Extremals in nonlinear potential theory

TL;DR

This paper characterizes solutions to a nonlinear PDE involving the p-Laplacian as extremals of a generalized Morrey inequality, providing insights into potential theory for signed measures.

Contribution

It introduces a novel characterization of PDE solutions as extremals of a generalized Morrey inequality for measures, extending potential theory.

Findings

Solutions are characterized as extremals of a generalized Morrey inequality.

The characterization applies to PDEs with signed Borel measures on .

The results connect nonlinear potential theory with variational extremals.

Abstract

We consider the PDE $-\Delta_pu=\rho$, where $\rho$ is a signed Borel measure on $\mathbb{R}^n$. For each $p>n$, we characterize solutions as extremals of a generalized Morrey inequality determined by $\rho$.