Hyperbolic-like solutions for singular Hamiltonian Systems

Abstract

We study the existence of unbounded solutions of singular Hamiltonian systems: q¨+∇V(q)=0,q¨+∇V(q)=0, where V(q)∼−1|q|αV(q)∼−1|q|α is a potential with a singularity. For a class of singular potentials with a strong force α>2α>2, we show the existence of at least one hyperbolic-like solutions. More precisely, for ven H>0H>0 and θ+,θ−∈SN−1θ+,θ−∈SN−1, we find a solution q(t) of (*) satisfying 12|q˙|2+V(q)=H,12|q˙|2+V(q)=H, |q(t)|⟶∞ast⟶±∞|q(t)|⟶∞ast⟶±∞ limt→±∞q(t)|q(t)|=θ±.