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Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

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Abstract

We consider the nonlinear Schrödinger equation -Δu + (λa(x) + 1)u = |u|p-1u on a locally finite graph G = (V,E). We prove via the Nehari method that if a(x) satisfies certain assumptions, for any λ > 1, the equation admits a ground state solution uλ. Moreover, as λ → 1, the solution uλ converges to a solution of the Dirichlet problem -Δu+u = |u|p-1u which is defined on the potential well Ω. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.

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Acknowledgements

The first author was supported by the Funding of Beijing Philosophy and Social Science (Grant No. 15JGC153) and the Ministry of Education Project of Humanities and Social Sciences (Grant No. 16YJCZH148). The second author was supported by the Fundamental Research Funds for the Central Universities. Both of the authors were supported by the Ministry of Education Project of Key Research Institute of Humanities and Social Sciences at Universities (Grant No. 16JJD790060). The authors thank members of Data Lighthouse for their helpful conversations and valuable suggestions. The authors are also thankful for the referees′ detailed and useful comments.

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Authors and Affiliations

  1. China Institute for Actuarial Science, Central University of Finance and Economics, Beijing, 100081, China

    Ning Zhang

  2. School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems of MOE, Beijing Normal University, Beijing, 100875, China

    Liang Zhao

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  1. Ning Zhang
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  2. Liang Zhao
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Correspondence to Liang Zhao.

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Zhang, N., Zhao, L. Convergence of ground state solutions for nonlinear Schrödinger equations on graphs. Sci. China Math. 61, 1481–1494 (2018). https://doi.org/10.1007/s11425-017-9254-7

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  • DOI: https://doi.org/10.1007/s11425-017-9254-7

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