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Topological degree for a mean field equation on Riemann surfaces. (English) Zbl 1032.58010
Summary: We consider the following mean field equations: \[ \Delta u+\rho \left({he^u\over\int_Mhe^u}-1\right)=0\text{ on }M,\tag{1} \] where \(M\) is a compact Riemann surface with volume \(1,h\) is a positive continuous function on \(M\), \(\rho\) is a real number, and \[ \begin{cases} \Delta u+ \rho {he^u\over\int_\Omega he^u}=0 \quad &\text{in }\Omega\\ u=0\quad &\text{on } \partial \Omega,\end{cases} \] where \(\Omega\) is a bounded smooth domain, \(h\) is a \(C^1\) positive function on \(\Omega\), and \(\rho\in\mathbb{R}\).
Based on our previous analytic work [the authors, Commun. Pure Appl. Math. 55, 728-771 (2002; Zbl 1040.53046)], we prove, among other things, that the degree-counting formula for (1) is given by \({m-\chi(M)\choose m}\) for \(\rho\in (8m\pi,8(m+1)\pi)\).

MSC:
58J05 Elliptic equations on manifolds, general theory
35J60 Nonlinear elliptic equations
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
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