# Ngô Quốc Anh

## August 6, 2010

### An upper bound for solutions via the maximum principle

Filed under: PDEs — Tags: , — Ngô Quốc Anh @ 1:29

It is known that [here] the following PDE

$\displaystyle\begin{cases}\Delta u =e^u, & {\rm in } \, \mathbb R^2,\\\displaystyle\int_{\mathbb R^2}e^u<\infty.\end{cases}$

has no $C^2$ solution. However, this is no longer true if we replace the whole space by a ball of radius $R$, say $B_R(0)$. In this entry, we show that if $u \in C^2(\overline B_R)$ is a solution of

$\Delta u=e^u, \quad {\rm in }\; B_R$

then

$\displaystyle u(0) \leqslant \log 8- 2\log R$.

To this purpose, let us recall the following

The Maximum Principle. Let assume $U\subset \mathbb R^2$ be open and bounded. We consider an elliptic operator $L$ of the form

$\displaystyle Lu = - \sum\limits_{i,j = 1}^2 {{a^{ij}}\frac{{{\partial ^2}u}}{{\partial {x_i}\partial {x_j}}} + \sum\limits_{k = 1}^2 {{b^k}\frac{{\partial u}}{{\partial {x_k}}}} } + cu$

where coefficients are continuous and the standard uniform ellipticity condition holds.