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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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