Mountain pass and Ekeland's principle for eigenvalue problem with variable exponent

In this article, we study the boundary value problem -div(vertical bar del u vertical bar(p(x)-2)del u) + vertical bar u vertical bar(alpha(x)-2)u=lambda vertical bar u vertical bar(q(x)-2)u, in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain in R-N and p, q, alpha are continuous functions on (Omega) over bar. We show that for any lambda > 0 there exists infinitely many weak solutions (respectively, if lambda > 0 and small enough, then there exists a non-negative, non-trivial weak solution). Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a Z(2) symmetric version for even functionals of the Mountain pass Theorem (respectively on simple variational arguments based on Ekeland's variational principle).