The existence of homoclinic solutions for second-order Hamiltonian systems with periodic potentials

In this paper, we study the existence of homoclinic solutions for the second-order Hamiltonian system ü−L(t)u+Wu(t,u)=0, where L(t)L(t) and W(t,u)W(t,u) are supposed to be periodic in tt. Under certain assumptions on LL and WW, we obtain two new existence results by using the variant mountain pass theorem and generalized linking theorem.