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

We consider the nonlinear eigenvalue problem

$-{\rm div}\left(\vert\nabla u\vert^{p(x)-2}\nabla u\right)=\lambda \vert u\vert^{q(x)-2}u$

in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded open set in $\mathbb{R}^N$ with smooth boundary and $p$, $q$ are continuous functions on $\overline\Omega$ such that $1<\inf_\Omega q< \inf_\Omega p<\sup_\Omega q$, $\sup_\Omega p<N$, and $q(x)<Np(x)/\left(N-p(x)\right)$ for all $x\in\overline\Omega$. 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.