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

In this paper, we study the following nonlinear Neumann boundary value problem -div(vertical bar del u vertical bar(p(x)-2)del u) + vertical bar u vertical bar(p(x)-2)u = lambda f(x, u), x is an element of Omega, t is an element of R partial derivative u/partial derivative v = 0, x is an element of partial derivative Omega, t is an element of R where Omega subset of R-n is a bounded domain with smooth boundary d Omega, partial derivative u/partial derivative v is the outer unit normal derivative on partial derivative Omega, lambda > 0 is a real number, p is a continuous function on sy with inf(x is an element of(Omega) over bar)p(x) > 1, f : Omega x R -> R is a continuous function. Using the three critical point theorem due to Ricceri, under the appropriate assumptions on f, we establish the existence of at least three solutions of this problem. Some known results are generalized. (c) 2009 Elsevier Ltd. All rights reserved.