On Solutions for Singular Toda System on Riemann Surfaces with Boundary

Abstract

This paper studies solutions to a singular $SU(3)$ Toda system with linear source terms on a compact Riemann surface $\hat{I}\pounds$ with smooth boundaries $\partial \hat{I}\pounds$. We establish the existence of solutions when the parameters are not critical, assuming that Euler characteristic $\ddot{I}‡(\hat{I}\pounds )<1$ via analyzing the sublevels. Furthermore, we find a sufficient condition that ensures multiple solutions for generic potentials by Morse inequalities and a transversality theorem.

The Moser-Trudinger Inequality applied in this paper is incorrect. By the approach in this paper, it cannot obtain the general results