# 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

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