# On the critical p-Laplace equation

created by catino on 16 Apr 2022
modified on 06 May 2022

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Submitted Paper

Inserted: 16 apr 2022
Last Updated: 6 may 2022

Year: 2022

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.