On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in \(\mathbb R^ 2\)
type: Item
instance of: scholarly article
title: On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in \(\mathbb R^ 2\)
zbMATH Open document ID: 0871.35014
author: Kuo-Shung Cheng
author: Chang-Shou Lin
published in: Mathematische Annalen
publication date: 1997-09-24T00:00:00Z
review text: We study the asymptotic behavior at infinity of a solution \(u\) of \(\Delta u+ K(x) e^{2u} =0\) in \(\mathbb R^2\). With some mild assumptions on \(K\) and \(u\), we conclude that \(u(x)= \alpha\log |x|+O(1)\) at infinity for some real \(\alpha\). Applying this result, we prove that solutions \(u\) of the equation must be radially symmetric provided that \(K=K(|x|)\) is radial and nonincreasing in \(|x|\). We also prove that the above equation has no solution with a finite total curvature provided that \(K\) is not identically a constant, bounded between two positive constants and monotonic along some direction.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 1031503
zbMATH Keywords: nonexistence
zbMATH Keywords: radially symmetric solutions
MaRDI profile type: Publication
full work available at URL: s002080050068
OpenAlex ID: W2076759540
DOI: 10.1007/S002080050068
description: scientific article; zbMATH DE number 1031503