New blow-up phenomena for SU(n+1) Toda system

Abstract

We consider the $SU(n+1)$ Toda system $$(S_\lambda) \quad \left\{ \begin{aligned} & \Delta u_1 + 2\lambda e^{u_1} - \lambda e^{u_2}- \dots - \lambda e^{u_k} = 0\quad \hbox{in}\ \Omega,\\ & \Delta u_2 - \lambda e^{u_1} + 2\lambda e^{u_2} - \dots - \lambda e^{u_k}=0\quad \hbox{in}\ \Omega,\\ &\vdots \hskip3truecm \ddots \hskip2truecm \vdots\\ & \Delta u_k -\lambda e^{u_1}-\lambda e^{u_2}- \dots+2\lambda e^{u_k}=0\quad \hbox{in}\ \Omega, &u_1 = u_2 = \dots = u_k =0 \quad \hbox{on}\ \partial\Omega.\\ \end{aligned}\right. $$ If $0\in\Omega$ and $\Omega$ is symmetric with respect to the origin, we construct a family of solutions $({u_1}_\lambda,\dots,{u_k}_\lambda)$ to $(S_\lambda )$ such that the $i-$th component ${u_i}_\lambda$ blows-up at the origin with a mass $2^{i+1}\pi $ as $\lambda$ goes to zero.