Let us start with the Pohozaev identity for semilinear elliptic equation with polygonal nonlinear terms of the form
over an open, star-shaped domain . We also assume is identical to zero on the boundary .
We multiply the PDE by and integrate over to find
The term on the left is just
On the other hand, since on , is parallel to the normal at each point . Thus
Using this equality we calculate
Combining all gives
We now evaluate the right hand side.
Therefore the identity becomes
Definition. The following identity
is called the Pohozaev identity for the problem .
Note that, if is the star-shaped domain, we then obtain on . Consequently,
But once we multiply the PDE by and integrate by parts, we produce the equality
We thus conclude
Therefore, if , we need .