Multiple solutions for inequality Dirichlet problems by the p(x)-Laplacian

In this paper we study the nonlinear elliptic problem driven by p(x)-Laplacian with a nonsmooth locally Lipschitz potential (hemivariational inequality), that is (P){−div(‖∇u(x)‖RNp(x)−2∇u(x))∈∂j(x,u(x)),a.a. x∈Ω,u=0,on ∂Ω, where Ω⊂RN is a bounded domain and p:Ω¯→R is a continuous function satisfying some given assumptions. The approach used in this paper is the variational method for locally Lipschitz functions. More precisely, Weierstrass Theorem and Mountain Pass Theorem are used to prove the existence of at least two nontrivial solutions. Finally, we obtain the existence of at least two nontrivial solutions of constant sign.