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Existence and multiplicity of homoclinic orbits for potentials on unbounded domains. (English) Zbl 0807.34058
Author’s abstract: “We study the system $\ddot q+ V'(q)= 0$, where $V$ is a potential with a strict local maximum at 0 and possibly with a singularity. First, using a minimizing argument, we can prove the existence of a homoclinic orbit when the component $\Omega$ of $\{x\in \bbfR\sp N: V(x)< V(0)\}$ containing 0 is an arbitrary open set; in the case $\Omega$ unbounded we allow $V(x)$ to go to 0 at infinity, although at a slow enough rate. Then we show that the presence of a singularity in $\Omega$ implies that a homoclinic orbit can also be found via a minimax procedure and, comparing the critical levels of the functional associated to the system, we see that these two solutions are distinct whenever the singularity is `not too far’ from 0”.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE 70K05 Phase plane analysis, limit cycles (general mechanics)
##### Keywords:
potential; singularity; homoclinic orbit
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