cases.


Introduction
The purpose of this paper is to deal with the following quasilinear problem involving variable critical growth with a nonsmooth potential −div(|∇u| p(x)−2 ∇u) ∈ λ|u| q(x)−2 + ∂F (x, u) for a.e. x ∈ Ω, u ∂Ω = 0, (P λ ) where Ω ⊂ R N is a bounded domain in R N , λ is a positive parameter and p, q : Ω → R are Lipschitz continuous functions and satisfy The study of variational problems and differential equations involving variable exponent conditions has been a very important and interesting topic. These problems are very useful in applications and lead many interesting mathematical problems. For example, p(x)-Laplacian problems can be found in the thermistor problem [40], the problem of electro-rheological fluid [36], or the problem of image recovery [6]. Of course, p(x)-Laplacian problems possess more complicated nonlinearities than p-Laplacian (a constant) problems, for instance, it is inhomogeneous and in general, it does not have the first eigenvalue. In other words, the infimum equals 0 (see [32]). Some related results can be found in [11,24,25,27] and references therein. Moreover the compact embedding theorem of the Lebesgue-Sobolev space W 1,p(x) (Ω) has more strict requirements.
If λ = 0, then problem (P λ ) becomes into the following form: u| ∂Ω = 0. (1.1) Problem (1.1) has been studied by several authors and obtained a few interesting results. For example, Dai and Liu [10] obtained the existence of three solutions for problem (1.1) by a nonsmooth version of three critical points theorem with ∂F (x, u) replaced by λ∂F (x, u). Qian and Shen [35], using the theory of nonsmooth critical points theory, derived the existence and multiplicity of solutions for problem (1.1). Ge et al. [15], employing variational methods combined with suitable truncation techniques based on nonsmooth critical points theory for locally Lipschitz functional, proved the existence of at least five solutions for problem (1.1) under suitable conditions. It is well known that when p(x) = p (a constant) p-Laplacian differential inclusion has been widely studied by lots of authors. Some related results can be found in [3, 7, 8, 14, 16-19, 29-31, 37, 39] and references therein. However, all the above results did not consider the critical growth of problem (1.1). Very little is known about critical growth nonlinearities for variable exponent problems with nonsmooth potentials. Motivated by this fact, we will consider problem (P λ ) involving critical Sobolev exponent and study the existence and multiplicity of solutions for (P λ ). Compared with the previous works, The critical case brings some new difficulties. In particular, there is no compact embedding W 1,p(x) (Ω) → L p * (x) (Ω). Then, it is not clear that the energy functional associated with (P λ ) satisfies the nonsmooth C-condition. To deal with this difficulty, we will employ a version of the concentration compactness lemma due to Lions for variable exponents found in Bonder and Silva [4] to overcome it. Furthermore, because of the non-differentiability of F , it is very important to find an efficient method to deal with problem (P λ ). In this paper our method relies on the theory of hemivariational inequalities [32][33][34] and differential inclusions (which involves the generalized gradient of a given locally Lipschitz functional).
In order to introduce our main results, we give our hypotheses on the nonsmooth potential function F (x, u).
for a.e. x ∈ Ω and all u ∈ R.
Here F 0 (x, u; v) denotes the partial generalized directional derivative of F (x, ·) at the point u ∈ R in the direction v ∈ R(see Section 2).
The main results are the following:  [1,38] for p(x)-Laplacian type problem with critical growth into nonsmooth cases. This means that our conditions are more wider than those in [1,38] and suit more practical applications.

Remark 1.2.
In this paper, we apply the concentration compactness principle in [4], which is slightly more general than those in [13] as we do not demand q(x) to be critical everywhere.  n |u| n − 2 n n + 2 m m , |u| > 2, where m, n ∈ (p + , q − ) and p + < α < min{m, n}. Then this function is locally Lipschitz and non-differentiable, and it satisfies hypothesis (H F ).
This paper is organized as follows: in Section 2, some necessary preliminary knowledge is presented. In Section 3, we prove our main results.

