Convergence of ground state solutions for nonlinear Schrödinger equations on graphs
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.
KeywordsSchrödinger equation locally finite graph ground state potential well
MSC(2010)35A15 35Q55 58E30
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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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