Existence of solutions to Chern-Simons-Higgs equations on graphs

TL;DR

This paper investigates the existence of solutions to generalized Chern-Simons-Higgs equations on finite graphs, establishing a critical parameter value for solution existence and completing previous results for the classical case.

Contribution

The paper proves the existence of solutions for a generalized Chern-Simons-Higgs equation on graphs, identifying a critical parameter and extending prior work to the classical equation at this critical value.

Findings

Existence of solutions for mbda  when mbda  is exceeded

Solutions exist at the critical value mbda  for the classical equation

Complete the theoretical framework for Chern-Simons-Higgs equations on graphs

Abstract

Let $G=(V,E)$ be a finite graph. We consider the existence of solutions to a generalized Chern-Simons-Higgs equation $$ \Delta u=-\lambda e^{g(u)}\left( e^{g(u)}-1\right)^2+4\pi\sum\limits_{j=1}^{N}\delta_{p_j} $$ on $G$, where $\lambda$ is a positive constant; $g(u)$ is the inverse function of $u=f(\upsilon)=1+\upsilon-e^{\upsilon}$ on $(-\infty, 0]$; $N$ is a positive integer; $p_1, p_2, \cdot\cdot\cdot, p_N$ are distinct vertices of $V$ and $\delta_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there is critical value $\lambda_c$ such that the generalized Chern-Simons-Higgs equation has a solution if and only if $\lambda\geq \lambda_c$ . We also prove the existence of solutions to the Chern-Simons-Higgs equation $$ \Delta u=\lambda e^{u}(e^{u}-1)+4\pi\sum\limits_{j=1}^{N}\delta_{p_j} $$ on $G$ when $\lambda$ takes the critical value $\lambda_c$ and this completes the results of An Huang, Yong Lin and Shing-Tung Yau (Commun. Math. Phys. 377, 613-621 (2020)).