Preliminaries
We firstly give some basic notations and some definitions.
• means weak convergence while → means strong convergence.
• c, C, c i and C i (i = 1, 2, · · · ) denote estimated constants (the exact value may be different from line to line). o n denotes a sequence whose limit is 0 as n → ∞.

Definition 2.1 ( [21]
). A function I: X → R is locally Lipschitz if for every u ∈ X there exist a neighborhood U of u and L > 0 such that for every ν, η ∈ U , Definition 2.2 ( [21]). Let I : X → R be a locally Lipschitz functional. The generalized derivative of I in u along the direction ν is defined by It is easy to see that the function ν → I 0 (u; ν) is sublinear, continuous and so is the support function of a nonempty, convex and w * -compact set ∂I(u) ⊂ X * , defined by Clearly, these definitions extend those of the Gâteaux directional derivative and gradient.

Definition 2.3 ( [21]
). We say that I satisfies the nonsmooth C-condition if every sequence {u n } ⊂ X satisfying has a strongly convergent subsequence, where m(u n ) = inf u * n ∈∂I(un) u * n X * . Definition 2.4. We say that u ∈ W 1,p 0 (Ω) is a weak solution of problem (P λ ), if for all v ∈ W 1,p 0 (Ω) the following hemivariational inequality is satisfied

function. Then h • j is locally Lipschitz and
Denote by S(Ω) the set of all measurable real functions defined on Ω. For any p ∈ C + (Ω) we define the variable exponent Lebesgue space by (Ω) as the closure of C ∞ 0 (Ω) in W 1,p(x) (Ω). We point out that when Ω is bounded, |∇u| p(x) is an equivalent norm on W 1,p(x) 0 (Ω). The following Hölder type inequality is very useful in the next section.
From Hölder inequality, we can easily obtain the following proposition: The following lemma is a variable exponent case of Brézis-Lieb Lemma.
Now, we give our main tools used in this paper.

Theorem 2.1 ( [22]).
Assume that X is a reflexive Banach space and ϕ : X → R is locally Lipschitz and satisfies the nonsmooth C-condition. Assume further that there exist u 1 ∈ X and r > 0 such that u 1 > r and Then ϕ has a nontrivial critical point u ∈ X such that ϕ(u) ≥ inf{ϕ(v) : v = r}.

Theorem 2.2 ( [20]).
Assume that X is a reflexive Banach space and ϕ : X → R is even locally Lipschitz and satisfies the nonsmooth C-condition and also (i) ϕ(0) = 0; (ii) There exists a subspace Y ⊆ X of finite codimension and numbers β, γ > 0, Then ϕ has at least dim V -codim Y pairs of nontrivial critical points.

Main results
(Ω). Since X is a reflexive and separable Banach space, there exist e j ⊂ X and e * j ⊂ X * such that X = span{e j |j = 1, 2, · · · }, X * = span{e * j |j = 1, 2, · · · }, and e * i , e j = 1, i = j, We define the function I on X by In order to prove our results, we need the following lemmas.
Proof. By virtue of hypothesis (H F )(iii), for given > 0, we can find for a.e. x ∈ Ω, all |u| ≥ M 1 and ω(x) ∈ ∂F (x, u). By Lebegue's mean value theorem, we obtain that From Hölder's inequality and the embedding theorem, we derive Hence F (u) is locally Lipschitz. Then I(u) is locally Lipschitz. Similar as that in [26, Lemma 2.1], we can obtain that F 0 (x, u; u) ≤ Ω F 0 (x, u; u)dx. Thus the proof is completed. □

Lemma 3.2. Assume that F satisfies hypotheses (H F )(i) − (iii), then every critical point of u 0 ∈ X of I λ is a weak solution of problem (P λ ).
Proof. Since u ∈ X is a critical point of I λ , for every v ∈ X, Lemma 3.1 gives i.e., u is a weak solution of problem (P λ ). □

