A Prescribed Energy Problem for a Singular Hamiltonian System with a Weak Force

We consider the existence of periodic solutions of a Hamiltonian system q + ∇V(q) = 0 (HS) such that 1 2| q (t)|²+ V(q(t)) = H for all t, where q ∈ R^N (N ≥ 3), H <0 is a given number, V(q) ∈ C²(R^N\(0), R) is a potential with a singularity, and ∇V/(q) denotes its gradient. We consider a potential V(q) which behaves like -l/|q|^α (α ∈ (0, 2)). In particular, in case V(q) satisfies ∇V(q) q ≤ -α₁V(q) for all q ∈ R^N\(0), and ∇V(q) q ≤ -α₂V(q) for all 0 < |q| ≤ R₀ for α₁, α₂ ∈ (0, 2), R₀ = 0, we prove the existence of a generalized solution that may enter the singularity 0. Moreover, under the assumption V(q) = - 1 |q|^α + W(q), where 0 < α < 2 and |q|^αW(q), |q|^{α + 1} ∇W(q), |q|^{α + 2} ∇²W(q) → 0 as |q| → 0, we estimate the number of collisions of generalized solutions. In particular, we get the existence of a classical (non-collision) solution of (HS) for α ∈ (l, 2) when N ≥ 4 and for α ∈ (4/3, 2) when N = 3.