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

2018/11/15 by Hsin-Yuan Huang, Huang, Hsin-Yuan
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1811.06463

arxiv created 2018/11/15 · arxiv updated 2018/11/16

Abstract

We study the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system \ \beginsplit Δu₁+(1)/(ε²)e^u₂(1-e^u₁)=4π∑_i=1^2kδ_p_1,i
Δu₂+(1)/(ε²)e^u₁(1-e^u₂)=4π∑_i=1^2kδ_p_2,i \endsplit in Ω., where k≥ 1 and Ω is a flat tours in ℝ². It continues the joint work with Zhang\citeHZ-2015, where we obtained the necessary conditions for the existence of bubbling solutions of Liouville type. Under nearly necessary conditions(see Theorem \refmain-thm), we show that there exist a sequence of solutions (u_1,ε, u_2,ε) to \eqrefe051 such that u_1,ε and u_2,ε blow up simultaneously at k points in Ω as ε→ 0.

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