Heteroclinic solutions between stationary points at different energy levels

Abstract

Consider the system of equations $$ -\ddot{q} = a(t)V'(q). $$ The main goal of this paper is to present a simple minimization method to find heteroclinic connections between isolated critical points of $V$, say $0$ and $\xi$, which are local maxima but do not necessarily have the same value of $V$. In particular we prove that there exist heteroclinic solutions from $0$ to $\xi$ and from $\xi$ to $0$ for a class of positive slowly oscillating periodic functions $a$ provided $\delta = |V(0) - V(\xi)|$ is sufficiently small (and another technical condition is satisfied). Note that when $V(0) \neq V(\xi)$, $a$ cannot be constant be conservation of energy. Existence of "multi-bump" solutions is also proved.