Periodic solutions of spatially periodic, even Hamiltonian systems
In this paper, the existence of periodic solutions of Hamiltonian systems on the cotangent bundle of the torus \(T^N\) is studied. The authors assume that the Hamiltonian function is even in spatial variables and superlinear in the tangential variables, its gradient with respect to the tangential variables has a polynomial growth, and all trivial solutions are nondegenerate. Then when there is one trivial solution possessing Conley-Zehnder index not in \([0,N]\), the existence of multiple nontrivial solutions of the system is proved. This result depends on a more general abstract critical point theorem, which is proved by measuring the change of the topologies of the level sets for the corresponding functional using the Borel cohomology method. This result is related to the earlier work Math. Z. 224, No. 1, 65-76 (1997) of the authors for multiple periodic solutions of Hamiltonian systems on tori.
scientific article; zbMATH DE number 1003899