Some existence results for solutions to \(\text{SU}(3)\) Toda system (Q2581026)

The present paper deals with \(\text{SU}(3)\) Toda systems in non-Abelian relativistic self-dual gauge theory given by \[ -\Delta_g u_1= 2\lambda_1\Biggl({e^{u_1}\over\int_M e^{u_1}}- {1\over|M|}\Biggr)- \lambda_2\Biggl({e^{u_2}\over \int_M e^{u_2}}- {1\over|M|}\Biggr),\tag{1} \] \[ -\Delta_g u_2=-\lambda_1 \Biggl({e^{u_1}\over \int_M e^{u_1}}- {1\over|M|}\Biggr)+ 2\lambda_2\Biggl({e^{u_2}\over \int_M e^{u_2}}- {1\over|M|}\Biggr)\tag{2} \] on \(M\) with \(\int_M u_1= \int_M u_2= 0\), where \(M\) denotes a compact Riemannian surface and \(\lambda_1\), \(\lambda_2\) are nonnegative constants. In the range of parameters where the corresponding Trudinger-Moser inequality fails, the authors show the existence of the solution of (1)--(2).