THE EXISTENCE OF HETEROCLINIC ORBITS FOR A SECOND ORDER HAMILTONIAN SYSTEM

Abstract

In this paper, via variational methods and critical point theory, we study the existence of heteroclinic orbits for the following second order nonautonomous Hamiltonian system $$ \ddot{u} - \triangledown{F(t,u)} = 0, $$ where $u \in R^{n}$ and $F \in C^{1}(R \times R^{n}, R)$, $F \geq 0$. $\mathcal{M} \subset R^{n}$ be set of isolated points and $\sharp{\mathcal{M}} \geq 2$. For each $\xi \in \mathcal{M}$, there exists a positive number $\rho_{0}$ such that if $y \in B_{\rho_{0}}(\xi)$, then $F(t,y) \geq F(t,\xi)$ for all $t \in R$, where $B_{\rho_{0}}(\xi) = \{y \in R^{n} \mid \mid y - \xi \mid \lt \rho_{0}\}$. Under some more assumptions on $F(t,x)$ and $\mathcal{M}$, we prove that each point in $\mathcal{M}$ is joined to another point in $\mathcal{M}$ by a solution of our system.