On periodic solutions of second-order differential equations with attractive-repulsive singularities
type: Item
instance of: scholarly article
title: On periodic solutions of second-order differential equations with attractive-repulsive singularities
zbMATH Open document ID: 1187.34049
author: Robert Hakl
author: Pedro J. Torres
published in: Journal of Differential Equations
publication date: 2009-12-22T00:00:00Z
review text: The paper studies the periodic problem \[ u''(t)=\frac{g(t)}{u^\mu(t)}-\frac{h(t)}{u^\lambda(t)}+f(t),\quad u(0)=u(\omega),\;u'(0)=u'(\omega), \tag{1} \] where \(g, h\in L(\mathbb{R}/\omega\mathbb{Z}; {\mathbb{R}}_+)\),\ \(f\in L(\mathbb{R}/\omega \mathbb{Z}; \mathbb{R})\), and \(\lambda, \mu >0\). Since the functions \(g\) and \(h\) are possibly zero on some sets of positive measure, the singularity may combine attractive and repulsive effects. The authors establish sufficient conditions for the existence of a solution to problem (1). Their main result can be applied to the original Lazer-Solimini equations both in the attractive and in the repulsive case. The proofs rely on the method of upper and lower functions. To illustrate the results, an application to the dynamics of a trapless Bose-Einstein condensate is given. Final sections are devoted to a comparative study of the equation with attractive (respectively repulsive) singularity. Along the paper some open problems are posed.
reviewed by: Irena Rachůnková
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34B18
Mathematics Subject Classification ID: 34B16
zbMATH DE Number: 5651184
zbMATH Keywords: second-order ordinary differential equation
zbMATH Keywords: singular equation
zbMATH Keywords: periodic solution
zbMATH Keywords: positive solution
MaRDI profile type: Publication
full work available at URL: j.jde.2009.07.008
OpenAlex ID: W1977310648
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DOI: 10.1016/J.JDE.2009.07.008
description: scientific article; zbMATH DE number 5651184