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 $\Delta$ be the usual graph Laplacian. Consider the following nonlinear Schr$\ddot{o}$dinger type equation of the form $$\{\begin{array}{l}-\Delta u-\alpha u=f(x,u),\\ u\in W^{1,2}(V),\end{array}$$ on graph $G$, where $f(x,u):V\times \mathbb{R}\to \mathbb{R}$ 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.

10 pages

References in corpus (2)

• The existence and nonexistence of global solutions for a semilinear heat equation on graphs
• Convergence of ground state solutions for nonlinear Schrödinger equations on graphs