## 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

### Citations:

Zbl 0296.58004; Zbl 0276.58005
Full Text:

### References:

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