Principal eigenvalues for the weighted p-Laplacian and antimaximum principle in ${\mathbb{R}}^N$

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 $\lambda \in \mathbb{R}$, $p>1$, $K:{\mathbb{R}}^N\to \mathbb{R}$, $L:{\mathbb{R}}^N\to {\mathbb{R}}^{+}$ 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.

14 pages