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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

q ¨+V q (t,q)=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)

37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
34C37Homoclinic and heteroclinic solutions of ODE
58E05Abstract critical point theory