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} &Δu_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=4π\sum_{i=1}^{2k}δ_{p_{1,i}}\\ &Δu_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=4π\sum_{i=1}^{2k}δ_{p_{2,i}} \end{split} \text{ in }\quad Ω\right., \end{equation} where $k\ge 1$ 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 $(u_{1,\epsilon },u_{2,\epsilon })$ to \eqref{e051} such that $u_{1,\epsilon }$ and $u_{2,\epsilon }$ blow up simultaneously at $k$ points in $\Omega$ as $\epsilon \to 0$.

References in corpus (6)

• Schrödinger Fields on the Plane with $[U(1){]}^N$ Chern-Simons Interactions and Generalized Self-dual Solitons
• Non-Abelian Chern-Simons Vortices
• Low energy dynamics of U(1)^{N} Chern-Simons solitons
• Non-Abelian Chern-Simons vortices with generic gauge groups
• Fractional and semi-local non-Abelian Chern-Simons vortices
• The domain geometry and the bubbling phenomenon of rank two Gauge theory