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.

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