MiS Preprint Repository

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser-Trudinger type inequality for this system. Let $\mathrm{\Sigma }$ be a closed surface with area 1 and $K=(a_{ij}{)}_{N\times N}$ the Cartan matrix for SU(N+1), i.e.,
$\left(\begin{array}{rcllll}2& -1& 0& ...& ...& 0\\ -1& 2& -1& 0& ...& 0\\ 0& -1& 2& -1& ...& 0\\ ...& ...& ...& ...& ...& ...\\ 0& ...& ...& -1& 2& -1\\ 0& ...& ...& 0& -1& 2\end{array}\right)$
We show that
${\mathrm{\Phi }}_M(u)=\frac{1}{2}\sum _{ij=1}^N{\int }_{\mathrm{\Sigma }}{i}_{ij}(\mathrm{\nabla }{u}_i\mathrm{\nabla }{u}_j=2M_i{u}_j)-\sum _{i=1}^N{M}_i\log {\int }_{\mathrm{\Sigma }}\exp \left(\sum _{j=1}^N{a}_{ij}{u}_j\right)$
has a lower bound in $(H^1(\mathrm{\Sigma }){)}^N$ if and only if
$M_j\le 4\pi ,forj=1,2,...,N.$
As a direct consequence, if $M_j<4\pi$ for $j=1,2,...,N,{\mathrm{\Phi }}_M$ has a minimizer u which satisfies
$-\mathrm{\Delta }{u}_i=M_i(\frac{\exp \left(\sum _{j=1}^N{a}_{ij}{u}_j\right)}{{\int }_{\mathrm{\Sigma }}\exp \left(\sum _{j=1}^N{a}_{ij}{u}_j\right)}-1),for1\le i\le N$.