Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems
type: Item
instance of: scholarly article
title: Computations of critical groups and periodic solutions for asymptotically linear Hamiltonian systems
zbMATH Open document ID: 1207.34049
author: Shiwang Ma
published in: Journal of Differential Equations
publication date: 2010-04-21T00:00:00Z
review text: The author gives some parabolic-like conditions which allow further computations of the critical groups both at degenerate critical points and at infinity, and the results generalize the well-known angle conditions for the computations of the critical groups. As an application, the author considers the second-order Hamiltonian system \[ u''(t)+\nabla H(t,u(t))=0,\quad t\in \mathbb R,\tag{1} \] where \(H:\mathbb R\times \mathbb R^N\longrightarrow \mathbb R\) is \(T\)-periodic in its first variable and \(H(t,x)\) is continuous in \(t\) for each \(x\in \mathbb R^N\) and twice continuously differentiable in \(x\) for each \(t\in [0,T]\). The authors prove the existence of \(T\)-periodic solutions for the system when \(\nabla H\) is asymptotically linear both at the origin and at infinity, that is, \[ |\nabla H(t,x)-A_\infty(t)x|=0(|x|),\text{ as }|x|\rightarrow \infty,\text{ uniformly in }t\in[0,T], \] \[ |\nabla H(t,x)-A_0(t)x|=0(|x|),\text{ as }|x|\rightarrow 0,\text{ uniformly in }t\in[0,T]. \] Let \[ H_0(t,x)=H(t,x)-\tfrac{1}{2}(A_0(t)x,x), \quad H_\infty(t,x)=H(t,x)-\tfrac{1}{2}(A_\infty(t)x,x), \] where \(A_\infty(t)\) and \(A_0(t)\) are \(N\times N\) continuous symmetric matrices and \(T\)-periodic in \(t\). The main conditions are the following: {\parindent13mm \begin{itemize}\item[{\((H_{\infty})\)}] There exists \(\alpha\in (0,1]\) such that \(\lim_{|x|\rightarrow \infty} \frac{|\nabla H_{\infty}(t,x)|}{|x|^{\alpha}}=0\) uniformly for \(t\in [0,T];\) \item[{\((H_{\infty}1)\)}] \(\limsup_{|x|\rightarrow {\infty}}\frac{(\nabla H_{\infty}(t,x),x )}{|x|^{2\alpha}}\leq h_{\infty}(t)\leq 0\) uniformly for \(t\in[0,T]\), where \(h_{\infty}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{\infty}(t)\,dt<0\); \item[{\((H_{\infty}2)\)}] \(\liminf_{|x|\rightarrow {\infty}}\frac{(\nabla H_{\infty}(t,x),x )}{|x|^{2\alpha}}\geq h_{\infty}(t)\geq 0\) uniformly for \(t\in[0,T]\), where \(h_{\infty}\in C([0,T],\mathbb R)\) with \(\int_{0}^{T}h_{\infty}(t)\,dt>0\); \item[{\((H_0)\…
reviewed by: Chun-Lei Tang
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E30
zbMATH DE Number: 5696611
zbMATH Keywords: critical group
zbMATH Keywords: Morse theory
zbMATH Keywords: periodic solution
zbMATH Keywords: second-order Hamiltonian system
MaRDI profile type: Publication
full work available at URL: j.jde.2009.11.013
OpenAlex ID: W1994566022
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DOI: 10.1016/J.JDE.2009.11.013
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description: scientific article; zbMATH DE number 5696611