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 $0\in \Omega$ 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.

arXiv admin note: text overlap with arXiv:1210.5719

References in corpus (4)

• Classification and nondegeneracy of $SU(n+1)$ Toda system with singular sources
• Multiple blow-up phenomena for the sinh-Poisson equation
• On Liouville systems at critical parameters, Part 1: one bubble
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Cited by in corpus (5)

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• A continuum of solutions for the SU(3) Toda System exhibiting partial blow-up
• A general existence result for the Toda system on compact surfaces
• Asymmetric blow-up for the SU(3) Toda System
• Blowup solutions of elliptic systems in two dimensional spaces