Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

Paper

Abstract

We consider the nonlinear Schrödinger equation

on a locally finite graph

. We prove via the Nehari method that if

satisfies certain assumptions, for any

, the equation admits a ground state solution

. Moreover, as

, the solution

converges to a solution of the Dirichlet problem

which is defined on the potential well

. We also provide a numerical experiment which solves the equation on a finite graph to illustrate our results.