论文标题
在图上存在解决方案的解决方案
Existence of solutions to Chern-Simons-Higgs equations on graphs
论文作者
论文摘要
令$ g =（v，e）$为有限图。我们考虑存在通用的Chern-Simons-higgs方程的解决方案$Δu=-λe^{g（u）} \ left（e^{g（u）} -1 \ right）^2+4π\ sum \ limits_ $ g（u）$是$ u = f（\ upsilon）= 1+\ upsilon-e^{\ upsilon} $（ - \ infty，0] $; $ n $是一个正整数；是$ P_J $的Dirac Delta质量。 $Δu=λe^{u}（e^{u} -1）+4π\ sum \ limits_ {j = 1}^{n}Δ_{p_j} $ g $时，$λ$当$λ$进行关键值$λ_c$并完成huang，yong yong yong yong yong y的关键值$λ_c$时。 613-621（2020））。
Let $G=(V,E)$ be a finite graph. We consider the existence of solutions to a generalized Chern-Simons-Higgs equation $$ Δu=-λe^{g(u)}\left( e^{g(u)}-1\right)^2+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$, where $λ$ 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 $δ_{p_j}$ is the Dirac delta mass at $p_j$. We prove that there is critical value $λ_c$ such that the generalized Chern-Simons-Higgs equation has a solution if and only if $λ\geq λ_c$ . We also prove the existence of solutions to the Chern-Simons-Higgs equation $$ Δu=λe^{u}(e^{u}-1)+4π\sum\limits_{j=1}^{N}δ_{p_j} $$ on $G$ when $λ$ takes the critical value $λ_c$ and this completes the results of An Huang, Yong Lin and Shing-Tung Yau (Commun. Math. Phys. 377, 613-621 (2020)).
