In this paper, we study the second-order Hamiltonian systems $$ \ddot{u}-L(t)u+\nabla W(t,u)=0, $$ where $t\in \mathbb{R}$ , $u\in \mathbb{R}^{N}$ , L and W depend periodically on t, 0 lies in a spectral gap of the operator $-d^{2}/dt^{2}+L(t)$ and $W(t,x)$ is locally superquadratic. Replacing the common superquadratic condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ uniformly in $t\in \mathbb{R}$ by the local condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $ a.e. $t\in J$ for some open interval $J\subset \mathbb{R}$ , we prove the existence of one nontrivial homoclinic soluiton for the above problem.

中文翻译：

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具有周期势的局部超二次哈密顿系统

在本文中，我们研究了二阶哈密顿系统$$ \ ddot {u} -L（t）u + \ nabla W（t，u）= 0，$$其中$ t \ in \ mathbb {R} $， $ u \ in \ mathbb {R} ^ {N} $，L和W周期性地取决于t，0表示算子$ -d ^ {2} / dt ^ {2} + L（t）的谱隙$和$ W（t，x）$是局部超二次的。替换$ \ lim_ {| x | \ rightarrow \ infty} \ frac {W（t，x）} {| x | ^ {2}} = + \ infty $统一在$ t \ in \ mathbb中的常见超二次条件{R} $由$ \ lim_ {| x | \ rightarrow \ infty} \ frac {W（t，x）} {| x | ^ {2}} = + \ infty $ ae $ t \ in对于某些开放区间$ J \ subset \ mathbb {R} $，我们证明了上述问题存在一个非平凡的同宿解。