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.

*Proof*. Let

Then . Fix any point and consider the function

Since as , is achieved at some . We have

By definition,

Thus,

Since is increasing we deduce that

Therefore,

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.

### Like this:

Like Loading...

*Related*

Thanks for interesting posts! I follow your blog.

In the second displayed equation, should there be ?

Comment by timur — June 8, 2011 @ 22:53

Thanks, you were correct. It is just equal to minus of the right hand side of the PDE.

Comment by Ngô Quốc Anh — June 8, 2011 @ 22:57

Hi! Good post again. 😉

I would say that when you define f(t) the second exponent should be -q-2. Am I right?

Greetings!!

Comment by Urko — June 8, 2011 @ 22:55

Urko, thanks. As I mentioned in the foregoing reply, you were absolutely correct. It is just equal to minus of the right hand side of the PDE. So function should read

Comment by Ngô Quốc Anh — June 8, 2011 @ 22:59

Ups sorry, I dont know why I didnt read the previous comments. My apologies!!!

Comment by Urko — June 9, 2011 @ 15:42