Existence and multiplicity of periodic solutions for some second order Hamiltonian systems. (English) Zbl 1316.34046

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).


34C25 Periodic solutions to ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E30 Variational principles in infinite-dimensional spaces
Full Text: Euclid