Construction of Multi-Bubble Solutions for a System of Elliptic Equations arising in Rank Two Gauge Theory

Abstract

We study the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system \begin{equation}\label{e051} \left\{ \begin{split} &\Delta u_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=4\pi\sum_{i=1}^{2k}\delta_{p_{1,i}}\\ &\Delta u_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=4\pi\sum_{i=1}^{2k}\delta_{p_{2,i}} \end{split} \text{ in }\quad \Omega\right., \end{equation} where $k\geq 1$ and $\Omega$ is a flat tours in $\mathbb{R}^2$. It continues the joint work with Zhang\cite{HZ-2015}, where we obtained the necessary conditions for the existence of bubbling solutions of Liouville type. Under nearly necessary conditions(see Theorem \ref{main-thm}), we show that there exist a sequence of solutions $(u_{1,\varepsilon}, u_{2,\varepsilon})$ to \eqref{e051} such that $u_{1,\varepsilon}$ and $u_{2,\varepsilon}$ blow up simultaneously at $k$ points in $\Omega$ as $\varepsilon\to 0$.