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

Abstract

We study some basic properties of generalized Morrey spaces ${\mathcal{M}}^{p,\varphi }({\mathbb{R}}^d)$. 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 ${\mathbb{R}}^d$, and potential $V$ is assumed to be not equivalent to zero and lies in ${\mathcal{M}}^{p,\varphi }(\Omega )$, is studied. Finally, we establish the strong unique continuation for the $p$-Laplace operator in the case $V\in {\mathcal{M}}^{p,\varphi }({\mathbb{R}}^d)$.