First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems

Abstract

and H:R x R 2 ~ R is smooth. By a homoclinic or a doubly asymptotic orbit to y, where y is a periodic solution of (1), we mean a solution x + y of (1) such that I x ( 0 y(t)l--, 0 as I tl--* oo. Finding homoclinic orbits in systems like (1) can be quite difficult. In the case when y is a hyperbolic orbit this is equivalent to showing that the stable and unstable manifold of y intersect. In some situation this can be done by Melnikov's method. Recently some progress has been made by applying variational methods. Coti Zelati and Ekeland showed in [3], using dual variational methods, that (1) has at least one homoclinic orbit, if H satisfies some convexity and growth assumptions. In their proof the convexity assumption enters in a crucial way already in the set up of the variational problem. Variational methods have been used by Benci and Giannoni [2], who studied homoclinic orbits on compact Riemannian manifolds and by P. Rabinowitz in [9] who proved existence of heteroclinic orbits on the n-dimensional torus as well as their multiplicity. In [2] and [9] only special Hamiltonian of "Lagrangian" type are studied. The associated variational problem is then at least semi-definite (the