We now consider the Pohozaev identity for some integral equations. We start with the following equation
where is the volume of the unit sphere in
and
and
are a smooth function in
and a constant, respectively.
Theorem. Suppose
is a
solution of the above integral equation such that
is absolutiely integrable over
. And if one sets
then
and the following identity holds
.
This theorem was due to Xu X.W. from the paper published in J. Funct. Anal. (2005). When , it was due to Cheng and Lin from this paper published in Math. Ann. (1997).
Finiteness for is just the assumption of the integrability of the function
. Here we mainly need to show the identity holds true.