A topological degree counting for some Liouville systems of mean field type

Let A = ( a ij ) n × n be an invertible matrix and A −1 = ( a ij ) n × n be the inverse of A . In this paper, we consider the generalized Liouville system 0.1$$\Delta_g u_i+\sum_{j=1}^n a_{ij}\rho_j\left({h_j e^{u_j}\over \int h_j e^{u_j}}-1\right)=0\quad {\rm in }\,M,$$ where 0 < h j ∈ C 1 ( M ) and \input amssym $\rho_j \in \Bbb R^+$ , and prove that, under the assumptions of ( H 1 ) and ( H 2 ) (see Introduction), the Leray‐Schauder degree of (0.1) is equal to$${(-\chi(M)+1)\cdots (-\chi(M)+N)\over N!}$$ if ρ = (ρ 1 , …, ρ 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 (0.1) is a natural generalization of the classic Liouville equation and is the Euler‐Lagrangian equation of the nonlinear function Φ ρ :$$\Phi_{\rho(u)}={1\over 2}\int\limits_M \sum_{1\leq i,j\leq n}a^{ij}\nabla_g u_i\cdot \nabla_{g} u_j+\sum_{i=1}^n\int\limits_M\rho_i u_i - \sum_{i=1}^{n} \rho_i {\rm log} \int\limits_M h_i e^{u_i}.$$The Liouville system (0.1) has arisen in many different research areas in mathematics and physics. Our counting formulas are the first result in degree theory for Liouville systems. © 2010 Wiley Periodicals, Inc.