A Topological Degree Counting for some Liouville Systems of Mean Field Equations

Abstract

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<\sum_{1\leq i,j\leq n}a_{ij}ρ_iρ_j<8π(N+1)\sum_{i=1}^nρ_i. Equation \eqref{abeq1} is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of Nonlinear function $\varPhi_ρ$: \varPhi_ρ(u)=1/2\int_M\sum_{1\leq i,j\leq n}a^{ij}\nabla_g u_i\cdot \nabla_g u_j+\sum_{i=1}^n\int_Mρ_iu_i -\sum_{i=1}^nρ_i\log \int_M h_i e^{u_i}. The Liouville system \eqref{abeq1} has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems.