Existence and Multiplicity of Solutions for a p(x)-biharmonic Problem with Neumann Boundary Conditions

In this paper, we study the following problem with Neumann boundary conditions {Delta(2)(p(x)) + alpha vertical bar u vertical bar(p(x) - 2) u = beta f (x,u) in Omega, partial derivative u/partial derivative v = partial derivative.partial derivative v (vertical bar Delta u vertical bar(p(x) - 2) Delta u) = 0 on partial derivative Omega Where Omega is a bounded domain in R-N with smooth boundary partial derivative Omega, N >= 1, Delta(2)(p(x))u := Delta (vertical bar Delta u vertical bar(p(x) - 2) Delta u), is the p(x)-biharmonic operator, alpha and beta are two positives reals numbers, p is a continuous function on (Omega) over bar with inf(x is an element of(Omega) over bar) p(x) > 1 and f : Omega x R -> R is a Caratheodory function such that f (x, 0) = 0. Using the three critical point Theorem, we establish the existence of at least three solutions of this problem.