On the critical $p$-Laplace equation

TL;DR

This paper classifies positive solutions to the critical p-Laplace equation on Euclidean space, extending previous results with weaker assumptions and discussing potential Riemannian generalizations.

Contribution

It provides a comprehensive classification of solutions to the critical p-Laplace equation under minimal assumptions, including cases with infinite energy and extensions to Riemannian manifolds.

Findings

Complete classification for n=2 or 3 with certain p ranges

Rigidity results without additional assumptions in specific dimensions

Extension of classification results to Riemannian settings

Abstract

In this paper we provide the classification of positive solutions to the critical $p-$Laplace equation on $\mathbb{R}^n$, for $1<p<n$, possibly having infinite energy. If $n=2$, or if $n=3$ and $\frac 32<p<2$ we prove rigidity without any further assumptions. In the remaining cases we obtain the classification under energy growth conditions or suitable control of the solutions at infinity. Our assumptions are much weaker than those already appearing in the literature. We also discuss the extension of the results to the Riemannian setting.