Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials
type: Item
instance of: scholarly article
title: Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials
zbMATH Open document ID: 0689.34034
DOI: 10.1016/0022-0396(89)90167-8
author: Vieri Benci
author: Fabio Giannoni
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
review text: Considering the equation (1) \(q''(t)+V''(q(t))=0,\) \(q\in C^ 2(R,R^ N\setminus \{0\})\), where \(V'(x)\) is the gradient of V at x and \(V(x)=- 1/| x|^{\alpha}+W,\) \(W\in C^ 1(R^ N,R)\), \(\alpha >0\), the following main result is given. Theorem. Let \(0<\rho <R\), \(h\in R\), \(V\in C^ 1(\{x\in R^ N:\rho \leq | x| \leq R\},R)\), \(N\geq 2\) such that (i) \(h-V(x)-<V'(x),x>\leq 0,\) \(\forall x:| x| =\rho,\) (ii) \(h-V(x)-<V'(x),x>\geq 0,\) \(\forall x:| x| =R,\) (iii) h-V(x)0, \(\forall x:\rho \leq | x| \leq R.\) Then there exists at least one nonconstant periodic solution q(t) of (1) having energy h and such that \(\rho\leq | q(t)| RoR\), \(\forall t\in R\).
reviewed by: Jan Andres
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 4129227
zbMATH Keywords: periodic solutions of prescribed energy
zbMATH Keywords: Hamiltonian system
MaRDI profile type: Publication
full work available at URL: 0022-0396(89)90167-8
OpenAlex ID: W2017520560
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description: scientific article; zbMATH DE number 4129227