Periodic solutions for a class of first order super-quadratic Hamiltonian system
type: Item
instance of: scholarly article
title: Periodic solutions for a class of first order super-quadratic Hamiltonian system
zbMATH Open document ID: 1122.37048
author: Anmin Mao
author: Shixia Luan
author: Yanheng Ding
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-04-10T00:00:00Z
review text: The authors consider the unbounded Hamiltonian system \[ \left\{ \begin{aligned} \partial_t u-\Delta_x u+V(x) u= H_v(t,x,u,v)\\ -\partial_t v-\Delta_xv+V(x)v=H_n(t,x,u,v) \end{aligned} \right. \] for \((t,x)\in\mathbb{R}\times\Omega\), \(\Omega\subset\mathbb{R}^N\) a bounded smooth domain. Here \(V\in C^0(\overline{\Omega})\) and \(0\) is not a Dirichlet eigenvalue of \(-\Delta_x+V(x)\). The nonlinearity \(H\in C^1(\mathbb{R}\times\overline{\Omega} \times\mathbb{R}^2)\) is \(T\)-periodic in \(t\) and superquadratic in \(z=(u,v)\) for \(z\) near \(0\) and near \(\infty\). The main theorem states the existence of a solution \(z(t,x)\not\equiv 0\) which is \(T\)-periodic in \(t\) and satisfies Dirichlet boundary conditions in the \(x\)-variable. The proof is based on variational methods for strongly indefinite functionals due to \textit{S. Luan} and \textit{A. Mao} [Nonlinear Anal., Theory Methods Appl. 61, No. 8 (A), 1413--1426 (2005; Zbl 1071.34039)] and \textit{Y. Ding} and the reviewer [Math. Z. 240, No. 2, 289--310 (2002; Zbl 1008.37040)].
Mathematics Subject Classification ID: 37K05
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 5141603
zbMATH Keywords: infinite-dimensional Hamiltonian system
zbMATH Keywords: local linking
reviewed by: Thomas J. Bartsch
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.003
OpenAlex ID: W2091963186
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DOI: 10.1016/J.JMAA.2006.08.003
description: scientific article; zbMATH DE number 5141603