We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equations with a sign-changing potential V that converges at infinity and a nonlinear term being asymptotically linear at infinity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite and the other is that, due to the convergency of V at infinity, the classical methods such as periodic translation technique and compact inclusion method cannot be employed directly to deal with the lack of compactness of the Cerami sequence. New techniques are developed in this work to overcome these two major difficulties. This enables us to establish the existence of a ground state solution and derive a necessary and sufficient condition for a special case. To the best of our knowledge, this is the first attempt in the literature on the existence of a ground state solution for the strongly indefinite problem under no periodicity condition on the bounded potential and the nonlinear term being asymptotically linear at infinity. Moreover, our conditions can also be used to significantly improve the well-known results of the corresponding continuous nonlinear Schrödinger equation.

中文翻译：

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具有界和非界势的离散渐近线性Schrödinger方程的基态解

我们研究了一类离散的非线性Schrödinger方程的基态解的存在性，这些方程的符号变化势V在无穷大处收敛，而非线性项在无穷大处渐近线性。由此产生的问题有两个主要困难：一个是相关的函数是非常不确定的，另一个是由于V的收敛性在无限远时，经典方法（例如，周期性翻译技术和紧凑包含方法）不能直接用于解决Cerami序列缺乏紧凑性的问题。在这项工作中开发了新技术来克服这两个主要困难。这使我们能够建立基态解的存在，并为特殊情况导出必要和充分的条件。据我们所知，这是在有界电势无周期性条件且非线性项渐近线性的无周期性条件下，针对强不确定问题的基态解的存在的文献中的首次尝试。此外，我们的条件还可用于显着改善相应的连续非线性Schrödinger方程的众所周知的结果。