Topological degree for a mean field equation on Riemann surfaces

Abstract We consider the following mean field equations: where M is a compact Riemann surface with volume 1, h is a positive continuous function on M , ρ is a real number, 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 ( ) for ρ ∈ (8 m π, 8( m + 1)π). © 2003 Wiley Periodicals, Inc.