Existence of homoclinic solution for the second order Hamiltonian systems.
type: Item
instance of: scholarly article
title: Existence of homoclinic solution for the second order Hamiltonian systems.
zbMATH Open document ID: 1057.34038
author: Zeng-Qi Ou
author: Chun-Lei Tang
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-03-14T00:00:00Z
review text: The authors apply a generalized version of the mountain pass theorem [\textit{P. H. Rabinowitz}, Minimax methods in critical point theory with applications to differential equations, Reg. Conf. Ser. Math. 65 (1986; Zbl 0609.58002)] for establishing the existence of a nontrivial homoclinic solution for the second-order Hamiltonian systems \[ \Ddot{u}(t)- L(t)u(t)+\nabla W(t,u(t))= 0, \qquad t\in \mathbb{R},\tag{\(*\)} \] where the symmetric-matrix-valued function \(L \in C(\mathbb{R},\mathbb{R}^{N^2})\) and \(W\in C^1(\mathbb{R}\times \mathbb{R}^N,\mathbb{R})\) satisfy some specific conditions. These conditions are weaker that those considered by other authors in proving the same type of results, namely \(L(t)\) is not supposed to be uniformly positive definite, and \(W\) satisfies a superquadratic condition. The authors consider the following standard functional \[ f(u)=\frac{1}{2}\int_{\mathbb{R}}(| \Dot{u}| ^2+ L(t)u,u))\,dt- \int_{\mathbb{R}}W(t,u)\,dt. \] The generalized mountain pass theorem gives the existence of a critical point \(u\) of \(f\) such that \(f(u)\geq \alpha_0 >0\). Furthermore, it is shown that \(u\in D(A)\), where \(A=-d^2/dt^2+L(t)\), and therefore \(| u(t)| \to 0\), \(| \Dot{u}(t)| \to 0\) as \(| t| \to \infty\). So, \(u\) is a homoclinic solution of \((*)\).
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 47J30
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 2056505
zbMATH Keywords: homoclinic solution
zbMATH Keywords: second-order Hamiltonian system
zbMATH Keywords: generalized mountain pass theorem
zbMATH Keywords: superquadratic potentials
reviewed by: Gabriela Liţcanu
MaRDI profile type: Publication
full work available at URL: j.jmaa.2003.10.026
OpenAlex ID: W2011223503
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DOI: 10.1016/J.JMAA.2003.10.026
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description: scientific article; zbMATH DE number 2056505