## Periodic solutions for Hamiltonian systems without Ambrosetti-Rabinowitz condition and spectrum 0.(English)Zbl 1218.37079

The authors consider the following second order Hamiltonian system $u''(t)+A(t)u(t)+ \nabla H(t,u(t))=0,\quad t\in \mathbb{R},$ where $$A(\cdot )$$ is a continuous $$T$$-periodic symmetric matrix, $$H:\mathbb{R}\times \mathbb{R}^{N}\rightarrow \mathbb{R}$$ is $$T$$-periodic in its first variable, $$H(t,x)$$ is continuous in $$t$$ for each $$x\in \mathbb{R}^{N}$$, and continuously differentiable in $$x$$ for each $$t\in [0,T]$$. The main result of the paper shows that if $$0$$ lies in a gap of $$\sigma(B)$$, where $$B=-\frac{d^2}{dt^2}-A(t)$$, and the technical work assumptions $$H_0 -H_6$$ hold, then the considered Hamiltonian system has at least one ground state $$T$$-periodic solution (Theorem 1.1).
The authors illustrate by examples that the assumptions $$H_0 -H_6$$ are reasonable and there are situations in which the well-known Ambrosetti-Rabinowicz superquadratic condition is not satisfied.

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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