Principal eigenvalues for the weighted p-Laplacian and antimaximum principle in $\mathbb{R}^N$
Anumol Joseph, Abhishek Sarkar
TL;DR
The paper analyzes the weighted $p$-Laplacian on $\mathbb{R}^N$ with a sign-changing weight $K$ and a positive weight $L$, establishing the existence and positivity of a principal eigenvalue $\lambda_1$ with a corresponding eigenfunction in $\mathcal{D}_L^{1,p}(\mathbb{R}^N)$. It develops the regularity theory, proving $u\in L^{\infty}(\mathbb{R}^N)$ and, under suitable assumptions, $u\in C^{1,\alpha}_{loc}$; in the radial case with $p=N=2$, the eigenfunction is increasing and can be bounded given decay conditions. The asymptotic behavior of principal eigenfunctions is explored in the radial setting, showing monotone growth and boundedness under additional decay. Finally, the authors establish local and global antimaximum principles for a perturbed problem $-\operatorname{div}(L|\nabla u|^{p-2}\nabla u)=\lambda K|u|^{p-2}u+h$, demonstrating that near $\lambda_1$ the sign of solutions is constrained in bounded sets and, with compactly supported $h$, globally constrained, leveraging eigenvalue isolation and variational techniques.
Abstract
We study the existence of principal eigenvalues and principal eigenfunctions for weighted eigenvalue problems of the form: \begin{equation*} - \mbox{div} ( L (x) |\nabla u|^{p-2} \nabla u ) = λK(x) |u|^{p-2} u \hspace{.1cm} \mbox { in } \hspace{.1cm} \mathbb{R}^N , \end{equation*} where $λ\in \mathbb{R}$, $p>1$, $K : \mathbb{R}^N \rightarrow \mathbb{R}$, $L : \mathbb{R}^N \rightarrow \mathbb{R}^+$ are locally integrable functions. The weight function $K$ is allowed to change sign, provided it remains positive on a set of nonzero measure. We establish the existence, regularity, and asymptotic behavior of the principal eigenfunctions. We also prove local and global antimaximum principles for a perturbed version of the problem.