4.4 Article

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

Journal

GEOMETRIC AND FUNCTIONAL ANALYSIS
Volume 30, Issue 3, Pages 770-803

Publisher

SPRINGER BASEL AG
DOI: 10.1007/s00039-020-00535-3

Keywords

Quasilinear anisotropic elliptic equations; Qualitative properties; Sobolev embedding; Convex cones

Categories

Funding

  1. Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM, Italy)
  2. PRIN 2017 Project Qualitative and quantitative aspects of nonlinear PDEs
  3. European Research Council [721675]
  4. European Research Council (ERC) [721675] Funding Source: European Research Council (ERC)

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Given n >= 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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