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Homoclinic orbits for a class of Hamiltonian systems. (English) Zbl 0759.58018
The Hamiltonian system under consideration is governed by equations of the form $\ddot q+V_ q(t,q)=\ddot q-L(t)q+W_ q(t,q)=0,$ where $$L(t)$$ is a positive definite matrix and further technical conditions, among other things, ensure that the origin is a local maximum of $$V$$ for all $$t$$. The authors first reconsider a theorem by Rabinowitz and Tanaka concerning the existence of a homoclinic orbit emanating from 0. Using a new compact imbedding theorem, they are able to show that the Palais- Smale condition is satisfied, which in turn makes it possible to prove the above cited theorem by the more traditional techniques relying on the Mountain Pass Theorem. If, in addition, $$W$$ is an even function for all $$t$$, they make use of the symmetric mountain pass theorem to prove the existence of an unbounded sequence of homoclinic orbits.
Reviewer: W.Sarlet (Gent)

MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces