Multiple periodic solutions of nonlinear second order differential equations

In this paper, we are interested in the existence of multiple nontrivial $T$-periodic solutions of the nonlinear second ordinary differential equation $\ddot{x}+V_x(t,x)=0$ in $N(\ge 1)$ dimensions. Using homological linking and morse theory, we get at least two critical points of the functional corresponding to our problem. And, we also prove that two critical points are different by critical groups. Then, we obtain there are at least two nontrivial $T$-periodic solutions of the problem.