Remarks on dynamical systems with weak forces
type: Item
instance of: scholarly article
title: Remarks on dynamical systems with weak forces
zbMATH Open document ID: 0606.58039
DOI: 10.1007/BF01168669
author: Vittorio Coti Zelati
published in: Manuscripta Mathematica
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155203
review text: In this paper the existence of T-periodic solutions for the forced system of differential equations -ÿ\(=\nabla V(y)+h(t)\) is studied, where h is T-periodic, V is defined in an open, bounded and convex subset \(\Omega \subset {\mathbb{R}}^ N\) and \(V\to -\infty\) as y tends to the boundary of the set \(\Omega\). The existence of at least one solution is proved, requiring convexity of -V near the boundary. Such a solution is found as a critical point of a functional \(\Phi\) defined via the dual action principle [see \textit{F. Clarke} and \textit{I. Ekeland}, Commun. Pure Appl. Math. 33, 103-116 (1980; Zbl 0403.70016)] and, for a situation very close to the one discussed here, see \textit{A. Ambrosetti} and the author [Solutions with minimal period for Hamiltonian systems in a potential well, Ann. Inst. Henri Poincaré, Anal. Non Linéaire (in print)]. The fact that the critical points of \(\Phi\) correspond to T-periodic solutions of class \(W^{2,1}\) is then exploited to prove that such a solution does not touch the boundary of \(\Omega\).
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 3980088
zbMATH Keywords: periodic solution
zbMATH Keywords: singular potential
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3980088