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Communications in Mathematical Physics

, Volume 349, Issue 1, pp 393–424 | Cite as

The Domain Geometry and the Bubbling Phenomenon of Rank Two Gauge Theory

  • Hsin-Yuan HuangEmail author
  • Lei Zhang
Article

Abstract

Let \({\Omega}\) be a flat torus and \({G}\) be the Green’s function of \({-\Delta}\) on \({\Omega}\). One intriguing mystery of \({G}\) is how the number of its critical points is related to blowup solutions of certain PDEs. In this article we prove that for the following equation that describes a Chern–Simons model in Gauge theory:
$$\begin{aligned} \left\{\begin{array}{ll}\Delta u_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=8\pi\delta_{p_{1}} \\ \Delta u_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=8\pi\delta_{p_{2}}\end{array}\text{ in } \quad \Omega\right., \quad p_1-p_2 {\text{ is a half period}}, \end{aligned}$$
(0.1)
if fully bubbling solutions of Liouville type exist, \({G}\) has exactly three critical points. In addition we establish necessary conditions for the existence of fully bubbling solutions with multiple bubbles.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of MathematicsNational Sun Yat-sen UniversityKaoshiungTaiwan
  2. 2.Mathematics Division, National Center for Theoretical SciencesNational Taiwan UniversityTaipeiTaiwan
  3. 3.Department of MathematicsUniversity of FloridaGainesvilleUSA

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