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

TL;DR

This paper investigates the asymptotic behavior of eigenvalues and eigenfunctions of a $p$-Laplacian problem with a sign-changing weight as $p$ approaches infinity, revealing new insights into the limit problem.

Contribution

It provides a detailed analysis of the limit of eigenvalues and eigenfunctions for the $p$-Laplacian with sign-changing weights as $p$ tends to infinity.

Findings

Eigenvalues converge to a limit as p→∞.

Normalized eigenfunctions have well-defined limits.

Results extend understanding of $p$-Laplacian eigenproblems with indefinite weights.

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 $$ \left\{ \begin{array} [c]{ll} -\Delta_{p}u=\lambda m(x)|u|^{p-2}u & \text{in }\Omega,\\ u=0 & \text{on }\partial\Omega, \end{array} \right. $$ 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 $k=1$.