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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 \(v'(0) = 0\).

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