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

TL;DR

This paper establishes a topological degree counting formula for a class of generalized Liouville systems on surfaces, extending classical results and providing new insights into their solution structure.

Contribution

It introduces the first degree theory counting formula for Liouville systems, generalizing classical equations and connecting topological invariants with solution counts.

Findings

Derived explicit Leray-Schauder degree formula under certain conditions.

Connected degree calculation with topological invariants like Euler characteristic.

Provided new tools for analyzing solution existence and multiplicity.

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} \Delta_g u_i+\sum_{j=1}^n a_{ij}\rho_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 $\rho_j\in \mathbb R^+$, 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{(-\chi(M)+1)... (-\chi(M)+N)}{N!} if $\rho=(\rho_1,..., \rho_n)$ satisfies 8\pi N\sum_{i=1}^n\rho_i<\sum_{1\leq i,j\leq n}a_{ij}\rho_i\rho_j<8\pi(N+1)\sum_{i=1}^n\rho_i. Equation \eqref{abeq1} is a natural generalization of the classic Liouville equation and is the Euler-Lagrangian equation of Nonlinear function $\varPhi_\rho$: \varPhi_\rho(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_M\rho_iu_i -\sum_{i=1}^n\rho_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.