The domain geometry and the bubbling phenomenon of rank two Gauge theory

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{equation}\label{e103} \left\{ \begin{array}{ll} Δu_1+\frac{1}{\varepsilon^2}e^{u_2}(1-e^{u_1})=8πδ_{p_{1}} Δu_2+\frac{1}{\varepsilon^2}e^{u_1}(1-e^{u_2})=8πδ_{p_{2}} \end{array} \text{ in }\quad Ω\right., \quad p_1-p_2 \mbox{ is a half period}, \end{equation} 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.

arXiv comment33 pages, to appear on Communications in Mathematical Physics, 2016

References in corpus (5)

• Schrödinger Fields on the Plane with $[U(1){]}^N$ Chern-Simons Interactions and Generalized Self-dual Solitons
• Profile of bubbling solutions to a Liouville system
• Low energy dynamics of U(1)^{N} Chern-Simons solitons
• Mean field equations, hyperelliptic curves and modular forms: II
• Uniqueness of topological multi-vortex solutions for a skew-symmetric Chern-Simons system

Cited by in corpus (3)

• Structure of bubbling solutions of Liouville systems with negative singular sources
• On Liouville systems at critical parameters, Part 2: Multiple bubbles
• Construction of Multi-Bubble Solutions for a System of Elliptic Equations arising in Rank Two Gauge Theory