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

In this paper, we consider the superquadratic second order Hamiltonian systemu″(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)σ(B), where B:=−d2dt2−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.