Existence of nontrivial periodic solutions for a class of resonance Hamiltonian systems
The author studies the existence of nontrivial periodic solutions to asymptotically linear Hamiltonian systems \[ \dot z= JH_z'(t, z),\quad z\in\mathbb{R}^{2N}, \] where \(H\) is \(2\pi\)-periodic in \(t\) and satisfies \[ | H_z'(t,z)- h_0z|/| z|\to 0\text{ as }| z|\to 0,\;| H_z'(t,z)- h_\infty z|/| z|\to 0\text{ as }| z|\to\infty, \] with two constant symmetric matrices \(h_0\), \(h_\infty\). Under some suitable assumptions on \(H\) and the indices \(i^-(h_\infty)\), \(i^0(h_\infty)\), \(i^-(h_0)\), \(i^0(h_0)\), he gives several existence theorems which extend previous results of \textit{S. Li} and \textit{J. Q. Liu} [J. Differ. Equations 78, No. 1, 53-73 (1989; Zbl 0672.34037)]. Comparing to previous works, the results of this paper are new in the following aspects: (i) the nonlinearity \(| H'(t,z)- h_\infty z|\) is unbounded, (ii) the relations between the indices \(i^-(h_\infty)\), \(i^0(h_\infty)\) and \(i^-(h_0)\), \(i^0(h_0)\) may be more relaxed. The basic idea is as follows. Based on the Morse theory, especially some new results on critical groups due \textit{T. Bartsch} and \textit{S. Li} [Nonlinear Anal., Theory Methods Appl. 28, No. 3, 419-441 (1997; Zbl 0872.58018)], the author first establishes several abstract critical point theorems in finite-dimensional space and then obtains some abstract critical point theorems in infinite-dimensional space by Galerkin methods. The proofs of the main results rely on applications of these abstract theorems.
asymptotically linear Hamiltonian systems
scientific article; zbMATH DE number 1289945
scientific article