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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.)
##### Keywords:
mean field equations; compact Riemann surface
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