# Ngô Quốc Anh

## May 24, 2013

### A proof of the uniqueness of solutions of the Lichnerowicz equations in the compact case with boundary

Filed under: PDEs — Ngô Quốc Anh @ 1:39

In this note, a method introduced by David Maxwell was considered. More precisely, we proved that

$\displaystyle -a\Delta_g u +\text{Scal}_g u =|\sigma|_g^2 u ^{-2\kappa-3}-b\tau^2 u^{2\kappa+1}$

on a compact manifold $(M,g)$ without boundary where $\kappa=\frac{2}{n-2}$, $a=2\kappa+4$, and $b=\frac{n-1}{n}$ admits at most one solution.

In this note, we consider the case when $(M,g)$ has boundary $\partial M$ and together with our PDE above, we have the following Neumann boundary condition

$\displaystyle \partial_\nu u+\frac{1}{\kappa}H_g u=-\frac{1}{2\kappa}\Theta_-u^{\kappa+1}$

where $\nu$ is the outward normal vector field on $\partial M$.

As before, we assume that $\phi_1$ and $\phi_2$ are solutions of the above PDE. Setting $\phi=\frac{\phi_2}{\phi_1}$. We wish to prove that at least $\phi$ is constant. Recall that the following holds

$\displaystyle - a{\Delta _{\widetilde g}}(\phi - 1) = |{\sigma _1}|_{\widetilde g}^2({\phi ^{ - 2\kappa - 3}} - \phi ) + b{\tau ^2})(\phi - {\phi ^{2\kappa + 1}})$

where $\widehat g=\phi_1^\frac{4}{n-2}g$ is our conformal change. For the detailed calculation, we refer the reader to the previous notes.

## May 20, 2013

### Super polyharmonic property of solutions: The case of single equations

Filed under: PDEs — Ngô Quốc Anh @ 9:39

Recently, I have read a paper by Chen and Li published in the journal CPAA [here or here] about the super polyharmonic property of solutions for some partial differential systems.

In this notes, we consider their result in a very particular case – the single equations. We shall prove the following.

Theorem 2.1. Let $p$ be a positive integer and $q>1$. For each positive solution $u$ of

$\displaystyle (-\Delta)^p u \geqslant u^q$

in $\mathbb R^n$, there holds

$\displaystyle (-\Delta)^i u >0 \quad i=\overline{1,p-1}.$

Proof. For simplicity, we write

$\displaystyle v_i = (-\Delta)^i u \quad i=\overline{1,p-1}.$

We must show that $v_i >0$ for all $i$. For simplicity, we divide the proof into two steps.

Step 1. Proving $v_{p-1}>0$.

Assume the contradiction, we then have two possible cases

Case 1. There is some $x^0 \in \mathbb R^n$ such that $v_{p-1}(x^0)<0$.

Case 2. $v_{p-1} \geqslant 0$ and there is a point $x_0$ such that $v_{p-1}(x_0)=0$.