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 $\varepsilon\to 0$.
Categories: math.AP