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MiS Preprint

28/2000

Analytic Aspects of the Toda system: I. A Moser-Trudinger inequality

Jürgen Jost and Guofang Wang

Abstract

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser-Trudinger type inequality for this system. Let $\Sigma$ be a closed surface with area 1 and $K=(a_{ij})_{N \times N}$ the Cartan matrix for SU(N+1), i.e., $\left( \begin{array}{rcl} 2&-1&0&...&...&0\\-1&2&-1&0&...&0\\0&-1&2&-1&...&0\\...&...&...&...&...&...\\0&...&...&-1&2&-1\\0&...&...&0&-1&2\end{array} \right)$ We show that $\Phi_M (u) = \frac{1}{2} \sum\limits^{N}_{ij=1} \int_\Sigma i_{ij} (\nabla u_i \nabla u_j =2M_i u_j ) -\sum\limits^{N}_{i=1} M_i \log \int_\Sigma \exp \left( \sum\limits^{N}_{j=1} a_{ij} u_j \right) $ has a lower bound in $(H^1 (\Sigma ))^N$ if and only if $M_j \leq 4 \pi , \ \ \ for \ j=1,2,...,N.$ As a direct consequence, if $M_j < 4 \pi $ for $j=1,2,...,N, \Phi_M$ has a minimizer u which satisfies $-\Delta u_i=M_i \left( \frac{\exp \left(\sum\limits^{N}_{j=1} a_{ij} u_{j} \right) }{\int_\Sigma \exp \left( \sum\limits^{N}_{j=1} a_{ij} u_{j} \right) } -1\right), \ for \ 1\leq i \leq N$.