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The Yamabe problem for subdomains of $$S^n$$. (Le problème de Yamabe sur des sous domaines de $$S^n$$.) (French) Zbl 1070.53501
This exposition in devoted to the Yamabe problem on subdomains of $$S^n$$: Given a closed subset $$\Lambda$$ of $$S^n$$, find a complete Riemannian metric $$g$$ on $$S^n\setminus\Lambda$$ which is conformally equivalent to the standard one and has constant scalar curvature. The author provides a survey of several known results, pays special attention to the case “$$\Lambda$$ finite” and gives the detailed proof of the following theorem 3 (due to Korevaar et al.): If $$u$$ is a solution in $$B\setminus\{ 0\}$$ of the equation $\Delta u + \frac14n(n-2)u^{\frac{n+2}{n-2}} = 0,$ then either $$u$$ extends to a regular solution in the whole $$B$$ or there exist $$\epsilon > 0$$, $$a\in \mathbb{R}^n$$ and $$R > 0$$ such that $u(x) = | x - | x| ^2a| ^{\frac{2-n}{2}}v_\epsilon (-2\log | x| + \log | x-| x| ^2a| + \log R) + o(| x| ^{\frac{2-n}{2} + \alpha} ),$ where $$\alpha > 1$$ and $$v_\epsilon$$ is the solution of $v'' - \frac14(n-2)^2v + \frac n4(n-2)v^{\frac{n+2}{n-2}} = 0$ with the initial conditions $$v_\epsilon (0) = \epsilon$$ and $$v'(0) = 0$$.

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 35J60 Nonlinear elliptic equations
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