Classification of proper solutions to some Emden-Fowler equations
type: Item
instance of: scholarly article
title: Classification of proper solutions to some Emden-Fowler equations
zbMATH Open document ID: 0946.34049
author: Ken-Ichi Kamo
published in: Hiroshima Mathematical Journal
publication date: 2000-10-17T00:00:00Z
review text: The author develops an asymptotic theory on the second-order Emden-Fowler equation \[ u''\pm (1+\varepsilon(t)) e^{bt}|u|^\lambda \text{sgn }u= 0,\quad t\geq t_0> 0,\tag{1} \] and he states analogous results to those obtained by Bellman, Chanturiya and others. Equation (1) is classified into four types according to the signs of the coefficient function and the exponent \(b\), namely: \[ u''+ (1+\varepsilon(t)) e^{\alpha t}|u|^\lambda\text{sgn }u= 0,\tag{E\(_{++}\)} \] \[ u''+ (1+\varepsilon(t)) e^{-\alpha t}|u|^\lambda\text{sgn }u= 0,\tag{E\(_{+-}\)} \] \[ u''- (1+ \varepsilon(t)) e^{\alpha t}|u|^\lambda\text{sgn }u= 0,\tag{E\(_{-+}\)} \] \[ u''- (1+\varepsilon(t)) e^{-\alpha t}|u|^\lambda \text{sgn }u= 0.\tag{E\(_{--}\)} \] The main goal is to determine the asymptotic forms of positive proper solutions to \((\text{E}_{+-})\), \((\text{E}_{-+})\), \((\text{E}_{--})\) and also to classify these solutions. To start with we mention that a nontrivial solution \(u\) to (1) is said to be proper if it is defined in some neighborhood of \(+\infty\), and is nontrivial in any neighborhood of \(+\infty\). Firstly, it is shown that if \(\lambda\in \mathbb{R}\setminus \{0,1\}\), \(\alpha> 0\) and \(\lim_{t\to\infty} \varepsilon(t)= 0\), then all positive proper solutions \(u\) to \[ u''+ (1+\varepsilon(t)) e^{-\alpha t} u^\lambda= 0\tag{E\(_{+-}\)} \] possess one of the following asymptotic forms \[ u= a_1t+ a_2- {(a_1t)^2\over \alpha^2} e^{-\alpha t} (1+ O(1)); \] \[ u= a-{a^\lambda\over \alpha^2} e^{-\alpha t} (1+ O(1)), \] where \(a>0\), \(a_1> 0\), \(a_2\) are constants. If \(\lambda> 1\), \(\varepsilon(t)\to 0\) as \(t\to\infty\) or if \(0< \lambda< 1\) and either \[ \lim_{t\to\infty} \varepsilon(t)= 0,\quad \int^\infty|\varepsilon'(t)|dt< \infty \] or \[ \lim_{t\to\infty} \varepsilon(t)= 0,\quad \int^\infty|\varepsilon(t)|^2 dt< \infty \] hold, then all positive proper solutions \(u\) to \[ u''- (1+ \varepsilon(t)) e^{\alpha t} u^\lambda= 0\tag{E\(_{-+}\)} \] possess the a…
Mathematics Subject Classification ID: 34E10
zbMATH DE Number: 1423814
zbMATH Keywords: Emden-Fowler equation
zbMATH Keywords: positive proper solution
zbMATH Keywords: asymptotic behavior
reviewed by: Marian Ivanovici
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206124852
OpenAlex ID: W1498965430
DOI: 10.32917/HMJ/1206124852
description: scientific article; zbMATH DE number 1423814