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

We are considered with the following nonlinear Schrödinger equation −∆ u +( λa ( x )+1) u = f ( u ) ,x ∈ V, on a locally finite graph G =( V,E ), where V denotes the vertex set, E denotes the edge set, λ> 1 is a parameter, f ( s ) is asymptotically linear with respect to s at infinity, and the potential a : V →[0 , +∞) has a nonempty well Ω. By using variational methods we prove that the above problem has a ground state solution u λ for any λ> 1. Moreover, we show that as λ →∞, the ground state solution u λ converges to a ground state solution of a Dirichlet problem defined on the potential well Ω.