On a class of nonlinear Schr\"odinger equation on finite graphs

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $\Delta$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$\left \{ \begin{array}{lcr} -\Delta u-\alpha u=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right.$$ on graph $G$, where $f(x,u):V\times\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear function and $\alpha$ is a parameter. Firstly, we prove the Trudinger-Moser inequality on graph $G$, and under the assumption that $G$ satisfies the curvature-dimension type inequality $CD(m, \xi)$, we prove an integral inequality on $G$. Then by using the two inequalities, we prove that there exists a positive solution to the nonlinear Schr$\ddot{o}$dinger type equation if $\alpha<\frac{2\lambda^{2}}{m(\lambda-\xi)}$, where $\lambda$ is the eigenvalue of the graph Laplacian. Our work provides remarkable improvements to the previous results.