Three Solutions for Inequalities Dirichlet Problem Driven by p(x)-Laplacian-Like

and Applied Analysis 3 Proposition 3 (see [19]). Set X = W 0 (Ω); A is as shown, then (1) A : X → X is a convex, bounded previously; and strictly monotone operator; (2) A : X → X is a mapping of type (S)+; that is, un w 󳨀→ u in X and lim sup n→∞ ⟨A(un), un − u⟩ ≤ 0 implies un → u inX; (3) A : X → X is a homeomorphism. Let (X, ‖ ⋅ ‖) be a real Banach space, and let X be its topological dual. A function f : X → R is called locally Lipschitz if each point u ∈ X possesses a neighborhood Ωu such that |f(u1) − f(u2)| ≤ L ‖u1 − u2‖ for all u1, u2 ∈ Ωu, for a positive constant L depending on Ωu. The generalized directional derivative of f at the point u ∈ X in the direction h ∈ X is f 0 (u; h) = lim sup V→u; t↓0 f (V + th) − f (V)


Introduction
Since many free boundary problems and obstacle problems may be reduced to partial differential equations with discontinuous nonlinearities, the existence of multiple solutions of the problems with discontinuous nonlinearities has been widely investigated in recent years.In 1981, Chang [1] extended the variational methods to a class of nondifferentiable functionals and directly applied the variational methods for nondifferentiable functionals to prove some existence theorems for PDE with discontinuous nonlinearities.Soon thereafter, Kourogenis and Papageorgiou [2] extend the nonsmooth critical point theory of Chang [1], by replacing the compactness and the boundary conditions.In [3], by using the Ekeland variational principle and a deformation theorem, Kandilakis et al. obtained the local linking theorem for locally Lipschitz functions.In the celebrated work [4], Ricceri elaborated a Ricceritype variational principle for Gateaux differentiable functionals.Later, Marano and Motreanu [5] extended Ricceri's result to a large class of nondifferentiable functionals and gave an application to a Neumann-type problem involving the -Laplacian with discontinuous nonlinearities.
The study of differential equations and variational problems with variable exponent has been a new and interesting topic.It arises from nonlinear elasticity theory, electrorheological fluids, and so forth (see [6,7]).The study on variable exponent problems attracts more and more interest in recent years.Many results have been obtained on this kind of problems, for example, [8][9][10][11][12][13][14].Neumann-type problems involving the ()-Laplacian have been studied, for instance, in [15][16][17][18].
Recently, Rodrigues [19] has considered the existence of nontrivial solution for the Dirichlet problem involving the ()-Laplacian-like of the type where Ω ⊂ R  is a bounded domain with smooth boundary Ω,  ∈ (Ω) with () > 2, for all  ∈ Ω, and  : Ω × R → R satisfies the Caratheodory condition.We emphasize that, in our approach, no continuity hypothesis will be required for the function  with respect to the second argument.So, (P) need not have a solution.
The aim of the present paper is to establish a threesolution theorem for the nonlinear elliptic problem driven by ()-Laplacian-like with nonsmooth potential (see Theorem 6) by using a consequence (see Theorem 4) of the three-critical-point theorem established firstly by Marano and Motreanu in [20], which is a non-smooth version of Ricceri's three-critical-point theorem (see [21]).The paper is organized as follows.In Section 2, we present some necessary preliminary knowledge on variable exponent Sobolev spaces and the generalized gradient of the locally Lipschitz function.In Section 3, we give the main result of this paper and use the non-smooth three-critical-point theorem to prove it.

Preliminary
In order to discuss problem (P), we need some theories on  1,() 0 (Ω) and the generalized gradient of the locally Lipschitz function.Firstly we state some basic properties of space  1,() 0 (Ω) which will be used later (for details, see [10][11][12]).Denote by (Ω) the set of all measurable real functions defined on Ω.Two functions in (Ω) are considered as the same element of (Ω) when they are equal almost everywhere.
We will use the equivalent norm in the following discussion and write ‖‖ = |∇| () for simplicity.
Consider the following function: We know that (see [1]).
For further details, we refer the reader to the work of Chang [1].
Finally, for proving our results in the next section, we introduce the following theorem.

Existence Results
In this part, we will prove that there exist three solutions for problem (P) under certain conditions.

(Ω).
We know that the critical points of  are just the weak solutions of (P).
We consider a non-smooth potential function  : Ω × R → R such that (, 0) = 0 a.e. on Ω satisfying the following conditions:
Step 1.We show that  is coercive.
Since Ω ⊆ R  is bounded, Ω is compact.