# Ngô Quốc Anh

## June 18, 2011

### A note on the equation involving the prescribing Gaussian curvature problem, 2

Filed under: PDEs, Riemannian geometry — Ngô Quốc Anh @ 17:28

Followed by this note, the trick use in that note also works for the following equation

$\displaystyle -\Delta u +K_0(x)=K_1(x)e^{2u}+K_2(x)e^{-2u}, \quad x \in M.$

In fact, by letting $\overline u=2u$, then our PDE becomes

$\displaystyle -\Delta \overline u +2K_0(x)=2K_1(x)e^{\overline u}+2K_2(x)e^{-\overline u}.$

Let $v$ be a solution of the following PDE

$\displaystyle -\Delta v=2K_0(x)-2\overline K_0$

where $\overline K_0$ is the average of $K_0$ over $M$. We let $w=\overline u+v$. Then it is easy to verify that $w$ solves the following

$\displaystyle -\Delta w +2\overline K_0=2K_1(x)e^{-v}e^w+2K_2(x)e^ve^{-w}.$

Finally, letting

$\alpha=2\overline K_0, \quad R_1(x)=2K_1(x)e^{-v(x)}, \quad R_2(x)=2K_2(x)e^{v(x)}$

we get that

$\displaystyle -\Delta w +\alpha = R_1(x)e^w+R_2(x)e^{-w}.$

Unfortunately, it is hard to see the relation between $\alpha$ and $\chi(M)$, the characteristic of $M$.