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

## TL;DR

This paper classifies solutions to critical anisotropic p-Laplacian equations within convex cones and establishes related anisotropic Sobolev inequalities, extending symmetry results beyond the whole space using new methods.

## Contribution

It introduces a novel approach to classify solutions in anisotropic settings and proves general anisotropic Sobolev inequalities in convex cones.

## Key findings

- Complete classification of solutions in convex cones
- Extension of symmetry results to anisotropic domains
- Establishment of new anisotropic Sobolev inequalities

## 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.