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 thatfor any solution of the PDE.

We prove this theorem using contradiction: There exists a sequence of solutions of

such that

as .

**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

Hence,

as claimed.

**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

i.e.

Using the Sobolev inequality, that is

we obtain

Thanks to Step 1, for large

holds. Therefore,

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

we obtain

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.

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