Following the previous topic where we was able to point out the serious mistake in the Yamabe paper found by Trudinger. Today, we discuss about the way Trudinger did correct that mistake. Trudinger published the result in a paper entitlde “Remarks concerning the conformal deformation of Riemannian structures on compact manifolds” in *Ann. Scuola Norm. Sup. Pisa* in 1968. The paper can be downloaded from this link.

In the paper, he proved the following result

Theorem 2. There exists a positive constant (depending on , ) such that if , there exists a positive, solution of the equationwith .

Let us discuss the proof of the above result. Again, the sub-critical approach was used in his argument and we refer the reader to the previous topic.

He said that we expect a subsequence of the converges in a certain sense to a smooth solution of the critical equation. However, the convergence is not strong enough to imply the non-triviality of the resulting solution. Fortunately, if is small enough, the convergence is sufficiently nice to guarantee a positive, smooth solution of the critical equation.

Recall that the function verifies the sub-critical equation in the weak sense, that is,

for all test functions .