Existence and multiplicity of solutions for a quasilinear equation involving the p(x)-Laplace operator

The purpose of this paper is to study the following nonlinear problem involving the p(x)-Laplace operator: (P-lambda){-Delta(u)(p(x)) = lambda f(x, u) in Omega, u > 0 in Omega, u = 0, on partial derivative Tau. where Omega subset of R-N, (N >= 2) is a bounded domain with C-2 boundary, lambda is a positive parameter, p(x), and f (x, u) are assumed to satisfy the assumptions (H1)-(H4) in the introduction. We employ variational techniques in order to show the existence of a number Lambda is an element of (0,infinity) such that problem (P lambda) admits two solutions for lambda is an element of (0, Lambda), one solution for lambda = Lambda, and no solutions for lambda > Lambda.