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 $B_{\rho_{0}}(\xi)=\{y\in R^{n}\mid \mid y-\xi \mid<\rho_{0}\}.$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.