Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line
type: Item
instance of: scholarly article
title: Positive, unbounded and monotone solutions of the singular second Painlevé equation on the half-line
zbMATH Open document ID: 1053.34028
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2004-09-23T00:00:00Z
review text: The authors consider the following singular boundary value problem \[ \begin{cases} \frac{\mathstrut 1}{p(t)}(p(t)x^\prime(t))^\prime+q(t)f(t,x(t),p(t)x^\prime(t))=0,\\ x(0)=a\geq 0,\\ \lim\limits_{t\to\infty}{p(t)x^\prime(t)}=b\geq 0, \end{cases} \] with \(f:[0,+ \infty)^2\times \mathbb{R}\to [0,+ \infty)\), \(p\in C([0,+ \infty),\mathbb{R})\cap C^1(0,+ \infty)\), \(p(t)>0\) and \(q(t)\geq 0\) for \(t\in (0,+\infty)\). The next two conditions are assumed throughout \[ \int_{0}^{1}\frac{\mathstrut ds}{p(s)}<\infty \text{ and }\int_{0}^{t}p(s)q(s)ds<\infty \text{ for any }t\geq 0. \] The existence of a global, monotone, positive and unbounded solution is proved. The approach is based on a corresponding vector field investigation. The article also provides two applications of the main result which are related to superconductivity theory and the theory of colloids correspondingly. The first application is the proof of the existence of strictly positive solutions of the following boundary value problems for Painlevé-type equations \[ x^{\prime\prime}=x^4-tx^2, \quad x^\prime(0)=0, \lim\limits_{t\to+\infty} \displaystyle\frac{\mathstrut x(t)}{\sqrt t}=1, \] and \[ x^{\prime\prime}=x^3-tx, \quad x(0)=a, \lim\limits_{t\to-\infty}x(t)=0=\lim\limits_{t\to-\infty} x^\prime(t). \] The second application is the proof of the existence of a strictly positive and decreasing solution of the problem \[ u^{\prime\prime}=2\sinh u(x), \quad u(0)=c, \lim\limits_{x\to+\infty}u(x)=0=\lim\limits_{x\to+\infty} u^\prime(t). \]
Mathematics Subject Classification ID: 34B40
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B16
Mathematics Subject Classification ID: 34B60
Mathematics Subject Classification ID: 34M55
zbMATH DE Number: 2102513
zbMATH Keywords: singular boundary value problem
zbMATH Keywords: positive monotone solution
zbMATH Keywords: shooting method
zbMATH Keywords: Kneser's properties
zbMATH Keywords: superconductivity
zbMATH Keywords: colloids theory
author: Panos K. Palamides
author: George N. Galanis
MaRDI profile type: Publication
full work available at URL: j.na.2004.02.021
OpenAlex ID: W2025149632
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DOI: 10.1016/J.NA.2004.02.021
description: scientific article; zbMATH DE number 2102513