The $\infty$-eigenvalue problem with a sign-changing weight

Abstract

Let $\Omega \subset {\mathbb {R}}^{n}$ be a smooth bounded domain and $m\in C(\overline{\Omega })$ be a sign-changing weight function. For $1<p<\infty$, consider the eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{p}u=\lambda m(x)|u|^{p-2}u &{}\quad \text {in}\;\; \Omega ,\\ u=0 &{}\quad \text {on}\;\; \partial \Omega , \end{array} \right. \end{aligned}$$

where $\Delta _{p}u$ is the usual p-Laplacian. Our purpose in this article is to study the limit as $p\rightarrow \infty$ for the eigenvalues $\lambda _{k,p}\left( m\right)$ of the aforementioned problem. In addition, we describe the limit of some normalized associated eigenfunctions when $k=1$.