Periodic solutions for singular Hamiltonian systems and closed geodesics on non-compact Riemannian manifolds

Abstract

We study the existence of periodic solutions of singular Hamiltonian systems as well as closed geodesics on non-compact Riemannian manifolds via variational methods.For Hamiltonian systems, we show the existence of a periodic solution of prescribed-energy problem: q••+∇V(q)=0,12|q•|2+V(q)=0under the conditions: (i) V(q)<0 for all q∈RN\\{0}; (ii) V(q)∼−1/|q|2 as |q|∼0 and |q|∼∞.For closed geodesics, we show the existence of a non-constant closed geodesic on (R×SN-1,g) under the condition: g(s,x)~ds2+h0ass~±∞,where h0 is the standard metric on SN−1.