A Topological Degree Counting for some Liouville Systems of Mean Field Equations
Chang‐Shou Lin, Lei Zhang · arXiv (Cornell University) · 2010
Let $A=(a_{ij})_{n\times n}$ be an invertible matrix and $A^{-1}=(a^{ij})_{n\times n}$ be the inverse of $A$. In this paper, we consider the generalized Liouville system: \label{abeq1} Δ_g u_i+\sum_{j=1}^n a_{ij}ρ_j(\frac{h_j e^{u_j}}{\int h_j e^{u_j}}-1)=0\quad\text{in \,}M, where $0< h_j\in C^1(M)$ and $ρ_j\in \mathbb R^+$, and prove that, under the assumptions of $(H_1)$ and $(H_2)$\,(see Introduction), the Leray-Schauder degree of \eqref{abeq1} is equal to \frac{(-χ(M)+1)... (-χ(M)+N)}{N!} if $ρ=(ρ_1,..., ρ_n)$ satisfies 8πN\sum_{i=1}^nρ_i