On the existence of homoclinic orbits for the asymptotically periodic Duffing Equation

Using variational methods, we show the existence of a homoclinic orbit for the Duffing equation $-ddot u+u=a(t)|u|^{p-1}u$, where $p>1$ and $ain L^infty(R)$ is a positive function of the form $a=a_0+a_infty$ with $a_infty$ periodic, and $a_0(t) o 0$ as $t opminfty$ satisfying suitable conditions. Under the same assumptions on $a$, we also prove that the perturbed equation $-ddot u+u=a(t)|u|^{p-1}u+alpha(t)g(u)$ admits a homoclinic orbit whenever $gin C^1(R)$ satisfies $g(u)=O(u)$ as $u o 0$ and $alphain L^infty(R)$, $alpha(t) o 0$ as $t opminfty$ and $|alpha|_{L^infty}$ is sufficiently small.