Heteroclinic solutions between stationary points at different energy levels - Coti Zelati, Vittorio; Rabinowitz, Paul H. (Topological Methods in Nonlinear Analysis, 2001)
Consider the system of equations$$ -\ddot{q} = a(t)V'(q).$$ The main goal of this paper is to present a simple minimization methodto find heteroclinic connections between isolated critical points of$V$, say $0$ and $ξ$, which are local maxima but do not necessarilyhave the same value of $V$. In particular we prove that there existheteroclinic solutions from $0$ to $ξ$ and from $ξ$ to $0$ for aclass of positive slowly oscillating periodic functions $a$ provided$δ = |V(0) - V(ξ)|$ is sufficiently small (and anothertechnical condition is satisfied). Note that when $V(0) eq V(ξ)$,$a$ cannot be constant be conservation of energy. Existence of``multi-bump'' solutions is also proved.