A note on homoclinic orbits for second-order Hamiltonian systems
type: Item
instance of: scholarly article
title: A note on homoclinic orbits for second-order Hamiltonian systems
zbMATH Open document ID: 0919.34045
author: Shaoping Wu
author: Haitao Yang
published in: Applied Mathematics. Series B (English Edition)
publication date: 1999-08-25T00:00:00Z
review text: In this note, nonautonomous Hamiltonian systems of the form \[ \ddot{x} - a(t) x + f(t,x) = 0, \quad x \in \mathbb{R}^n, \] are considered. Homoclinic orbits satisfying \(x(t) \to 0\), \(\dot{x}(t) \to 0\), as \(t \to \infty\), are sought for. The type of results obtained are that such homoclinic orbits exist if \(a\) and \(f\) satisfy a number of conditions. In nontechnical terms, these include that \(f\) is sublinear at the origin and asymptotically linear at infinity. Further, \(a(t)\) is large for \(| t| \) large. The results generalize work of \textit{P. H. Rabinowitz} and \textit{K. Tanaka} [Math. Z. 206, No. 3, 473-499 (1991; Zbl 0707.58022)] among others and are proved by variational methods such as the mountain pass theorem.
reviewed by: Ale Jan Homburg
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 46N20
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 1234933
zbMATH Keywords: homoclinic orbits
zbMATH Keywords: nonautonomous Hamiltonian systems
zbMATH Keywords: mountain pass theorem
MaRDI profile type: Publication
full work available at URL: s11766-998-0017-3
OpenAlex ID: W2953218675
cites work: Homoclinic orbits for second order Hamiltonian system with quadratic growth
cites work: Some existence results of solutions for the semilinear elliptic equations on \(\mathbb{R}^ N\)
cites work: Homoclinic orbits for a class of Hamiltonian systems
cites work: Existence and multiplicity results for homoclinic solutions to a class of Hamiltonian systems
cites work: Q3999077
cites work: Critical point theory and Hamiltonian systems
cites work: Solutions in spectral gaps for a nonlinear equation of Schrödinger type
DOI: 10.1007/S11766-998-0017-3
description: scientific article; zbMATH DE number 1234933