Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems (Q874937)

In this paper \(2\pi\)-periodic solutions for the Hamiltonian system \(\dot z= JH(z,t)\) are investigated, where \(H(z,t)\in C^1(\mathbb{R}^{2N},\mathbb{R})\) is \(2\pi\)-periodic for \(t\) and \(J\) is the standard symplectic matrix. As in the classical infinite-dimensional Morse theory, the efficient application of the \({\mathcal E}\)-Morse theory depends on the computation of the critical groups at given critical points or infinity. By computing the \({\mathcal E}\)-critical groups at \(\theta\) and infinity of the corresponding functional of Hamiltonian systems, the existence of nontrivial periodic solutions for the systems which may be resonant at \(\theta\) and infinity under some new conditions is proved. Some results in the literature are extended and some new type of theorems are proved. The main tool is the \({\mathcal E}\)-Morse theory developed by \textit{W. Kryszewski} and \textit{A. Szulkin} [Trans. Am. Math. Soc.349, No. 8, 3181--3234 (1997; Zbl 0892.58015)] and Zou.