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Topological degree for mean field equations on $$S^2$$. (English) Zbl 0964.35038
The very interesting paper under review is devoted to the topological degree theory for the mean field equation on the unit sphere $$S^2$$ of $$\mathbb{R}^3$$ equipped with the metric $$g_0$$ induced from the flat metric of $$\mathbb{R}^3.$$ Precisely, let $$f$$ be a positive smooth function of $$S^2,$$ $$\Delta$$ the Laplace-Beltrami operator on $$(S^2,g_0),$$ $$d\mu$$ the volume form with respect to $$g_0$$ and $$\rho>0$$ a constant. The mean field equation $\Delta \phi +\rho\left( {{f(y) e^\phi} \over {\int_{S^2} f(y) e^\phi d\mu}} - {1\over {4\pi}}\right)=0\quad \text{on} S^2 \tag{1}$ provides the Euler-Lagrange equation of the nonlinear functional $J_\rho(\phi)={{1}\over{2}} \int_{S^2} |\nabla \phi|^2 d\mu -\rho\log\left( \int_{S^2} f(y)e^\phi d\mu\right)$ for $$\phi$$ satisfying the normalizing condition $\int_{S^2} \phi(y) d\mu(y)=0$ and lying in the Sobolev space $$H^1(S^2)$$ of functions with $$L^2$$-integrable first derivatives.
The author studies symmetric properties of the concentrated solutions to $$(1)$$ and apply them to the calculation of the Leray-Schauder degree in the cases $$f(y)\equiv 1$$ and $$f(y)\equiv \exp (-\gamma\langle n, y\rangle)$$ with positive constant $$\gamma$$ and $$n$$ being a unit vector in $$\mathbb{R}^3.$$

##### MSC:
 35J60 Nonlinear elliptic equations 58J05 Elliptic equations on manifolds, general theory 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
##### Keywords:
topological degree theory; mean field equation
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##### References:
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