Existence of solutions for a p(x)-biharmonic problem under Neumann boundary conditions

In this paper, we consider for a given smooth bounded domain Omega of R-N, (N >= 3), the following p(x)-biharmonic type problem Delta(xi(x)vertical bar Delta u vertical bar(p(x)-2) Delta u) + a(x)vertical bar u vertical bar(p(x)-2)u = lambda partial derivative F/partial derivative u (x, u) in Omega partial derivative u/partial derivative n = 0 on partial derivative Omega partial derivative/partial derivative n (xi(x)vertical bar Delta u vertical bar(p(x)-2) Delta u) = 0 on partial derivative Omega, where lambda is a positive parameter, p is an element of C+((Omega) over bar) with p- := inf(x epsilon(Omega) over bar) p(x) > 1, xi is a function which satisfies the condition 0 < xi 1 <= xi(x) <= xi 2, a epsilon L-infinity (Omega) with essinf(x epsilon<(Omega)over bar>) a(x) > 0 and F : ((Omega) over bar) x R -> R is a C1 function. We prove the existence of a continuous family of eigenvalues in a neighbourhood of the origin, under some suitable conditions by using Ekeland's principle and variational method.