# Ngô Quốc Anh

## June 28, 2010

### A non-existence result for PDE Delta u=exp(u) in R^2

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 0:28

We provide a proof of the following well known fact.

Theorem. There is no $C^2$ solution to

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

I found this proof in a paper due to Y.Y. Li published in Commun. Math. Phys. in 1999 [here]. Before deriving the proof, let us recall the following notation known as the sphere mean in the literature. In $\mathbb R^n$ we denote the integral

$\displaystyle\displaystyle\frac{1}{\omega _n}{r^{n - 1}}\int_{\partial B\left( {0,r} \right)} {f\left( x \right)dS_x}$

by $\overline f(r)$. We call $\overline f$ the average of $f$ on the sphere $S(0,r)$ of radius $r$, or sphere mean of a function around the origin. In this context, we simply have

$\displaystyle\displaystyle\frac{1}{2\pi r}\int_{\partial B\left( {0,r} \right)} {f\left( x \right)dS_x}$.

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