Convergence of ground state solutions for nonlinear Schrodinger equations on graphs

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