On the asymptotic symmetry of singular solutions of the scalar curvature equations
type: Item
instance of: scholarly article
title: On the asymptotic symmetry of singular solutions of the scalar curvature equations
zbMATH Open document ID: 0927.35034
author: Chang-Shou Lin
author: Chiun-Chuan Chen
published in: Mathematische Annalen
publication date: 1999-11-29T00:00:00Z
review text: We study the asymptotic behavior of positive singular solutions of the scalar curvature equations near its singular set. When the scalar curvature function is a positive constant, it was proved by Caffarelli-Gidas-Spruck that any positive singular solution with an isolated singular point at the origin asymptotically approaches to a positive entire solution in \(\mathbb{R}^n\) with a nonremovable singular point at the origin. We extend their result to the case when nonconstant scalar curvature function \(K\) is considered. When the vanishing order of \(K\) at the origin is no less then \({n-2\over 2}\), we prove that the asymptotic result of Caffarelli-Gidas-Spruck still holds. However, if the vanishing order of the scalar curvature function \(K\) is less than \({n-2\over 2}\), we show by examples that the asymptotic result does not hold.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 45G10
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 1336981
zbMATH Keywords: positive solutions
zbMATH Keywords: singular set
zbMATH Keywords: removable singularity
zbMATH Keywords: conformal metric
MaRDI profile type: Publication
full work available at URL: s002080050259
OpenAlex ID: W1985895902
DOI: 10.1007/S002080050259
description: scientific article; zbMATH DE number 1336981