Ngô Quốc Anh

July 15, 2011

Stereographic projection, 4

Filed under: PDEs, Riemannian geometry — Tags: — Ngô Quốc Anh @ 19:31

It turns out that, via the stereographic projection, equation

\displaystyle - {\Delta _{{\mathbb S^n}}}u = \lambda u + {u^{\frac{{n + 2}}{{n - 2}}}}

on \mathbb S^n with u>0 becomes

\displaystyle - {\Delta _{{\mathbb R^n}}}u = V(x) u + {u^{\frac{{n + 2}}{{n - 2}}}}, \quad x \in \mathbb R^n

with the following property

u(x) \to 0, \quad |x|\to +\infty

where

\displaystyle V(x) = \frac{{n(n - 2) + 4\lambda }}{{{{(1 + |x|^2)}^2}}}.

A very simple consequence is that for the prescribing scalar curvature equation, the term V disappears as we already notice that

\displaystyle\lambda = -\frac{n(n-2)}{4}.

See also:

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