On the eigenvalue problem involving the weighted $p$-Laplacian in radially symmetric domains

TL;DR

This paper studies the eigenvalue problem involving the weighted p-Laplacian in radially symmetric domains, establishing existence, regularity, positivity, and asymptotic behavior of eigenfunctions with sign-changing weights.

Contribution

It proves the existence of the first eigenpair and analyzes the regularity, positivity, and asymptotic estimates of eigenfunctions in weighted p-Laplacian problems.

Findings

Existence of the first eigenpair is established.

Eigenfunctions are shown to be regular and positive under certain conditions.

Asymptotic behavior of eigenfunctions and their gradients near domain boundaries is characterized.

Abstract

We investigate the following eigenvalue problem \begin{align*} \begin{cases} -\operatorname{div}\left( L(x) |\nabla u| ^{p-2}\nabla u\right)=\lambda K(x)|u|^{p-2}u \quad \text{in } A_{R_1}^{R_2} , u=0\quad \text{on } \partial A_{R_1}^{R_2} , \end{cases} \end{align*} where $A_{R_1}^{R_2}:=\{x\in\mathbb{R}^N: R_1<|x|<R_2\}$ $(0< R_1<R_2\leq\infty)$, $\lambda>0$ is a parameter, the weights $L$ and $K$ are measurable with $L$ positive a.e. in $A_{R_1}^{R_2}$ and $K$ possibly sign-changing in $A_{R_1}^{R_2}$. We prove the existence of the first eigenpair and discuss the regularity and positiveness of eigenfunctions. %apriori bounds of any eigenfunction as well as local boundedness. The asymptotic estimates for $u(x)$ and $\nabla u(x)$ as $|x|\to R_1^+$ or $R_2^-$ are also investigated.