论文标题
存在有限图上的广义自我偶义Chern-Simons方程的解决方案
Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs
论文作者
论文摘要
令$ g =（v，e）$为连接的有限图。我们研究了以下广义Chern-Simons方程的解决方案的存在， ΔU=λ\ Mathrm {e}^{ $δ_{p_ {s}} $是vetex $ p_s $的零质量，而$ p_1，p_2，\ dots，p_n $是图表上任意选择的不同顶点。我们表明，存在批评价值$ \hatλ$，因此当$λ> \hatλ$时，广义的Chern-Simons方程至少具有两个解决方案，当$λ= \hatλ$时，广义的Chern-Simons方程就会有一个解决方案，当$λ<\hatλ$时，Chern-simimizized chern-simons simons emimanseequeation nes nessimate equalation ne sore。
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*} Δu=λ\mathrm{e}^{u}\left(\mathrm{e}^{u}-1\right)^{5}+4 π\sum_{s=1}^{N} δ_{p_{s}} \quad , \end{equation*} where $λ>0$, $δ_{p_{s}}$ 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 $\hatλ$ such that when $λ> \hatλ$, the generalized Chern-Simons equation has at least two solutions, when $λ= \hatλ$, the generalized Chern-Simons equation has a solution, and when $λ< \hatλ$, the generalized Chern-Simons equation has no solution.
