Existence of homoclinic solutions for nonlinear second-order coupled systems
type: Item
instance of: scholarly article
title: Existence of homoclinic solutions for nonlinear second-order coupled systems
zbMATH Open document ID: 1409.34042
author: Robert de Sousa
author: Feliz Minhós
published in: Journal of Differential Equations
publication date: 2018-12-03T00:00:00Z
review text: In this work, the authors consider the second order coupled system on the real line \[ \begin{aligned} u''(t)- k_1^2 u(t)=f\big(t, u(t), v(t),u'(t), v'(t)\big), \;t\in \mathbb R, \\ v''(t)- k_2^2 v(t)=h\big(t, u(t), v(t),u'(t), v'(t)\big), \;t\in \mathbb R, \end{aligned}\eqno(1) \] where \( f, h : \mathbb R^5 \longrightarrow \mathbb R \) are \(L^1\)-Carathéodory functions, \( k_1, k_2 > 0\), the boundary conditions \[ \begin{aligned} u(\pm \infty) = u'(\pm \infty) = 0, \\ v(\pm \infty) = v'(\pm \infty) = 0. \end{aligned}\eqno(2) \] Under conditions on the monotonicity of the non-linearities \(f\) and \(h\) mainly, the authors establish the existence and the localization of a homoclinic solution for (1), that is, a non-trivial solution of the problem (1)--(2). The proof of the main result is topological based on the Schauder fixed-point theorem and a criterion of compactness, previously established by the first author. Also, it combines Green's functions technique with the notion of lower and upper solutions. The last section of the paper presents an application to a family of a coupled stationary nonlinear Schrödinger system (NLS).
reviewed by: Adel Daouas
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34B27
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 34B40
zbMATH DE Number: 6987665
zbMATH Keywords: homoclinic solutions
zbMATH Keywords: coupled systems
zbMATH Keywords: \(L^1\)-Carathéodory functions
zbMATH Keywords: Green's functions
zbMATH Keywords: Schauder's fixed-point theorem
zbMATH Keywords: lower and upper solutions
MaRDI profile type: Publication
full work available at URL: j.jde.2018.07.072
OpenAlex ID: W2885770980
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Wikidata QID: Q129433355
DOI: 10.1016/J.JDE.2018.07.072
description: scientific article; zbMATH DE number 6987665