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Register a new userConstruction of Multi-Bubble Solutions for a System of Elliptic Equations arising in Rank Two Gauge Theory
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Hsin-Yuan Huang
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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})=4pisum_{i=1}^{2k}delta_{p_{1,i}} &Delta u_2+frac{1}{varepsilon^2}e^{u_1}(1-e^{u_2})=4pisum_{i=1}^{2k}delta_{p_{2,i}} end{split} text{ in }quad Omegaright., end{equation} where $kgeq 1$ and $Omega$ is a flat tours in $mathbb{R}^2$. It continues the joint work with Zhangcite{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 $varepsilonto 0$.
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