Degree counting and shadow system for Toda system of rank two: one bubbling
type: Item
instance of: scholarly article
title: Degree counting and shadow system for Toda system of rank two: one bubbling
zbMATH Open document ID: 1394.35166
author: Youngae Lee
author: Chang-Shou Lin
author: Wen Yang
author: Juncheng Wei
published in: Journal of Differential Equations
publication date: 2018-02-06T00:00:00Z
review text: The authors initiate a program for computing the Leray-Schauder topological degree for Toda systems of rank two: \[ \begin{cases} \Delta u_1 +K_{11}\varrho_1 \left(\frac{h_1e^{u_1}}{\int_M h_1e^{u_1}\,dv_g} -1 \right) +K_{12}\varrho_2 \left(\frac{h_2e^{u_2}}{\int_M h_2e^{u_2}\,dv_g} -1\right)= 0 \cr \Delta u_2 +K_{21}\varrho_1 \left(\frac{h_1e^{u_1}}{\int_M h_1e^{u_1}\,dv_g} -1 \right) +K_{22}\varrho_2 \left(\frac{h_2e^{u_2}}{\int_M h_2e^{u_2}\,dv_g} -1 \right) = 0\end{cases}\quad\text{in}\; M, \tag{1.3} \] where \(K = \begin{pmatrix} K_{11} & K_{12} \cr K_{21} & K_{22} \end{pmatrix}\) is the Cartan matrix of rank two of the Lie algebra \(\mathfrak{g}\), \(h_1, h_2 \geq 0\) are \(C^{2,\sigma}\) non-negative functions on \(M\), \((M, \mathfrak{g})\) is a compact Riemann surface with volume 1. The authors determine the set of critical parameters \((\varrho_1, \varrho_2)\) for the system (1.3) such that the a priori bounds for the solutions of (1.3) fail, and prove that if a sequence of solutions \((u_{1k}, u_{2k})\) to (1.3) blows up, then one of \[ \frac{h_i e^{u_{ik}}}{\int_M h_i e^{u_{ik}} \,dv_g} ,\quad i = 1, 2, \] tends to a sum of Dirac measures. The authors introduce a Green function \(G(x, p)\), which satisfies \[ -\Delta G(x,p) = \delta_p - 1\quad\text{in}\; M,\quad\text{and}\quad \int_M G(x,p)\,dv_g = 0, \] where \(\delta_q\) is the Dirac measure at \(q\in M\), and investigate the following shadow systems: \[ \begin{cases} \Delta w +2\varrho_2 \left( \frac{h_2e^{w+K_{21}4\pi G(x,p)}}{\int_M h_2e^{w+K_{21}4\pi G(x,p)}\,dv_g}-1 \right)= 0, \cr \nabla \left(\log h_1e^{\frac{K_{12}}{2} w} +4\pi R(x,x)\right)|_{x=p}= 0,\end{cases} \tag{1.5} \] \[ \begin{cases} \Delta w +2\varrho_1 \left( \frac{h_1e^{w+K_{12}4\pi G(x,p)}}{\int_M h_1e^{w+K_{12}4\pi G(x,p)}\,dv_g}-1 \right)= 0, \cr \nabla \left(\log h_2e^{\frac{K_{21}}{2} w} +4\pi R(x,x)\right)|_{x=p}= 0.\end{cases} \tag{1.6} \] The authors show how to calculate the topological degree of the Toda systems by computi…
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35J47
Mathematics Subject Classification ID: 35B44
zbMATH DE Number: 6834793
zbMATH Keywords: Leray-Schauder topological degree
zbMATH Keywords: Toda system
zbMATH Keywords: shadow system
zbMATH Keywords: blow up
MaRDI profile type: Publication
full work available at URL: j.jde.2017.12.018
OpenAlex ID: W2781050708
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DOI: 10.1016/J.JDE.2017.12.018
description: scientific article; zbMATH DE number 6834793