In this paper we study the first-order Hamiltonian system $\dot{z}=\mathcal{J}{H}_z(t,z).$ Moreover precisely, assuming that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition, we obtain new existence results of ground state homoclinic orbits and infinitely many geometrically distinct homoclinic orbits by using a variational method. An interesting problem is that the nonlinearity may be super-quadratic on some domains and asymptotically quadratic on other domains under local super-quadratic condition. Since we are without more global information on the nonlinearity, we apply a perturbation approach and some special techniques in the proofs.

本文研究一阶哈密顿系统 $\dot{ž}=\mathcal{Ĵ}{H}_{ž}（Ť，ž）。$更精确地说，假设非线性满足局部超二次条件，该条件比通常的整体超二次条件弱，我们使用变分方法获得了基态同宿轨道和无限多个几何上不同的同宿轨道的新存在结果。一个有趣的问题是，在局部超二次条件下，非线性可能在某些域上是超二次的，而在其他域上是渐近二次的。由于我们没有关于非线性的更多全局信息，因此我们在证明中应用了摄动方法和一些特殊技术。