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: We identify necessary and sufficient conditions on the parameter and the "flux" pair: which ensure the radial solvability of Since for problem reduces to the (integrable) 2 2 Toda system, 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 this case. Our method relies on a blow-up analysis for solutions of , which (even in the radial setting) takes new turns compared to the single equation case. We mention that our approach also permits handling the non-symmetric case, where in each of the two equations in , the parameter is replaced by two different parameters and respectively, and also when the second equation in includes a Dirac measure supported at the origin.