On the critical $p$-Laplace equation

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 ob...

Full description

Saved in:
Bibliographic Details
Main Authors Catino, Giovanni, Monticelli, Dario Daniele, Roncoroni, Alberto
Format Journal Article
LanguageEnglish
Published 14.04.2022
Subjects
Mathematics - Analysis of PDEs
Mathematics - Differential Geometry
Online AccessGet full text

Cover

More Information
Summary: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.
DOI:10.48550/arxiv.2204.06940