期刊
GEOMETRIC AND FUNCTIONAL ANALYSIS
卷 30, 期 3, 页码 770-803出版社
SPRINGER BASEL AG
DOI: 10.1007/s00039-020-00535-3
关键词
Quasilinear anisotropic elliptic equations; Qualitative properties; Sobolev embedding; Convex cones
类别
资金
- Gruppo Nazionale per l'Analisi Matematica, la Probabilita e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM, Italy)
- PRIN 2017 Project Qualitative and quantitative aspects of nonlinear PDEs
- European Research Council [721675]
- European Research Council (ERC) [721675] Funding Source: European Research Council (ERC)
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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