Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

Abstract

We consider the nonlinear Schrödinger equation $-Δu+(λa(x)+1)u=|u|^{p-1}u$ on a locally finite graph $G=(V,E)$. We prove via the Nehari method that if $a(x)$ satisfies certain assumptions, for any $λ>1$, the equation admits a ground state solution $u_λ$. Moreover, as $λ\rightarrow \infty$, the solution $u_λ$ converges to a solution of the Dirichlet problem $-Δu+u=|u|^{p-1}u$ 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.