On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent

We consider the nonlinear eigenvalue problem - div (vertical bar del u|(p(x)-2)del u) = lambda|u|(q(x)-2) u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded open set in R-N with smooth boundary and p, q are continuous functions on (Omega) over bar such that 1 < inf(Omega) q < inf (Omega) p < sup(Omega) q, sup(Omega) p < N, and q(x) < Np(x)/(N-p(x)) for all x is an element of (Omega) over bar. The main result of this paper establishes that any lambda > 0 sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.