# Ngô Quốc Anh

## April 14, 2010

### The Pohozaev identity: Integral equation with exponential nonlinearity

Filed under: PDEs — Tags: — Ngô Quốc Anh @ 1:15

We now consider the Pohozaev identity for some integral equations. We start with the following equation

$\displaystyle u(x) = \frac{2}{{{\omega _n}}}\int_{{\mathbb{R}^n}} {\log \frac{{|y|}}{{|x - y|}}K(y){e^{nu(y)}}dy} + {C_0}$

where $\omega_n$ is the volume of the unit sphere in $\mathbb R^{n+1}$ and $K$ and $C_0$ are a smooth function in $\mathbb R^n$ and a constant, respectively.

Theorem. Suppose $u$ is a $C^1$ solution of the above integral equation such that $K(x)e^{nu(x)}$ is absolutiely integrable over $\mathbb R^n$. And if one sets

$\displaystyle\alpha = \frac{2}{{{\omega _n}}}\int_{{\mathbb{R}^n}} {K(y){e^{nu(y)}}dy}$

then

$\displaystyle- \infty < \alpha < \infty$

and the following identity holds

$\displaystyle\alpha (\alpha - 2) = \frac{4}{{n{\omega _n}}}\int_{{\mathbb{R}^n}} {\left\langle {y,\nabla K(y)} \right\rangle {e^{nu(y)}}dy}$.

This theorem was due to Xu X.W. from the paper published in J. Funct. Anal. (2005). When $n=2$, it was due to Cheng and Lin from this paper published in Math. Ann. (1997).

Finiteness for $\alpha$ is just the assumption of the integrability of the function $K(x)e^{nu(x)}$. Here we mainly need to show the identity holds true.