Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation
type: Item
instance of: scholarly article
title: Uniqueness of rapidly decaying solutions to the Haraux-Weissler equation
zbMATH Open document ID: 0856.34058
author: Eiji Yanagida
published in: Journal of Differential Equations
publication date: 1996-06-09T00:00:00Z
review text: The author uses the Haraux-Weissler equation \(\Delta U+ {1\over 2} \langle x, \text{grad } U\rangle+ {1\over p- 1} U+ |U|^{p- 1} U= 0\) in \(\mathbb{R}^n\) as a motivation (via radial solutions) to study the ODE problem \(u''+ ({n- 1\over r}+ {r\over 2}) u'+ {1\over p- 1} u+ |u|^{p- 1} u= 0\) in \(\mathbb{R}^+\) with initial conditions \(u(0)= \alpha> 0\), \(u'(0)= 0\). He looks for solutions \(u\) with a prescribed number \(i\) of zeroes and a certain exponential decay at infinity. The main result is a uniqueness theorem (for \((n -2) p\leq n\)) saying that given \(i\) there is at most one \(\alpha\) such that \(u\) meets these conditions. Its proof is based on two propositions: i) The number of zeroes of \(u\) increases with \(\alpha\) and ii) in fact strictly when \(\alpha\) passes \(\varphi(0)\), where \(\varphi\) is a decaying solution.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 885515
zbMATH Keywords: Haraux-Weissler equation
zbMATH Keywords: exponential decay
zbMATH Keywords: uniqueness
reviewed by: Norbert Weck
MaRDI profile type: Publication
full work available at URL: jdeq.1996.0083
OpenAlex ID: W2004565844
DOI: 10.1006/JDEQ.1996.0083
description: scientific article; zbMATH DE number 885515