Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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G. Ciraolo - A. Figalli - A. Roncoroni

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

created by roncoroni on 17 Oct 2023


Published Paper

Inserted: 17 oct 2023
Last Updated: 17 oct 2023

Journal: Geom. Funct. Anal.
Year: 2020

ArXiv: 1906.00622 PDF


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.

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