$p$-Laplacian operator with potential in generalized Morrey Spaces

TL;DR

This paper investigates properties of generalized Morrey spaces and analyzes a $p$-Laplacian type PDE with potential in these spaces, establishing unique continuation results for the operator.

Contribution

It introduces and studies the properties of generalized Morrey spaces and proves strong unique continuation for the $p$-Laplacian with potential in these spaces.

Findings

Properties of generalized Morrey spaces are characterized.

Existence and behavior of solutions to the PDE with potential in Morrey spaces are analyzed.

Strong unique continuation property is established for the $p$-Laplacian with potential.

Abstract

We study some basic properties of generalized Morrey spaces $\mathcal{M}^{p,\phi}(\R^{d})$. Also, the problem $-\mbox{div}(|\nabla u|^{p-2}\nabla u)+V|u|^{p-2}u=0$ in $\Omega$, where $\Omega$ is a bounded open set in $\R^d$, and potential $V$ is assumed to be not equivalent to zero and lies in $\mathcal{M}^{p,\phi}(\Omega)$, is studied. Finally, we establish the strong unique continuation for the $p$-Laplace operator in the case $V\in\mathcal{M}^{p,\phi}(\R^d)$.