Some existence results for the Toda system on closed surfaces

Abstract

Given a compact closed surface $\Sigma$, we consider the generalized Toda system of equations on $\Sigma$: $-\Delta u_i={\sum }_{j=1}^2{\rho }_j{a}_{ij}\left(\frac{h_j{e}^{u_j}}{{\int }_{\Sigma }{h}_j{e}^{u_j}d{V}_g}-1\right)$ for $i=1,2$, where ${\rho }_1,{\rho }_2$ are real parameters and $h_1,h_2$ are smooth positive functions. Exploiting the variational structure of the problem and using a new minimax scheme we prove existence of solutions for generic values of ${\rho }_1$ and for ${\rho }_2<4\pi$.