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On a quasilinear elliptic equation and a Riemannian metric invariant under Möbius transformations. (English) Zbl 0763.35033
The maximal solutions of the equation \(\Delta u=u^{(n+2)/(n-2)}\) arising from the conformal change from a flat metric in a domain (in Euclidean space) to a metric of negative scalar curvature are determined. These solutions are obtained on the upper half ball (and therefore on anything conformally equivalent to it). It is shown that the associated metric induced by a maximal solution is comparable with the quasihyperbolic metric for “nice” domains. An analog of the isoperimetric inequality is found for the “harmonic radius”.

MSC:
35J60 Nonlinear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
51B10 Möbius geometries
53B21 Methods of local Riemannian geometry
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References:
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