Today, we talk about an inequality of the form for solutions of the so-called prescribed scalar curvature problem, i.e.
where the scalar curvature is fixed but the function which satisfies
for any . Here is the Sobolev critical exponent and is the dimension of . In addition, the manifold is compact and has no boundary.
Theorem (Bahoura). There exists a positive constant depending on such that
for any solution of the PDE.
We prove this theorem using contradiction: There exists a sequence of solutions of
Step 1. There holds as .
To conclude this step, we make use of the Green function associated to the operator
It is worth noticing that is invertible and
for some positive constants and . Using , we have the following representation
WLOG, we may assume that . Thanks to , for any , there holds
Step 2. There holds as .
For , we multiply the PDE by
and integrate by parts to get
Thanks to we eventually get
The positivity of plus the bounds for imply
Using the Holder inequality, we know that
Hence we have proved that
Using the Sobolev inequality, that is
Thanks to Step 1, for large
Repeatedly using for we can conclude from Step 1 that strongly in any with . Then the presentation using the Green function plus the Holder inequality immediately imply that uniformly in . This certainly proves Step 2.
Step 3. A contradiction.
Again, WLOG, we assume . Then
Since the Green function verifies
Thus we have a contradiction in view of Step 2.
Note that when , the inequality we have just discussed become the inequality . This type of result is bascially due to Brezis-Li-Shafrir.