Convergence of ground state solutions for nonlinear Schrödinger equations on graphs

Abstract

We consider the nonlinear Schrödinger 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 $\lambda \to \infty$, 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.

17 pages, 5 figures

References in corpus (2)

• The existence and nonexistence of global solutions for a semilinear heat equation on graphs
• On-diagonal lower estimate of heat kernel on graphs

Cited by in corpus (2)

• On a class of quasilinear elliptic equation with indefinite weights on graphs
• On a class of nonlinear Schrödinger equation on finite graphs