PDE’s involving variable exponents

Mihai Mihăilescu
University of Craiova, Romania
3rd September 2012 2-3pm, AGR Carslaw 829

Abstract

We discuss some aspects regarding the eigenvalue problem \(-\Delta_{p(x)}u=\lambda|u|^{p(x)-2}u\) if \(x\in\Omega\), \(u=0\) if \(x\in\partial\Omega\), where \(\Omega\subset\mathbf R^N\) is a bounded domain, \(p\colon\bar\Omega\rightarrow(1,\infty)\) is a continuous function and \(\Delta_{p(x)}u:=\nabla\cdot\bigl(|\nabla u|^{p(x)-2}\nabla u\bigr)\) stands for the \(p(x)\)-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 \(p\in C^1(\Omega)\) some extensions will be presented. In a related context some connections with a maximum principle will be pointed out.

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