Almost homoclinic solutions for the second order Hamiltonian systems - Janczewska, Joanna (Topological Methods in Nonlinear Analysis, 2008)
The second order Hamiltonian system $\ddot{q}+V_{q}(t,q)=f(t)$, where $t\in\mathbb R$ and$q\in\mathbb R^n$, is considered. We assume that a potential $V\in C^{1}(\mathbb R×\mathbb R^n,\mathbb R)$is of the form $V(t,q)=-K(t,q)+W(t,q)$, where $K$ satisfies the pinching conditionand $W_{q}(t,q)=o(|q|)$, as $|q|→ 0$ uniformly with respect to $t$. It is alsoassumed that $f\in C(\mathbb R,\mathbb R^n)$ is non-zero and sufficiently small in $L^{2}(\mathbb R,\mathbb R^n)$.In this case $q\equiv 0$ is not a solution. Therefore there are no orbits homoclinic to $0$ in a classical sense. However, we show that there is a solution emanating from $0$ and terminating at $0$. We are to call such a solution almost homoclinic to $0$. It is obtained here as a weak limit in $W^{1,2}(\mathbb R,\mathbb R^n)$ of a sequence of almost critical points.