Symmetry and asymmetry: the method of moving spheres
type: Item
instance of: scholarly article
title: Symmetry and asymmetry: the method of moving spheres
zbMATH Open document ID: 1201.35099
published in: Advances in Differential Equations
publication date: 2009-06-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0703808
review text: The first part of the paper deals with symmetry properties of positive solutions of the semilinear elliptic equation \( \Delta u+ (c/|x|^2)u + u ^{(n+2)/(n-2)}=0\) in \(\mathbb R^n\setminus \{0\}\). In partially answering a problem raised by L. Veron, the authors prove, among other results, the following theorem: if \(c \geq (n-2)^2/4\), then the problem has no smooth solutions. If \(c>0\), then any \({\mathcal C}^\infty(\mathbb R^n\setminus \{0\})\) solution must be radially symmetric about the origin and there is a constant \(C>0\) such that \( u(x) \leq C|x|^{-(n-2)/2}\). If \(c \leq (n-2)^2/4\), then the problem has infinitely many smooth radial solutions. If \(c \leq -(n-2)^2/4\), then the problem has non-radial solutions. The second part of the paper is concerned with some nonlinear elliptic equations on the unit sphere.
reviewed by: Otto Liess
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 58J05
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35B06
zbMATH DE Number: 5572590
zbMATH Keywords: nonlinear elliptic equations
zbMATH Keywords: method of moving spheres
zbMATH Keywords: radially symmetric solutions
author: Qinian Jin
author: Yan Yan Li
author: Hao-yuan Xu
MaRDI profile type: Publication
arXiv ID: math/0703808
description: scientific article