The existence of solutions homoclinic to an equilibrium \(0\in \mathbb{R}^N\) for second-order time-dependent Hamiltonian systems of the type \[ \ddot q- L(t) q+ {\partial\over \partial q} W(t, q)= 0,\quad q\in \mathbb{R}^N,\tag{1} \] where \(L\in C(\mathbb{R}, \mathbb{R}^{N^2})\) is a symmetric matrix, \(W\in C^1(\mathbb{R}\times \mathbb{R}^N, \mathbb{R})\), is studied. As usual, a solution \(q(t)\) of system (1) is said to be homoclinic (to 0) if \(q(t)\neq 0\), \(q(t)\to 0\), and \(\dot q(t)\to 0\) as \(|t|\to \infty\).
The existence and multiplicity of homoclinic solutions for Hamiltonian systems of this type have been studied in many recent papers via the critical point theory under the assumptions that \(L(t)\) is positive definite for all \(t\in \mathbb{R}\), \(W(t, q)\) is globally superquadratic in \(q\) and the potential \(V(t, q)= -{1\over 2} L(t)q\cdot q+ W(t, q)\) is periodic in \(t\).
The author studies the existence of homoclinic (to 0) solutions for a class of systems (1) when the global positive definiteness of \(L(t)\) is not necessarily satisfied and \(L\) and \(W\) are not periodic in \(t\). Both the case that \(W(t, q)\) is superquadratic in \(q\) and the one that \(W(t, q)\) is of subquadratic growth as \(|q|\to \infty\) are considered.