In this note, we are interested in the following Lichnerowicz type equation in
As we have already seen from the previous note that solutions for the above equation are always bounded from below for certain .
Theorem (Brezis). Any solution of the Lichnerowicz equation with satisfies in .
Remarkably, we are able to prove that solutions for the Lichnerowicz type equation are also bounded from above if we require instead of . This result is basically due to L. Ma and X. Xu, see this paper.
Theorem (Ma-Xu). Any solution of the Lichnerowicz type equation with is uniformly bounded from above in .
Combining the two theorem above, we conclude that any solution of the Lichnerowicz equation, i.e. , is uniformly bounded in .
The idea of the proof for Ma-Xu’s theorem is as follows: Denote
Fix but arbitrary, we then look for a positive radial super-solution of the Lichnerowicz type equation on the ball with positive infinity boundary condition for some to be specified. This is equivalent to finding in such a way that
for any and any . Then the comparison lemma holds in the ball which tells us that
in . To construct such a super-solution , we initially choose
for some large to be determined later. Since is radially symmetric, we calculate to find
where the function is given as follows
A simple calculation shows that
Hence, to obtain , it is necessary to obtain the following
However, provided and is such that and , we obtain
for any . Hence, in , we obtain the following key estimate
Since is arbitrary and is fixed, the preceding estimate guarantees that is bounded from above.
I thank Mr. Vu Van Khu from NTU for useful discussion concerning this note.