Symmetry results for positive solutions of some elliptic equations on manifolds
type: Item
instance of: scholarly article
title: Symmetry results for positive solutions of some elliptic equations on manifolds
zbMATH Open document ID: 0968.58015
DOI: 10.1023/A:1006644025734
author: Yanyan Li
published in: Annals of Global Analysis and Geometry
publication date: 2001-09-11T00:00:00Z
review text: Adapting the method of moving planes (of A. Alexandrov), the authors study positive solutions of the (nonlinear) Laplace-Beltrami equation \(Lu=-\text{div}(a'|\nabla|^2)\nabla u=f(u)\), on an open submanifold \(\mathcal M\) in an \(n\)-dimensional manifold \(\mathcal N\) in the presence of technical conditions which ensure ellipticity and of some symmetry assumptions. The main regularity assumptions are that \(a\in W^{2,\infty}((0,\infty))\cap{\mathcal C}([0,\infty))\) and that \(f\) is locally Lipschitz. Applications are given for annular domains in \({\mathbb R}^2\), convex geodesic balls in \(S^n\) and in (the hyperbolic space) \(H^n\), subgroups of the polarized Heisenberg group, and for monotonicity results.
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B50
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1436918
zbMATH Keywords: moving plane method
zbMATH Keywords: positive solutions
zbMATH Keywords: elliptic PDE
reviewed by: Otto Liess
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1436918