Let us consider the following so-called Lichnerowicz equation in
Recently, Brezis [here] proved the following
Theorem. Any solution of the Lichnerowicz equation with satisfies in .
Let us study the trick used in his paper.
Then . Fix any point and consider the function
Since as , is achieved at some . We have
Since is increasing we deduce that
By sending we deduce that . In other words, .
As can be seen, he only uses the fact that is monotone increasing in his argument, therefore, this approach can be used for a wider class of nonlinearity.