Homoclinic orbits of positive definite Lagrangian systems

AbstractIf the Aubry set A˜(c) satisfies some topological hypothesis, such as H1(M×T,A(c),R)≠0, then the α-function has a flat. In this paper, we will prove that A˜(c′) has infinitely many M˜-minimal homoclinic orbits when c′ is on the boundary of the maximal flat of the α-function. These homoclinic orbits are different from the usually called multi-bump solutions