Ngô Quốc Anh

May 16, 2011

Some properties of the Yamabe equation in the null case

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 19:56

Let us consider the Yamabe equation in the null case, that is

\displaystyle -\Delta u = f u^\frac{n+2}{n-2}, \quad x \in M

where M is a compact manifold of dimension n without boundary. We assume that u>0 is a smooth positive solution.

Since the manifold is compact without the boundary, the most simple result is

\displaystyle\int_M f u^\frac{n+2}{n-2}=0

by integrating both sides of the equation. Now we prove that

\displaystyle\int_M f <0.

Indeed, multiplying both sides of the PDE with u^{-\frac{n+2}{n-2}} and integrating over M, one obtains

\displaystyle -\int_M(\Delta u) u^{-\frac{n+2}{n-2}} = \int_M f .

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