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
and the following identity holds
Finiteness for is just the assumption of the integrability of the function . Here we mainly need to show the identity holds true.
Since is assumed to be , exists and is continuous. It is not hard to justify that we can take the derivative under the integral sign. First, differentiate the equation to get
Now one multiplies both sides by and integrate the resulting equation both sides over the ball for any , one obtains
Now on the right hand side, one uses the divergence theorem
Letting , one has
since is finite, the second term goes to zero as .
While on the left hand side, by using the fact that
one has the following identity
Now it is easy to see that the last term will vanish when one takes the limit simply by changing variables and , the Fubini theorem, and the fact that
The theorem follows.
The following is a very beautiful result.
Corollary. Positive solution to the following integral equation
verifies the following condition