On a class of nonlinear Schrödinger equation on finite graphs

Abstract

Suppose that $G=(V, E)$ is a finite graph with the vertex set $V$ and the edge set $E$. Let $Δ$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$ \left \{ \begin{array}{lcr} -Δu-αu=f(x,u),\\ u\in W^{1,2}(V),\\ \end{array} \right. $$ on graph $G$, where $f(x,u):V\times\mathbb{R}\rightarrow\mathbb{R}$ is a nonlinear function and $α$ 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 $CD(m, ξ)$, 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 $α<\frac{2λ^{2}}{m(λ-ξ)}$, where $λ$ is the eigenvalue of the graph Laplacian. Our work provides remarkable improvements to the previous results.