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Homoclinic orbits for a singular second order Hamiltonian system. (English) Zbl 0712.58026

Summary: We consider the second order Hamiltonian system: (HS) \(q''+V'(q)=0\) where \(q=(q_1,\dots,q_N)\in {\mathbb{R}}^N\) (\(N\geq 3)\) and \(V:{\mathbb{R}}^N\setminus \{e\}\to {\mathbb{R}}\) \((e\in {\mathbb{R}}^N)\) is a potential with a singularity, i.e., \(| V(q)| \to \infty\) as \(q\to e\). We prove the existence of a homoclinic orbit of (HS) under suitable assumptions. Our main assumptions are the strong force condition of W. B. Gordon [Trans. Am. Math. Soc. 204, 113–135 (1975; Zbl 0276.58005)] and the uniqueness of a global maximum of \(V\).

MSC:

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:

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