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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 \(\{x\in \mathbb{R}^ N: V(x)< V(0)\}\) 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 to ordinary differential equations
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
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