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Ground State Solutions of Discrete Asymptotically Linear Schrödinger Equations with Bounded and Non-periodic Potentials

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Abstract

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.

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Acknowledgements

We are thankful to the anonymous reviewer for the thoughtful and constructive comments which largely help improve this article. This research was partially supported by the National Natural Science Foundation of China (Grant No. 11571084), the Program for Changjiang Scholars and Innovative Research Team in University (Grant No. IRT_16R16) and the Guangzhou Postdoctoral International Training Program Funding Project.

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Authors and Affiliations

  1. Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, China

    Genghong Lin, Zhan Zhou & Jianshe Yu

  2. School of Mathematics and Information Science, Guangzhou University, Guangzhou, 510006, China

    Genghong Lin & Zhan Zhou

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  1. Genghong Lin
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  2. Zhan Zhou
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  3. Jianshe Yu
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Correspondence to Jianshe Yu.

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Lin, G., Zhou, Z. & Yu, J. Ground State Solutions of Discrete Asymptotically Linear Schrödinger Equations with Bounded and Non-periodic Potentials. J Dyn Diff Equat 32, 527–555 (2020). https://doi.org/10.1007/s10884-019-09743-4

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  • DOI: https://doi.org/10.1007/s10884-019-09743-4

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