EIGENVALUES FOR A FOURTH ORDER ELLIPTIC PROBLEM

We study the fourth order nonlinear eigenvalue problem with a p(x)-biharmonic operator {Delta(2)(p(x))u + a(x)vertical bar u vertical bar p((x)-2)u = lambda w(x)f(u) in Omega, u = Delta u = 0 on partial derivative Omega where Omega is a smooth bounded domain in R-N, p is an element of C((Omega) over bar) with p(x) > 1 on (Omega) over barO, Delta(2)(p(x)) u - Delta(vertical bar Delta u vertical bar(p(x)-2 Delta)u) is the p(x)-biharmonic operator, and lambda > 0 is a parameter. Under some appropriate conditions on the functions p, a, w, f, we prove that there exists (lambda) over bar > 0 such that any lambda is an element of (0,(lambda) over bar) is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland's variational principle and some recent theory on the generalized Lebesgue-Sobolev spaces L-p(x)(Omega) and W-k,W-p(x)(Omega).