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

We consider the nonlinear eigenvalue problem $-{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=\lambda |u|^{q(x)-2}u$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in $\RR^N$ with smooth boundary and $p$, $q$ are continuous functions on $\bar\Omega$ such that $1<\inf\_\Omega q< \inf\_\Omega p<\sup\_\Omega q$, $\sup\_\Omega p<N$, and $q(x)0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.