# Ngô Quốc Anh

## September 30, 2013

### The Lichnerowicz equation under some variable changes

Filed under: Linh Tinh, Nghiên Cứu Khoa Học, PDEs — Tags: — Ngô Quốc Anh @ 4:54

Let us consider the so-called Lichnerowicz equation

$-\Delta_g u + hu = fu^{2^\star-1}+au^{-2^\star-1} \quad u>0$

on $(M,g)$, a Riemannian manifold of dimension $n \geq 3$. Here $h$, $f$, and $a$ are smooth function with $a \geq 0$.

• We first use the the following variable change

$\displaystyle v=\log u \quad u=e^ v.$

Clearly,

$\displaystyle\Delta v = \frac{\Delta u}{u} - \frac{|\nabla u|^2}{u^2}$

and

$\displaystyle |\nabla v|^2 = \frac{|\nabla u|^2}{u^2}.$

Therefore, we can write

$\displaystyle -\Delta v =-\frac{\Delta u}{u} +|\nabla v|^2.$

Using this rule, we can rewrite the equation as follows

$\displaystyle \boxed{-\Delta v = -h+fu^{2^\star-2}+au^{-2^\star-2}+|\nabla v|^2=-h+fe^{(2^\star-2)v}+ae^{-(2^\star+2)v}+|\nabla v|^2. }$

Clearly, under this variable change, we have killed $u$ after $h$ in the original equation.

• We now use the change

$\displaystyle v=e^u \quad u=\log v.$

Clearly,

$\displaystyle\Delta v = v\Delta u + v|\nabla u|^2$

and

$\displaystyle |\nabla v|^2 = v^2|\nabla u|^2.$

Therefore, we can write

$\displaystyle -\Delta v =-\frac{|\nabla v|^2}{v}-v\Delta u.$

Again, our PDE becomes

$\displaystyle \boxed{-\Delta v = huv-fvu^{2^\star-1}-avu^{-2^\star-1}-\frac{|\nabla v|^2}{v}. }$

This seems that this variable change is not useful and is useful only when we deal with PDEs with nonlinearities of power type.

• Finally, we use the change

$\displaystyle v=u^\alpha \quad u=v^{1/\alpha}.$

Clearly,

$\displaystyle\Delta v = \alpha(\alpha-1)u^{\alpha-2}|\nabla u|^2+\alpha u^{\alpha-1}\Delta u=\alpha(\alpha-1)v\frac{|\nabla u|^2}{u^2}+\alpha v\frac{\Delta u}{u}$

and

$\displaystyle |\nabla v|^2 = \alpha^2u^{2(\alpha-1)}|\nabla u|^2=\alpha^2v^2\frac{|\nabla u|^2}{u^2}.$

Therefore, we can write

$\displaystyle -\Delta v =-\frac{\alpha-1}{\alpha}\frac{|\nabla v|^2}{v}-\alpha v\frac{\Delta u}{u}.$

Again, our PDE becomes

$\displaystyle \boxed{-\Delta v =-\frac{\alpha-1}{\alpha}\frac{|\nabla v|^2}{v}-\alpha hv+\alpha fvu^{2^\star-2}+\alpha avu^{-2^\star-2}. }$

Depending on each problem, we may choose a suitable variable change.