Yamabe’s approach was to consider first the perturbed functional
By using a direct minimizing procedure, it can be shown that for , there exists a smooth positive function such that its -norm is equal to one, , and satisfies the equation
The direct method does not work when because the Sobolev embedding is continuous but not compact. However, if one can show that is uniformly bounded, i.e. there exists a positive constant such that in for , then there exists a sequence such that and converges to a smooth positive function which satisfies the Yamabe equation .
We discuss a blow-up argument. Suppose that no such upper bound exists. It follows that there exist sequences and such that
As is compact, we may assume that as . For a normal coordinate system centered at and with radius , let the coordinates of be , . In the local coordinates,
From the equation, we know that satisfies
The idea here is to consider the normalized function
where . We have and as . Here is defined on a ball in of radius as . Obviously,
Under the choice of , we have
where those limits are taken as . By the argument of diagonal subsequence and the property of normal coordinates, one observes that a subsequence of converges to a smooth positive function which is a nonnegative solution of the equation
where , and is the standard Laplacian on . By the strong maximum principle, . It is known that
where is an invariant depending only on the conformal class of the metric . Let be the diameter of . By a change of variables we have
where denotes the open ball in with center at and radius equal to . we note that
From (2) the Fatou lemma and , we obtain
A similar argument implies
Let be a cutoff function satisfies
Defined , then
Multiplies (1) by and integration by parts, we obtain
Taking in above equation and thanks to (4) we get
- If , then , and (3) implies , which is a contradiction with .
- If , . (2) (5) and the best Sobolev imbedding implies
We are led to the contradiction with
Therefore, is uniformly bounded.