Homoclinic Orbits for Second-Order Hamiltonian Systems with Some Twist Condition

Abstract

We study the existence and multiplicity of homoclinic orbits for second-order Hamiltonian systems $\ddot{q}-L\left(t\right)q+{\nabla }_{q}W\left(t,q\right)=0$, where $L\left(t\right)$ is unnecessarily positive definite for all $t\in \mathrm{\backslash Bbb\; R}$, and ${\nabla }_{q}W\left(t,q\right)$ is of at most linear growth and satisfies some twist condition between the origin and the infinity.