Existence and multiplicity of nonzero periodic solution with saddle point character for some nonautonomous second order systems

The purpose of this paper is to study the existence and multiplicity of nonzero periodic solutions with saddle point character for the following nonautonomous second order systems: {u¨(t)=∇F(t,u(t)),a.e. t∈[0,T],u(0)−u(T)=u˙(0)−u˙(T)=0. Some new existence and multiplicity theorems are obtained by using saddle point reduction method and a three-critical-point theorem proposed by Brezis and Nirenberg.