Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs

TL;DR

This paper proves the existence and multiplicity of solutions for a generalized self-dual Chern-Simons equation on finite graphs, depending on a parameter threshold, extending understanding of such equations in discrete settings.

Contribution

It establishes the existence, non-existence, and multiplicity of solutions for the generalized Chern-Simons equation on finite graphs based on a critical parameter value.

Findings

Existence of solutions when parameter exceeds a critical value

Unique solution at the critical parameter

No solutions when parameter is below the critical value

Abstract

Let $G=(V,E)$ be a connected finite graph. We study the existence of solutions for the following generalized Chern-Simons equation on $G$ \begin{equation*} \Delta u=\lambda \mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 \pi \sum_{s=1}^{N} \delta_{p_{s}} \quad , \end{equation*} where $\lambda>0$, $\delta_{p_{s}}$ is the Dirac mass at the vetex $p_s$, and $p_1, p_2,\dots, p_N$ are arbitrarily chosen distinct vertices on the graph. We show that there exists a critial value $\hat{\lambda}$ such that when $\lambda > \hat{\lambda}$, the generalized Chern-Simons equation has at least two solutions, when $\lambda = \hat{\lambda}$, the generalized Chern-Simons equation has a solution, and when $\lambda < \hat\lambda$, the generalized Chern-Simons equation has no solution.