On a p(x)-biharmonic problem with Navier boundary condition

In this paper, we study a p(x)-biharmonic equation with Navier boundary condition {Delta(2)(p(x))u + a(x)vertical bar u vertical bar(p(x)-2)u = lambda f(x, u) + mu g(x,u) in Omega, u = Delta u = 0 on partial derivative Omega. Here Omega subset of R-N (N >= 1) is a bounded domain with smooth boundary partial derivative Omega, Delta(2)(p(x))( )u is a p(x)-biharmonic operator with p(x) is an element of subset of ((Omega) over bar, p(x) > 1. lambda, mu, is an element of R, a is an element of L-infinity(Omega) such that inf(x is an element of Omega) a(x) = a(-) > 0. By variational methods, we establish the results of existence and non-existence of solutions.( )