In this paper, we study the nonlinear Choquard equation

$$-\mathrm{\Delta }u+V(x)u=\left(\sum _{\genfrac{}{}{0ex}{}{y\ne x}{y\in {\mathbb{Z}}^N}}\frac{|u(y){|}^p}{|x-y{|}^{N-\alpha }}\right)|u{|}^{p-2}u$$

on lattice graph ${\mathbb{Z}}^N$. Under some suitable assumptions, we prove the existence of a ground state solution of the equation on the graph when the function $V$ is periodic or confining. Moreover, when the potential function $V(x)=\lambda a(x)+1$ is confining, we obtain the asymptotic properties of the solution $u_{\lambda }$ which converges to a solution of a corresponding Dirichlet problem as $\lambda \to \mathrm{\infty }$.

In this paper we study the existence of multiple nontrivial solutions of the coupled Schrödinger system with external sources terms as perturbations. This type of the system arises from Bose-Einstein condensate. As these external sources terms are nonlinear functions and small in some sense, we use fibre map to divide the Nehari manifold into threes parts, and then prove the existence of a nontrivial ground state solution and a bound state solution.