Periodic Orbits and Subharmonics of Dynamical Systems on Non-Compact Riemannian Manifolds

✍ Scribed by Silvia Cingolani; Elvira Mirenghi; Maria Tucci

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 778 KB
• Volume: 130
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0136

✦ Synopsis

Let (M, ( } , } ) R ) be a Riemannian manifold and V: M Ä R a C 2 potential function. The research of periodic solutions of the system

where D t (x* (t)) is the covariant derivative of x* along the direction of x* and { R the Riemannian gradient, has been studied when M is a noncontractible manifold (see [2,8,9,14]), assuming, if M is non-compact, the existence of a function on M convex at infinity . When V is bounded the difficulties arise from the lack of compactness of M; indeed, in this case the action functional does not satisfy the Palais Smale compactness condition.

On the other side, if V is unbounded the action functional is unbounded both from below and from above. Therefore neither min max methods and linking arguments can be used since the loop space is not linear nor the Ljusternik Schnirelmann category theory can be applied although M has a non-trivial topology.