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 $$ \left\{ \begin{array} [c]{ll} -Δ_{p}u=λm(x)|u|^{p-2}u & \text{in }Ω,\\ u=0 & \text{on }\partialΩ, \end{array} \right. $$ where ${\Delta }_{p}u$ is the usual $p$-Laplacian. Our purpose in this article is to study the limit as $p\to \infty$ 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 $k=1$.