Theorem(Aubin) Let be a compact Riemannian manifold with , where is called the Yamabe invariant and defined by
Then the infimum of the functional is attained. Namely, the Yamabe problem can be solved.
The original proof of Theorem used the subcritical equations. There is another simple proof by Druet and Hebey using the Brezis and Lieb’s lemma.
Proof: After passing to a subsequence, we may also assume that there exists such that weakly in , applying the imbedding theorem, we obtain strongly in , furthermore almost everywhere as . In particular, is nonnegative. It easy form the weakly convergence that
for all , where as . We also have that() by Brezis and Lieb’s lemma
Thanks to the sharp Sobolev inequality of Hebey and Vaugon, there exists such that for any ,
Since , it follows that
Since , and since in , we also have that
We assume that and note that
this implies that
as . and since
We obtain that strongly in as . In particular, is a minimizer for and is a weak nonnegative solution of the Yamabe equation
Regularity argument and the maximum principle then give that is smooth and positive. This prove the Theorem.