# Ngô Quốc Anh

## August 19, 2010

### L^infinity-boundedness for a single solution of -Delta u = Vexp(u)

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 7:00

The aim of this entry is to derive the $L^\infty$-boundedness for a single solution of the following PDE

$\displaystyle -\Delta u = V(x) e^u$

over a domain $\Omega$. This elegant result had been done by Brezis and Merle around 1991 published in Comm. Partial Differential Equations [here].

There are two possible cases.

The case of bounded domain. Let us assume $u$ a solution of the following PDE

$\displaystyle\begin{cases}- \Delta u = V(x){e^u}& \text{ in }\Omega , \hfill \\ u = 0&\text{ on }\partial \Omega ,\end{cases}$

where $\Omega \subset \mathbb R^2$ is a bounded domain and $V$ is a given function on $\Omega$.

Theorem. If $V \in L^p$ and $e^u \in L^{p'}$ for some $1 then $u \in L^\infty$.

Proof. It first follows from the Brezis-Meler inequality that

$e^{ku} \in L^1, \quad \forall k>0$

which by the Holder inequality gives

$e^{u} \in L^r, \quad \forall r<\infty$.

Therefore, if $p<\infty$

$Ve^u \in L^{p-\delta}, \quad \forall \delta>0$

while if $p=\infty$

$Ve^u \in L^r, \quad \forall r<\infty$.

Thus, a standard $L^p$-estimate argument from the elliptic theory implies that $u$ is bounded.

## August 2, 2010

### The Brezis-Merle inequality

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 0:25

I am going to talk about uniform estimates and blow-up phenomena for solutions of

$\displaystyle -\Delta u=V(x)e^u$

in two dimensions done by Brezis and Merle around 1991 published in Comm. Partial Differential Equations [here]. As a first step, I am going to derive some inequality that we need later.

Assume $\Omega \subset \mathbb R^2$ is bounded domain and let $u$ be a solution of

$\displaystyle -\Delta u=f(x)$

together with Dirichlet boundary condition. Here function $f$ is assumed to be of class $L^1(\Omega)$.

Theorem (Brezis-Merle). For every $\delta \in (0,4\pi)$ we have

$\displaystyle\int_\Omega {\exp \left[ {\frac{{(4\pi - \delta )|u(x)|}}{{{{\left\| f \right\|}_1}}}} \right]dx} \leqslant \frac{{4{\pi ^2}}}{\delta }{\rm diam}{(\Omega )^2}$

where $\|\cdot\|_1$ denotes the $L^1$-norm and $u$ a solution to our PDE.