Periodic solutions for Hamiltonian systems without Ambrosetti–Rabinowitz condition and spectrum 0

In this paper, we consider the superquadratic second order Hamiltonian system u ″ ( t ) + A ( t ) u ( t ) + ∇ H ( t , u ( t ) ) = 0 , t ∈ R . Our main results here allow the classical Ambrosetti–Rabinowitz superlinear condition to be replaced by a general superquadratic condition, and 0 lies in a gap of σ ( B ) , where B : = − d 2 d t 2 − A ( t ) . We will study the ground state periodic solutions for this problem. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou.