PDE’s involving variable exponents

Abstract

We discuss some aspects regarding the eigenvalue problem $-{\Delta }_{p\left(x\right)}u=\lambda |u{|}^{p\left(x\right)-2}u$ if $x\in \Omega$, $u=0$ if $x\in \partial \Omega$, where $\Omega \subset R^N$ is a bounded domain, $p:\bar{\Omega }\to \left(1,\infty \right)$ is a continuous function and ${\Delta }_{p\left(x\right)}u:=\nabla \cdot \left(|\nabla u{|}^{p\left(x\right)-2}\nabla u\right)$ stands for the $p\left(x\right)$-Laplace operator. Let $\Lambda$ be the set of eigenvalues of the above problem and ${\lambda }_{*}=inf\Lambda$. In particular, we will emphasize, on the one hand, situations when ${\lambda }_{*}$ vanishes, and, on the other hand, we will advance some sufficient conditions when ${\lambda }_{*}$ is positive. In the case when $p\in C^1\left(\Omega \right)$ some extensions will be presented. In a related context some connections with a maximum principle will be pointed out.