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.
Categories: math.DG