In this paper, the existence and multiplicity of homoclinic solutions for discrete nonlinear Schrödinger equations $-\Delta u_k+v_k{u}_k-\omega u_k=f_k\left(u_k\right)$are considered. With asymptotical nonlinearity and unbounded potential $V=\{v_k:k\in \mathbb{Z}\}$ satisfying ${lim}_{\mid k\mid \to \infty }{v}_k=\infty$, we achieve the existence of nontrivial homoclinic solution via the saddle point reduction method and Conley index theory. Furthermore, we can prove there are at least two nontrivial homoclinic solutions with extra assumptions about the nonlinearity $f_k$.

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具有无限势的离散非线性薛定谔方程的同宿解

本文研究了离散非线性薛定谔方程同宿解的存在性和多重性 $-\Delta {你}_{克}+v_{克}{你}_{克}-\omega {你}_{克}=F_{克}\left({你}_{克}\right)$被考虑。具有渐近非线性和无限潜力$伏=\{v_{克}：克\in \mathbb{Z}\}$ 满意 ${林}_{\mid 克\mid \to \infty }{v}_{克}=\infty$，我们通过鞍点约简方法和康利指数理论实现了非平凡同宿解的存在。此外，我们可以证明至少有两个非平凡的同宿解，并且对非线性有额外的假设$F_{克}$.