Lemma 3.3. If hypothesis (H F ) hold, then any nonsmooth C-condition sequence of I λ is bounded in X.
Proof. Let {u n } n≥1 ⊂ X be a sequence such that I λ (u n ) → c, and (1+ u n )m(u n ) → 0 as n → +∞. Let A : X → X * be the nonlinear operator defined by From [23] we know that A is maximal monotone and where ω n ∈ ∂F (x, u n ) and u * n ∈ ∂I λ (u n ) for n ≥ 1.
as n → +∞. We claim that the sequence {u n } is bounded. Indeed, by virtue of (3.2) and (3.3), we derive that for n sufficiently large. Using Lemma 3.1, the above estimation and (H F )(iv), we obtain that Once that u n > 1, it follows from Proposition 2.1 that Since p − > 1, the above inequality means that {u n } is bounded in X. □ As a consequence of the last result, if {u n } is a nonsmooth C-condition of I λ , we can extract a subsequence of {u n }, still denoted by {u n }, and u ∈ X such that u n u in X, u n u in L q(x) (Ω), By virtue of the concentration compactness lemma of Lions for variable exponents in [4], there exist two nonnegative measures µ, ν ∈ Λ(Ω), a countable set idex set E, points {x j } i∈E in D and sequences {µ j } j∈E , {ν j } j∈E ⊂ [0, +∞), such that and In the following, we will prove an important estimate for {ν j }. With this aim in mind, we have to prove a technical lemma. Let φ ∈ C ∞ 0 (Ω) such that

Lemma 3.4. For each y ∈Ω and u ∈ L p(x) (Ω)
, where C is a constant independent of and y.

Proof. Observe that
.

Making a change of variable, we derive
Then (3.5) is satisfied. □ Proof. For ∀ > 0, we set φ ϵ ∈ C ∞ 0 (Ω) as in Lemma 3.4. Then {φ ϵ (· − x j )u n } ⊂ X for any j ∈ E. After a direct computation, we derive that (3.7) Passing to the limit of n → +∞ in (3.7), we have (3.9) Furthermore, it follows from hypothesis (H F )(iii), for ∀ > 0, where C ϵ > 0 and ω(x, u n ) ∈ ∂F (x, u n ). Then (3.10) On the other hand, from the compactness lemma of Strauss [9] lim n→∞ Ω Noting that {u n } is bounded in X, from the Sobolev embedding theorem, setting → 0, we have where C is a constant independent of and j. Recall that Consequently, or ν j = 0. □ Lemma 3.6. If hypothesis (H F ) holds and λ < 1, then I λ satisfies the nonsmooth C-condition for c < λ

Proof.
Since By virtue of hypothesis (H F )(iv), we have Noting that if ν s > 0 for some s ∈ E, we infer that which is a contradiction. Then, we must have ν j = 0 for any j ∈ E, leading to From the above equation we derive Ω |u n − u| q(x) dx → 0 as n → ∞.
Then u n → u in L q(x) (Ω). (3.14) Since u * n , u n = o n (1), we obtain In the following, let us denote by {P n } the following sequence The definition of {P n } means that Since u n u in W 1,p(x) 0 (Ω), we obtain Ω |∇u n | p(x)−2 ∇u n ∇(u n − u)dx → 0 as n → ∞, which means that Furthermore, from u * n , u n = o n (1), we have where u * n ∈ ∂I λ (u n ) and ω(x, u n ) ∈ ∂F (x, u n ). Combining (3.14) with the compactness lemma of Strauss [9], we infer that Ω P n dx → 0 as n → ∞.
Next, let us discuss the sets
By Lemma 3.6, the nonsmooth C-condition is fulfilled. It follows from Theorem 2.1 we obtain that I λ has at least one nontrivial critical pointû ∈ X, i.e., a nontrivial solution of problem (P λ ). □ Proof of Theorem 1.2. We claim that I λ (u) → −∞ as u → +∞, for any u ∈ Y k . Assume that u > 1. By virtue of (3.17), setting 0 < < λ, we derive Noting that Y k is a finite dimensional space, then all norms in Y k are equivalent. Since p + < q − , we obtain that I λ (u) → −∞ as u → +∞. Recalling that I λ (0) = 0 and I λ is even with V = Y k (dim Y k = k) and Y = X (codim Y = 0), from Lemma 3.6 and Claim 1 in Theorem 1.1, we infer that I λ has k-pairs of nontrivial solutions for problem (P λ ). □