A Variational Analysis of the Toda System on Compact Surfaces

In this paper we consider the following Toda system of equations on a compact surface:\documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*} \begin{cases} - \Delta u_1 = 2 \rho_1 \Bigl( \frac{h_1 e^{u_1}}{\int \sum h_1 e^{u_1} dV_g} - 1 \Bigr) - \rho_2 \Bigl( \frac{h_2 e^{u_2}}{\int\sum h_2 e^{u_2} dV_g} - 1 \Bigr), \\[2\jot] - \Delta u_2 = 2 \rho_2 \Bigl( \frac{h_2 e^{u_2}}{\int\sum h_2 e^{u_2} dV_g} - 1 \Bigr) - \rho_1 \Bigl( \frac{h_1 e^{u_1}}{\int\sum h_1 e^{u_1} dV_g} - 1 \Bigr) \end{cases}. \end{align*} \end{document} We will give existence results by using variational methods in a noncoercive case. A key tool in our analysis is a new Moser‐Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components u 1 and u 2 . © 2012 Wiley Periodicals, Inc.