Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

By Ning Zhang, Liang Zhao. Science China Mathematics, 2017

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0027d4671ddaa399322c3495688aa068da45f1d9