Topological Methods in Nonlinear Analysis

Heteroclinic solutions between stationary points at different energy levels

Vittorio Coti Zelati and Paul H. Rabinowitz

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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.

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Topol. Methods Nonlinear Anal., Volume 17, Number 1 (2001), 1-21.

First available in Project Euclid: 22 August 2016

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Coti Zelati, Vittorio; Rabinowitz, Paul H. Heteroclinic solutions between stationary points at different energy levels. Topol. Methods Nonlinear Anal. 17 (2001), no. 1, 1--21.

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