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 a general class of planar Liouville-type system in the presence of singular sources. We identify necessary and sufficient conditions on the given physical parameters which ensure the radial solvability. In particular we recover the existence result of Lin et al. (Invent Math 190(1):169–207, 2012) and Jost and Wang (Int Math Res Not 6:277–290, 2002), concerning the integrable 2 × 2 Toda system. Our method relies on a blow-up analysis, which (even in the radial setting) takes new turns compared to the case of the single equation. We mention that our approach also permits to handle the non-symmetric cases, and when both equations include a Dirac measures supported at the origin.