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

We consider the nonlinear eigenvalue problem $-{\rm div}\left(|\nabla u|^{p(x)-2}\nabla u\right)=\lambda |u|^{q(x)-2}u$ 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 $\overline\Omega$ such that $1<\inf_\Omega q< \inf_\Omega p<\sup_\Omega q$, $\sup_\Omega p