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Non-collision periodic solutions for singular Hamiltonian systems

Authors:	Email author" target="_blank">Morched?Boughariou Email author

Institution:	(1) Faculté des Sciences de Tunis, Département de Mathématiques, Campus Universitaire, 1060 Tunis, Tunisie

Abstract:	We study the existence of classical (non-collision) T-periodic solutions of the Hamiltonian system $\openup2pt \displaylines{ \dot q = H_p (t,p, q),\cr \dot p = -H_q (t,p, q),\cr p(t+T) = p(t),\quad q(t+T)=q(t),\cr }$ where $p,q :{\bf R} \rightarrow {\bf R}^N (N \geq 3)$ and is a T-periodic function in t which has a singularity at like $H(t,p,q)\sim {1 \over \beta}\|p\|^{\beta}-{1 \over \|q\|^{\alpha}} \hbox{ with } 0<\alpha < \beta \hbox{ and} \beta \geq 2.$ Under suitable conditions on H, we prove that if $\alpha \in ]{\beta \over 2}, \beta$ then (HS) possesses at least one non-collision solution and if $\alpha \in ]0,{\beta \over 2}]$ then the generalized solution of (HS) obtained in 5] has at most one time of collision in its period.

Keywords:	35D10 35B40 58E05
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