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

Abstract

We consider the $SU(n+1)$ Toda system $$(S_λ) \quad \left\{ \begin{aligned} & Δu_1 + 2λe^{u_1} - λe^{u_2}- \dots - λe^{u_k} = 0\quad \hbox{in}\ Ω,\\ & Δu_2 - λe^{u_1} + 2λe^{u_2} - \dots - λe^{u_k}=0\quad \hbox{in}\ Ω,\\ &\vdots \hskip3truecm \ddots \hskip2truecm \vdots\\ & Δu_k -λe^{u_1}-λe^{u_2}- \dots+2λe^{u_k}=0\quad \hbox{in}\ Ω, &u_1 = u_2 = \dots = u_k =0 \quad \hbox{on}\ \partialΩ.\\ \end{aligned}\right. $$ If $0\inΩ$ and $Ω$ is symmetric with respect to the origin, we construct a family of solutions $({u_1}_λ,\dots,{u_k}_λ)$ to $(S_λ)$ such that the $i-$th component ${u_i}_λ$ blows-up at the origin with a mass $2^{i+1}π$ as $λ$ goes to zero.

arXiv admin note: text overlap with arXiv:1210.5719