Given n≥ 2 and 1 < p< n, we consider the critical p-Laplacian equation Δpu+up∗-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.

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

Roncoroni A.
2020-01-01

Abstract

Given n≥ 2 and 1 < p< n, we consider the critical p-Laplacian equation Δpu+up∗-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.
2020
Convex cones
Qualitative properties
Quasilinear anisotropic elliptic equations
Sobolev embedding
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1220907
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