Topological degree for a mean field equation on Riemann surfaces

We consider the following mean field equations: 0.1$$\Delta u + \rho \left( {{he^u}\over{\int_M he^u}} - 1\right) = 0 \quad \hbox{on } M$$ where M is a compact Riemann surface with volume 1, h is a positive continuous function on M , ρ is a real number, 0.2$$\cases{\Delta u + \rho \left( {{he^u}\over{\int_M he^u}} - 1\right) = 0 \quad \hbox{in } \Omega \cr u=0 \qquad \qquad \qquad \qquad \qquad \hbox{on } \partial \Omega,}$$ and where Ω is a bounded smooth domain, h is a C 1 positive function on Ω, and ρ ∈ ℝ. Based on our previous analytic work [14], we prove, among other things, that the degree‐counting formula for ( 0.1) is given by (   m m − χ( M ) ) for ρ ∈ (8 m π, 8( m + 1)π). © 2003 Wiley Periodicals, Inc.