LAUSR

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