Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems

We study the existence and multiplicity of periodic solutions of the following second-order Hamiltonian system ü(t)+∇[−K(t,u(t))+W(t,u(t))]=0. The existence of a nontrivial periodic solution is obtained when ∇W∇W is asymptotically linear at infinity, and the existence of infinitely many periodic solutions is also obtained when ∇W∇W is superlinear.