# Ngô Quốc Anh

## September 7, 2010

### Asympotic behavior of integrals

Filed under: Giải Tích 6 (MA5205), PDEs — Tags: — Ngô Quốc Anh @ 10:52

Long time ago, we studied [here] the following fact

Suppose $f \in L^1(\mathbb R^n) \cap L_{loc}^\infty (\mathbb R^n)$ with $f \geq 0$. Define

$\displaystyle Sf\left( x \right) = \int_{\mathbb{R}^n } {\log \frac{{\left| y \right|}}{{\left| {x - y} \right|}}f\left( y \right)dy}$.

Show that $Sf(x)$ is finite for all $x \in \mathbb R^n$ and $Sf \in L_{loc}^1(\mathbb R^n)$.

In this entry, from now on we continue to prove several useful results appearing in PDE. We shall prove the following

Theorem. Assume $u$ is a solution to

$\displaystyle (-\Delta)^\frac{3}{2} u(x)=-2e^{3u(x)}, \quad x \in \mathbb R^3$

with finite energy

$\displaystyle \int_{{\mathbb{R}^3}} {{e^{3u(x)}}dx} < \infty$.

Then

$\displaystyle\mathop {\lim }\limits_{|x| \to \infty } \frac{{u(x)}}{{\log |x|}} = - \frac{1}{{{\pi ^2}}}\int_{{\mathbb{R}^3}} {{e^{3u(y)}}dy}$.