Almost homoclinics for nonautonomous second order Hamiltonian systems by a variational approach

Abstract

In this paper we shall be concerned with the existence of almost homoclinic solutions of the Hamiltonian system $\ddot{q}+V_q(t,q)=f(t)$, where $t\in\mathbb{R}$, $q\in\mathbb{R}^n$ and $V(t,q)=-\frac{1}{2}(L(t)q,q)+W(t,q)$. It is assumed that $L$ is a conti\-nuous matrix valued function such that $L(t)$ are symmetric and positive definite uniformly with respect to $t$. A map $W$ is $C^1$-smooth, $W_q(t,q)=o(|q|)$, as $q\to 0$ uniformly with respect to $t$ and $W(t,q)|q|^{-2}\to\infty$, as $|q|\to\infty$. Moreover, $f\neq 0$ is continuous and sufficiently small in $L^2(\R,\mathbb{R}^n)$. It is proved that this Hamiltonian system possesses a solution $q_{0}\colon\mathbb{R}\to\mathbb{R}^n$ such that $q_{0}(t)\to 0$, as $t\to\pm\infty$. Since $q\equiv 0$ is not a solution of our system, $q_{0}$ is not homoclinic in a classical sense. We are to call such a solution almost homoclinic. It is obtained as a weak limit of a sequence of almost critical points of an appropriate action functional $I$.