Non-collision solutions for a second order singular Hamiltonian system with weak force

Under a weak force type condition, we consider the existence of time periodic solutions of singular Hamiltonian systems: q¨+V_q(q,t)=0q(t+T)=q(t).}We assume V (q, t) < 0 for all q, t and V (q, t), V_q(q, t) → 0 as |q| → ∞. Moreover we assume V (q, t) is of a form: V(q,t)=−1|q|α+U(q,t)where 0 < α <2 and U(q, t) ∈ C² ((R^N\{0}) × R, R) is a T-periodic funetion in t such that |q|^α U (q, t), |q|^{α + 1} U_q(q, t), |q|^α+2 U_qq, (q, t), |q|^α U_t, (q, t) → 0 as |q| → 0. For α ∈ (1, 2], we prove the existence of a non-collision solution of (HS). For α ∈ (0, 1], we prove that the generalized solution of (HS), which is introduced in [BR], enters the singularity 0 at most one time in its period. Our argument depends on a minimax argument due to [BR] and an estimate of Morse index of corresponding functional, which will be obtained via re-scaling argument.