*Analysis and PDE*

**Inserted:** 18 mar 2015

**Last Updated:** 5 nov 2015

**Year:** 2015

**Abstract:**

We consider a Toda system of Liouville equations on a compact surface $\Sigma$ which arises as a model for non-abelian Chern-Simons vortices.

For the first time the range of parameters $\rho_1 \in (4k\pi , 4(k+1)\pi)$, $k\in\mathbb{N}$, $\rho_2 \in (4\pi, 8\pi )$ is studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by means of a new improved Moser-Trudinger type inequality and introducing a topological join construction in order to describe the interaction of the two components.

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