On Non-Topological Solutions for Planar Liouville Systems of Toda-Type Academic Article uri icon

abstract

  • We identify necessary and sufficient conditions on the parameter \({\tau}\) and the “flux” pair: \({(\beta_1, \beta_2),}\) which ensure the radial solvability of \({(1)_\tau.}\) Since for \({\tau=\frac{1}{2},}\) problem \({(1)_\tau}\) reduces to the (integrable) 2 \({\times}\) 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 \({(1)_\tau}\), 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 \({(1)_\tau}\), the parameter \({\tau}\) is replaced by two different parameters \({\tau_1 > 0}\) and \({\tau_2 > 0}\) respectively, and also when the second equation in \({(1)_\tau}\) includes a Dirac measure supported at the origin.

publication date

  • January 1, 2016