Homoclinic solutions for a class of second-order Hamiltonian systems with locally defined potentials

In this article, we establish sufficient conditions for the existence of homoclinic solutions for a class of second-order Hamiltonian systems $$ \ddot u(t)-L(t)u(t)+\nabla W\bigl(t,u(t)\bigr)=f(t), $$ where L(t) is a positive definite symmetric matrix for all $t\in\mathbb{R}$. It is worth pointing out that the potential function W(t,u) is locally defined and can be superquadratic or subquadratic with respect to u.