## Collision and non-collision solutions for a class of Keplerian-like dynamical systems.(English)Zbl 0832.70009

Based on one-to-one correspondence between periodic solutions of the second order Hamiltonian system $$(*)$$ $$q'' = - \nabla V(t,q (t))$$ and the critical points of the functional $$f(u) = {1 \over 2} \int^T_0 |u'(t) |^2 dt - \int^T_0 V(t,u (t))dt$$, where $$q(t) \in R^N$$, $$u(t) \in H^1(S^1, R^N \backslash \{0\})$$ is $$T$$- periodic in $$t$$, and $$V \in C^1 (R \times R^N \backslash \{0\},R)$$, $$V(t,x) \to - \infty$$ as $$|x |\to 0$$, the authors look for conditions on $$V$$ which ensure the existence of a non-collisional (classical) solution of $$(*)$$. First of all, they prove that every generalized solution of an autonomous system has only finitely many collisions and then, extending this result to non-autonomous systems and using Morse index theory, provide a bound on the number of collisions. Finally, they prove the existence of a non-collisional solution of $$(*)$$ when $$V$$ behaves near the origin like $$- |x |^\alpha$$ with $$\alpha \in (1,2)$$.