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

Abstract

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