In this paper, we consider a class of second-order Hamiltonian systems of the form
$ \ddot{u}(t)-L(t) u(t)+\nabla W(t,u(t)) = 0 $
where $ L:R\rightarrow R^{N^2} $ and $ W \in C^1(R\times R^N, R) $ are asymptotically periodic in $ t $ at infinity. Under the reformative perturbation conditions and weaker superquadratic conditions on the nonlinearity, the existence of a ground state homoclinic orbit is established. The main tools employed here are the local mountain pass theorem and the concentration-compactness principle.
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