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

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### 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