### 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.

$$

-\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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