On the classification of radial solutions for \(\Delta u+K(| x|)e^{2u}=0\) in \(\mathbb{R}^ n\)
type: Item
instance of: scholarly article
title: On the classification of radial solutions for \(\Delta u+K(| x|)e^{2u}=0\) in \(\mathbb{R}^ n\)
zbMATH Open document ID: 0802.34036
author: Kuo-Shung Cheng
author: Ting-Kung Luo
published in: Journal of Differential Equations
publication date: 1994-08-22T00:00:00Z
review text: This paper mostly concerns the existence and asymptotic behavior at \(\infty\) of solutions \(u: \mathbb{R}_ +\to \mathbb{R}\) of the differential equation (1) \(u''+ {n-1\over x} u'+ f(x) e^{2u}= 0\) for \(n\geq 3\) and a nontrivial locally Hölder continuous function \(f\) in \(\mathbb{R}_ +\) satisfying various additional technical conditions. First existence and nonexistence theorems are proved. Then sufficient conditions on \(f\) are given for every solution \(u(x)\) to have various asymptotic behavior at \(\infty\), including \(C+ o(1)\), \(-C\log x+ O(1)\), and \(-{1\over 2} \log(\log x)+ O(1)\) as \(x\to\infty\) for a positive constant \(C\). Specific asymptotic formulas are obtained if \(f(x)\) is a power function.
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 34A34
zbMATH DE Number: 599892
zbMATH Keywords: existence
zbMATH Keywords: asymptotic behavior
reviewed by: Charles A. Swanson
MaRDI profile type: Publication
full work available at URL: jdeq.1994.1067
OpenAlex ID: W2056410420
DOI: 10.1006/JDEQ.1994.1067
description: scientific article; zbMATH DE number 599892