Multiple Solutions to a Class of Inclusion Problems with Operator Involving P(x)-laplacian

In this paper, we prove the existence of at least two nontrivial solutions for a non-linear elliptic problem involving p(x)-Laplacian-like operator and nonsmooth potentials. Our approach is variational and it is based on the nonsmooth critical point theory for locally Lipschitz functions. In this paper we are concerned with the following Dirichlet-type differential inclusion problems    −div (1 + |∇u| p(x) √ 1+|∇u| 2p(x))|∇u| p(x)−2 ∇u ∈ λ∂F (x, u), a.e. in Ω, u = 0, on ∂Ω, (P) where Ω ⊆ R N is a bounded domain, λ > 0 is a real number, p(x) ∈ C(Ω), 1 < p − ≤ p(x) < +∞ and F : Ω × R → R is a locally Lipschitz with respect to the second variable (in general it can be nonsmooth), and ∂F (x, t) is the subdifferential with respect to the t-variable in the sense of Clarke [1]. Parabolic and elliptic problems with variable exponents have attracted in recent years a lot of interest of mathematicians around the world. For example, [2–14] and the references therein. The wide study of such kind of problems is motivated by various applications related *

In a recent paper [20], by using the nonsmooth three critical points theorem and assuming suitable conditions for nonsmooth potential F , we proved the existence of three solutions of (P ).In this paper our goal is to prove the existence of at least two solutions for the problem (P ) as the parameter λ > λ 0 for some constant λ 0 .
Next, we assume that F (x, t) satisfies the following general conditions: (f 1 ) |w| ≤ c 1 + c 2 |t| α(x)−1 , for almost all x ∈ Ω, all t ∈ R and w ∈ ∂F (x, t); uniformly for almost all x ∈ Ω and all w ∈ ∂F (x, t); The paper is organized as follows.We first introduce some basic preliminary results and a well-known lemma in Section 2, including the variable exponent Lebesgue and Sobolev spaces.In Section 3, we give the main result and its proof.In Section 4, we give the summary of this paper.§2 Preliminaries In this part, we introduce some definitions and results which will be used in the next section.
Denote by U (Ω) the set of all measurable real functions defined on Ω.Two functions in U(Ω) are considered to be one element of U(Ω), when they are equal almost everywhere.
Hereafter, let ) ≥ N. We remember that the variable exponent Lebesgue spaces are separable and reflexive Banach spaces.Denote by L q(x) (Ω) the conjugate Lebesgue space of L p(x) (Ω) with 1 p(x) + 1 q(x) = 1, then the Hölder type inequality then the following relations hold . Proposition 2.1 [21] If q ∈ C + (Ω) and q(x) < p * (x) for any x ∈ Ω, then the embedding from W 1,p(x) (Ω) to L q(x) (Ω) is compact and continuous.
We know that (see [1]), J ∈ C 1 (W 1,p(x) 0 (Ω), R).If we denote A= J : W 1,p(x) 0 (Ω) → (W 1,p(x) 0 (Ω)) * , then (Ω), A is as above, then (1) A : X → X * is a convex, bounded and strictly monotone operator; (2) A : X → X * is a mapping of type (S) + , i.e., u n w → u in X and lim sup n→∞ Let X be a Banach space and X * be its topological dual space and we denote •, • as the duality bracket for 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 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 If ϕ is also convex, then ∂ϕ(x) coincides with subdifferential in the sense of convex analysis, defined by A locally Lipschitz function ϕ : If this condition is satisfied at every level c ∈ R, then we say that ϕ satisfies the nonsmooth C-condition.
Finally, in order to prove our result in the next section, we introduce the following lemma: Lemma 2.1 [25] Let ϕ : X → R be locally Lipschitz function and The main results and proof of the theorem In this part, we will prove that for (P ) there also exist two weak solutions for the general case.
Our hypotheses on nonsmooth potential F (x, t) are as follows.H(F): F : Ω × R → R is a function such that F (x, 0) = 0 a.e. on Ω and satisfies the following facts: (1) for all t ∈ R, x → F (x, t) is measurable; (2) for almost all x ∈ Ω, t → F (x, t) is locally Lipschitz.We consider the energy function ϕ : W 1,p(x) 0 (Ω) → R for the problem (P ), defined by Then we obtain that there exists a positive constant M , such that J (u) W −1,q(x) (Ω) ≤ M , for sufficiently small r.
Proof: The proof is divided into five steps as follows.
Step 1.We will show that ϕ is coercive in the step.
Firstly, on account of (f 1 ), we have for almost all x ∈ Ω and t ∈ R.
In view of condition (f 3 ), there exists
Step 4. We will check the C-condition in the following.
(Ω).Hence by passing to a subsequence if necessary, we may assume that u n u weakly in W 1,p(x) 0 (Ω).Next we will prove that So using Proposition 2.2, we have u n → u as n → ∞.Thus ϕ satisfies the nonsmooth Ccondition.
Step 5. We will show that there exists another nontrivial weak solution of problem (P ).
Remark 3.1.The result in this paper is different from the one in [20] since the assumption on the nonsmooth potential function F is different.In fact, our conditions (f 1 )-(f 3 ) are weaker than conditions (h 1 )-(h 3 ) in [20].For example, we can find a nonsmooth potential function satisfying the hypothesis of our Theorem 3.1.But the function does not satisfy conditions Theorem 3.1 of Zhou and Ge [20].For more details, please see (2) in the Summary.
So far the results involved potential functions exhibiting p(x)-sublinearity.The next theorem concerns problems where the potential function is p(x)-superlinear.Theorem 3.2.Let us suppose that H(F), (f 1 ), (f 2 ), (f 3 ) hold α − > p + and the following condition (f 4 ) hold, (f 4 ) For almost all x ∈ Ω and all t ∈ R, we have Then there exists a λ 0 > 0 such that for each λ > λ 0 , the problem (P ) has at least two nontrivial solutions.
Proof: The steps are similar to those of Theorem 3.1.In fact, we only need to modify Step 1 and Step 4 as follows: (1 ) Show that ϕ is coercive under the condition (f 4 ); (4 ) Show that there exists a second nontrivial solution under the conditions (f 1 ) and (f 2 ).Then from Steps (1 ), 2, 3 and (4 ) above, the problem (P ) has at least two nontrivial solutions.EJQTDE, 2013 No. 63, p. 8 Step 1 .Due to the assumption (f 4 ), for all u ∈ W 1,p(x) 0 (Ω), u > 1, we have Step 4 .By hypothesis (f 1 ) and the mean value theorem for locally Lipschitz functions, we have F (x, t) ≤c 1 |t| + c 2 |t| α(x) for a.e.x ∈ Ω and all t ∈ R.
So, u 1 is second nontrivial critical point of ϕ.

( 1 )−
If F : Ω × R → R satisfies the Carathéodory condition, then ∂F (x, t) = {f (x, t)}.Therefore by Theorem 3.1 we can show the existence of two weak solutions of the following Dirichlet problem involving the p(x)-Laplacian-like  div (1 + |∇u| p(x)