On Non-Topological Solutions for Planar Liouville Systems of Toda-Type

Motivated by the study of non-abelian Chern Simons vortices of non-topological type in Gauge Field Theory, see e.g. Gudnason (Nucl Phys B 821:151–169, 2009), Gudnason (Nucl Phys B 840:160–185, 2010) and Dunne (Lecture Notes in Physics, New Series, vol 36. Springer, Heidelberg, 1995), we analyse the solvability of the following (normalised) Liouville-type system in the presence of singular sources: (1)τ-Δu1=eu1-τeu2-4Nπδ0,-Δu2=eu2-τeu1,β1=12π∫R2eu1andβ2=12π∫R2eu2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1)_\tau \begin{cases}-\Delta u_1 = e^{u_1} - \tau e^{u_2} - 4N \pi \, \delta_0,\\-\Delta u_2 = e^{u_2} - \tau e^{u_1},\\ \beta_1 = \frac{1}{2\pi} \int_{\mathbb{R}^{2}} e^{u_1} \, {\rm and } \, \beta_2 = \frac{1}{2\pi} \int_{\mathbb{R}^{2}} e^{u_2},\end{cases}$$\end{document}with τ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tau > 0}$$\end{document} and N>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N > 0}$$\end{document}.