Today we continue to discuss the second fundamental lemma in the method of moving spheres. This lemma also comes from a paper due to Y.Y. Li published in J. Eur. Math. Soc. (2004).
Recall from the previous entry where the first lemma was considered if the following
holds then is constant or . We now consider the equality case. Precisely,
Lemma. Let , and . Suppose that for every there exists such that
Then for some , and
Proof. It follows from the continuity of that
If then and we are done. On the other hand, the case can easily be reduced to the case if we let
in the given identity. It is then sufficient to consider the only case . If then and we are also done. Otherwise, replacing by a nonzero multiple of , we may assume that . Since as and since is continuous and positive, has a maximum point and we may assume that has a maximum point at the origin.
For , we have for large
Taking in the above and using the fact that has a maximum point at the origin, we obtain
Claim. For any there exists such that for any there exists satisfying
Proof of claim. Remember that
for all . For any pick so that
Then for all
Thus by a degree argument using the continuity of , there exists the desired . The proof of claim follows.
With in hand we obtain
Sending to 0 we have
For any , and let defined by
Taking this and sending to 0, we obtain
By the continuity of we know that is in and the proof follows by writing the above system of PDEs as
and solving it.