Stereographic Projection, 4

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}.

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