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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