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Ground States for DNLS Equation with Periodic or Asymptotically Periodic Potential

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Abstract

We study the existence of ground state solutions for a class of discrete nonlinear Schrödinger equation with a sign-changing potential which is periodic or asymptotically periodic. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the ℤ-translation invariance, and many effective methods for periodic problems cannot be applied to asymptotically periodic ones. These enables us to develop a direct approach to find ground state solutions with asymptotically periodic potential. Two types of ground state solutions are obtained with some new super-quadratic conditions on nonlinearity which are weaker that some well-known ones. 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

The authors are grateful to the anonymous referees for their contribution to improve the quality of the manuscript. This work was supported in part by the National Natural Science Foundation of China (No. 11301297) and Natural Science Foundation of Hubei Province and Yichang City.

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Authors and Affiliations

  1. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, 443002, China

    Peng Chen

  2. College of Science, China Three Gorges University, Yichang, 443002, China

    Peng Chen & Li Meng

  3. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Xianhua Tang

Authors
  1. Peng Chen
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  2. Li Meng
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  3. Xianhua Tang
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Correspondence to Peng Chen.

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Chen, P., Meng, L. & Tang, X. Ground States for DNLS Equation with Periodic or Asymptotically Periodic Potential. Front. Math 19, 467–494 (2024). https://doi.org/10.1007/s11464-021-0271-8

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  • DOI: https://doi.org/10.1007/s11464-021-0271-8

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