Degree counting and shadow system for Toda system of rank two: one bubbling
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 computing the topological degree of the general shadow systems and calculate the topological degree of the shadow system for one blow up point.
scientific article; zbMATH DE number 6834793
scientific article