Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems
type: Item
instance of: scholarly article
title: Homoclinic solutions for a class of subquadratic second-order Hamiltonian systems
zbMATH Open document ID: 1230.37079
author: Juntao Sun
author: Haibo Chen
author: Juan J. Nieto
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-10-22T00:00:00Z
review text: Consider the second order Hamiltonian system \[ \ddot{u}(t)-L(t)u(t)+\nabla W(t,u(t))=0\tag{HS}, \] where \(L\in C(R,R^N)\) is a symmetric matrix valued function and \(W \in C^1(R\times R^N, R)\). A nonzero solution \(u\) of (HS) is said to be homoclinic (to 0) if \(u(t)\rightarrow 0\) as \(|t|\rightarrow\infty\). The authors prove that problem (1) has infinitely many homoclinic orbits under the following conditions: {\parindent=8mm \begin{itemize}\item[(H] \(L \in C(R,R^{N{^2}})\) is a symmetric and positively definite matrix for all \(t\in R\) and there exists a continuous function \(l:R\rightarrow R\) such that \(l(t)>0\) for all \(t\in R\) and \[ (L(t)x,x)\geq l(t)|x|^2,\;l(t)\rightarrow\infty\;\text{as}\;|t|\rightarrow\infty. \] \item[(H] \(W(t,x)=a(t)|x|^r\), where \(a:R\rightarrow R^+\) is a continuous function such that \[ a\in L^{\frac{2}{2-r}}(R,R) \] and \(1<r<2\) is a constant. \end{itemize}} In fact, in Theorem 1.2 the condition respect to \(a\) is not sufficient and the condition that \(a\) is positive is also used in the proof of the Lemma 3.1. Theorem 1.2 in this paper generalizes the result in [\textit{Z. Zhang} and \textit{R. Yuan}, Nonlinear Anal., Theory Methods Appl. 71, No. 9, A, 4125--4130 (2009; Zbl 1173.34330)], in which \(a\) is a positive continuous function such that \[ a\in L^2(R, R)\cap L^{\frac{2}{2-r}}(R, R). \]
reviewed by: Chun-Lei Tang
Mathematics Subject Classification ID: 37J45
zbMATH DE Number: 5804436
zbMATH Keywords: homoclinic solutions
zbMATH Keywords: Hamiltonian systems
zbMATH Keywords: subquadratic
zbMATH Keywords: variational methods
Wikidata QID: Q60199363
MaRDI profile type: Publication
full work available at URL: j.jmaa.2010.06.038
OpenAlex ID: W1983257378
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DOI: 10.1016/J.JMAA.2010.06.038
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description: scientific article; zbMATH DE number 5804436