# Symmetry results for critical anisotropic $p$-Laplacian equations in convex cones

created by figalli on 24 Feb 2020
modified by roncoroni on 05 Nov 2020

[BibTeX]

Accepted Paper

Inserted: 24 feb 2020
Last Updated: 5 nov 2020

Journal: Geometric and Functional Analysis
Year: 2020

ArXiv: 1906.00622 PDF

Abstract:

Given $n \geq 2$ and $1<p<n$, we consider the critical $p$-Laplacian equation $\Delta_p u + u^{p^*-1}=0$, which corresponds to critical points of the Sobolev inequality. Exploiting the moving planes method, it has been recently shown that positive solutions in the whole space are classified. Since the moving plane method strongly relies on the symmetries of the equation and the domain, in this paper we provide a new approach to this Liouville-type problem that allows us to give a complete classification of solutions in an anisotropic setting. More precisely, we characterize solutions to the critical $p$-Laplacian equation induced by a smooth norm inside any convex cone. In addition, using optimal transport, we prove a general class of (weighted) anisotropic Sobolev inequalities inside arbitrary convex cones.