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)


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