MiS Preprint Repository

In this paper, we continue to consider the 2-dimensional (open) Toda system (Toda lattice) for $SU(N+1)$ \begin{equation}- \Delta u_i = \sum_{j=1}^N a_{ij} e^{u_j}, \end{equation} for $i=1,2,\cdots, N$, where $K=(a_{ij})_{N\times N}$ is the Cartan matrix for $SU(N+1)$ given by \[ \left(\begin{array}{rrrrrr} 2 & -1 & 0 & \cdots & \cdots & 0\cr -1& 2 &-1& 0 & \cdots & 0\cr 0&-1&2&-1& \cdots & 0\cr \cdots&\cdots&\cdots&\cdots&\cdots&\cdots\cr 0 &\cdots&\cdots& -1&2&-1\cr 0&\cdots&\cdots & 0 &-1&2\cr\end{array}\right).\nonumber\] we give a much more precise bubbling behavior of solutions and consider its existence in some critical cases