Ground state solutions for asymptotically linear Schrodinger equations on locally finite graphs

We are considered with the following nonlinear Schrodinger equation: -Delta u + (lambda a(x) + 1)u = (u), x is an element of V, on a locally finite graph G = (V, E), where V denotes the vertex set, E denotes the edge set lambda > 1 is a parameter, f(s) is asymptotically linear with respect to sat infinity, and the potential a V -> [0, +infinity) has a nonempty well Omega. Byusing variational methods, we prove that the above problem has a ground state solution u(lambda) for any lambda > 1. Moreover, we show that as lambda -> infinity, the ground state solution u(lambda) converges to a ground state solution of a Dirich let problem defined on the potential well Omega.