Interacting Near-Solutions to a Hamiltonian System

A Hamiltonian system with a superquadratic potential is examined. The system is asymptotic to an autonomous system. The difference between the Hamiltonian system and the “problem at infinity,” the autonomous system, may be large, but decays exponientially. The existence of a nontrivial solution homoclinic to zero is proven. Many results of this type rely on a monotonicity condition on the nonlinearity, not assumed here, which makes the problem resemble in some sense the special case of homogeneous (power) nonlinearity. The proof employs variational, minimax arguments. In some similar results requiring the monotonicity condition, solutions inhabit a manifold homeomorphic to the unit sphere in a the appropriate Hilbert space of functions. An important part of the proof here is the construction of a similar set, using only the mountain-pass geometry of the energy functional. Another important element is the interaction between functions resembling widely separated solutions of the autonomous problem.