We know consider another type of equation, precisely, we consider the positive solution to the following
in . Following is what we need to prove.
Theorem. The following identity
Proof. We multiply the PDE by and integrate over to find
Followed by a calculation in this topic, the LHS is already calculated which is nothing but the following
In this situation,
which helps us to write down
Regarding to , clearly
Furthermore, the Green identity tells us that
We next evaluate the RHS. Indeed,
Combining all gives
In , the above identity is just
The left hand side of the previous identity can be simplified as follows
(Cf. reference: On the asymptotic behavior of solutions of the conformal Gaussian curvature equations in , Math. Ann. 308 (1997), pp. 119-139).
NB: I thank Prof. Fontana Luigi for pointing out a missing term in the above derivation.