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 $a\in L^\infty(\mathbb{R})$ is a positive function of the form $a=a_0+a_\infty$ with $a_\infty$ periodic, and $a_0(t)\to 0$ as $t\to\pm\infty$ 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 $g\in C(\mathbb{R})$ satisfies $g(u)=O(u)$ as $u\to 0$ and $\alpha\in L^\infty(\mathbb{R})$, $\alpha(t)\to 0$ as $t\to\pm\infty$ and $\|\alpha\|_{L^\infty}$ is sufficiently small.