Multiple periodic solutions for lattice dynamical systems with superquadratic potentials

In this paper, we consider one dimensional lattices consisting of infinitely many particles with nearest neighbor interaction. The autonomous dynamical system is described by the following infinite system of second order differential equationsq¨i=Φi−1′(qi−1−qi)−Φi′(qi−qi+1),i∈Z, where Φi denotes the interaction potential between two neighboring particles and qi(t) is the state of the i-th particle. Supposing Φi is superquadratic at infinity, for all T>0, we obtain a nonzero T-periodic solution of finite energy which may be nonconstant in some range of period. If in addition Φi(x) is even in x, we also obtain infinitely many geometrically distinct solutions for any period T>0. In particular, a prescribed number of geometrically distinct nonconstant periodic solutions is obtained for some range of period. Since the functional associated to the above system is invariant under the actions of the non-compact group Z and the continuous compact group S1 under our assumptions, in order to prove our results, we need to extend the abstract critical point theorem about strongly indefinite functional developed by Bartsch and Ding [Math. Nachr. 279 (2006) 1267-1288] to a more general class of symmetry.