有限图上广义self-dual Chern-Simons方程解的存在性
Existence of solutions to a generalized self-dual Chern-Simons equation on finite graphs
摘要：
设$G=(V,E)$是一个连通有限图.本文研究了$G$ \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*}上的广义Chern-Simons方程解的存在性，其中$λ>0$，$δ_{p_{s}}$是$p_s$上的Dirac质量，$p_1, p_2,\dots, p_N$是图上任意选取的不同顶点.证明了存在一个临界值$\hatλ$，使得当$λ> \hatλ$时，广义Chern-Simons方程至少有两个解，当$λ= \hatλ$时，广义Chern-Simons方程有解，当$λ< \hatλ$时，广义Chern-Simons方程无解.
英文摘要：
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.
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