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

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