## Homoclinics for a singular Hamiltonian system.(English)Zbl 0936.37035

Jost, Jürgen (ed.), Geometric analysis and the calculus of variations. Dedicated to Stefan Hildebrandt on the occasion of his 60th birthday. Cambridge, MA: International Press. 267-296 (1996).
The paper studies the existence of homoclinics for a family of Hamiltonian systems described by the differential equation $$\ddot q+V_{q}(t,q)=0$$, where $$q\in \mathbb{R}^{2}$$ and the potential $$V$$ satisfies some conditions. In particular $$V(t,x)$$ has a singularity as $$x\to \xi$$ uniformly in $$t$$. Using elementary minimization arguments the author shows that the system has a pair of solutions that are homoclinic to $$0$$ and wind about $$\xi$$ in a positive and a negative sense. Some criteria for the existence of further homoclinic solutions are given. When $$V$$ is autonomous these solutions can be represented by a simple curve. Besides, the more general situation when the potential possesses multiple singularities is treated. It is shown that if $$V$$ has $$k$$ strong singularities at $$\xi_{1},\dots,\xi_{k}$$ there exist at least $$k$$ geometrically distinct solutions of the system homoclinic to $$0$$.
For the entire collection see [Zbl 0914.00109].

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations