Existence and multiplicity of periodic solutions for second order Hamiltonian systems depending on a parameter

The existence of at least one nontrivial periodic solution for a class of second order Hamiltonian systems depending on a parameter is obtained, under an algebraic condition on the nonlinearity G and without requiring any asymptotic behavior neither at zero nor at infinity. The existence is still deduced in the particular case when G is subquadratic at zero. Finally, two multiplicity results are proved if G, in addition, is required to fulfill some different Ambrosetti-Rabinowitz type superquadratic conditions at infinity. The approach is fully variational.