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Multiple homoclinic solutions for a class of autonomous singular systems in $$\mathbb{R}^2$$. (English) Zbl 0907.58014
The paper deals with the study of a class of autonomous second order planar Hamiltonian systems involving a potential $$V$$ which has a strict global maximum at the origin and such that there exists a finite set $$S\subset \mathbb{R}^2$$ of singularities, namely, $$V(x)\rightarrow -\infty$$ as $$\text{dist } (x,S)\rightarrow 0$$. The aim of this paper is to find homoclinics with an arbitrary winding number. More precisely, if $$V$$ satisfies a suitable geometrical property, then for any integer $$k\geq 1$$ the system admits a homoclinic orbit turning $$k$$ times around a singularity $$\xi \in S$$. The geometrical property imposed in the paper is satisfied by a large class of potentials, including those with discrete rotational symmetry, or potentials given by the sum of a smooth radial term and a localized singular perturbation. The proofs use in an essential manner the Ekeland variational principle in order to find a Palais-Smale sequence at an appropriate level.

##### MSC:
 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 58E30 Variational principles in infinite-dimensional spaces
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##### References:
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