Heteroclinic solutions between stationary points at different energy levels

Vittorio Coti Zelati, Paul H. Rabinowitz

DOI: http://dx.doi.org/10.12775/TMNA.2001.001

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.

Keywords


Heteroclinic solutions; variational methods; Lagrangian systems

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