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

Abstract:

We consider the nonlinear eigenvalue problem \[ -\textrm {div}\left (|\nabla u|^{p(x)-2}\nabla u\right )=\lambda |u|^{q(x)-2}u\] in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a bounded open set in $\mathbb R^N$ 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.