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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.

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