Existence of solutions to 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 $$\Delta u=-\lambda e^{g(u)}{\left(e^{g(u)}-1\right)}^2+4\pi \sum _{j=1}^N{\delta }_{p_j}$$ 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 $\lambda \ge {\lambda }_c$ . We also prove the existence of solutions to the Chern-Simons-Higgs equation $$\Delta u=\lambda e^u(e^u-1)+4\pi \sum _{j=1}^N{\delta }_{p_j}$$ 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)).

13 pages