Existence and multiplicity of periodic solutions for some second order Hamiltonian systems (Q475783)

The authors consider the existence of \(T\)-periodic solutions for the second order Hamiltonian system \[ \ddot u(t)+\nabla F(t,u(t)) = 0. \] There are two sets of assumptions on \(F\). In the first set \(F\) is supposed to satisfy a certain nonquadraticity condition as \(|u|\to\infty\) and it is shown that there exists a nontrivial \(T\)-periodic solution. In the second set of assumptions \(F\) is superquadratic as \(|u|\to\infty\) but may not satisfy the Ambrosetti-Rabinowitz condition. It is then shown that there exists a nontrivial \(T\)-periodic solution (always), and infinitely many such solutions if in addition \(F\) is even in \(u\). These results extend and complement some recent work. The proofs are effected by showing that the underlying functional satisfies the Cerami condition and then using the Mountain Pass Theorem of Ambrosetti and Rabinowitz (for 1 solution) and the Fountain Theorem of Bartsch (for infinitely many solutions).