On the existence of infinitely many periodic solutions of non-complete Riemannian manifolds
type: Item
instance of: scholarly article
title: On the existence of infinitely many periodic solutions of non-complete Riemannian manifolds
zbMATH Open document ID: 0832.58010
published in: Journal of Differential Equations
publication date: 1996-03-07T00:00:00Z
review text: Let \(M\) be a Riemannian manifold and \(V : M \times \mathbb{R} \to \mathbb{R}\) a \(C^1\) potential function. The author considers curves \(x : [0,T] \to {\mathcal M}\) satisfying the equation \[ D_t \dot x(t) = -\nabla_\mathbb{R} V(x (t), t), \tag{*} \] where \(D_t \dot x(t)\) denotes the covariant derivative of \(\dot x(t)\) along the direction \(\dot x(t)\) and \(\nabla_\mathbb{R} V(x(t), t)\) the Riemannian gradient with respect to \(x\) of \(V\) at \((x(t), t)\). In the present paper the author studies two problems: (a) for periodic \(V\) find the periodic solutions of (*); (b) for given \(x_1, x_2 \in {\mathcal M}\) find the solutions of (*) when \(x(0) = x_1\) and \(x(T) = x_2\).
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E10
zbMATH DE Number: 790888
zbMATH Keywords: Palais-Smale condition
zbMATH Keywords: Lyusternik-Schnirelman category
zbMATH Keywords: periodic solutions
author: Addolorata Salvatore
reviewed by: Witold Mozgawa
MaRDI profile type: Publication
full work available at URL: jdeq.1995.1110
OpenAlex ID: W2083836670
DOI: 10.1006/JDEQ.1995.1110
description: scientific article; zbMATH DE number 790888