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

2019/06/03 by Giulio Ciraolo, Alessio Figalli, Ciraolo, Giulio +3 · 11 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering

paper · pdf · doi:10.48550/arxiv.1906.00622

Abstract

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

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• The Ambrosetti-Prodi problem for anisotropic p-Laplacian
• Classification results for bounded positive solutions to the critical p-Laplace equation
• Bifurcation from bubbles in nonconvex cones
• Radial symmetry of positive solutions to quasilinear Hardy-Sobolev doubly critical systems
• The weighted isoperimetric inequality and Sobolev inequality outside convex sets
• Liouville type theorem for a class of quasilinear p-Laplace type equations in the half space
• A Liouville theorem for the 2-Hessian equation on the Heisenberg group
• Non-radial solutions for the critical quasi-linear Hénon equation involving p-Laplacian in \R^N
• Liouville theorems for anisotropic p-Laplace equations with a semilinear term
• Global Lipschitz regularity in anisotropic elliptic problems with natural gradient growth
• A New proof of Liouville type theorems for a class of semilinear elliptic equations

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