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

• Published in: Geometric and functional analysis Vol. 30; no. 3; pp. 770 - 803
• Main Authors: Ciraolo, Giulio, Figalli, Alessio, Roncoroni, Alberto
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.06.2020; Springer Nature B.V
• Subjects: Analysis; Cones; Critical point; Laplace equation; Mathematics; Mathematics and Statistics; Quasilinear anisotropic elliptic equations; Convex cones; 35B06; Sobolev embedding; 35J92; 35B33; Qualitative properties
• ISSN: 1016-443X; 1420-8970
• DOI: 10.1007/s00039-020-00535-3

Given n ≥ 2 and 1 < p < n , we consider the critical p -Laplacian equation Δ 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.