I believe there may be a gap towards the end of the proof of proposition 4.6 in the Bulletin of the AMS paper The Yamabe Problem by Lee and Parker : https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-17/issue-1/The-Yamabe-problem/bams/1183553962.full. They are trying to produce a minimizer to the Yamabe quotient on the standard round sphere. They proceed by considering the corresponding sub-critical problem for which it is standard to produce a smooth minimizer. This produces a family $\{\varphi_{s}\}$, which (since we are on the sphere), may not stay uniformly bounded as $s \to p= \frac{2n}{n-2}. $ They proceed by renormalizing the sequence using a family of chosen conformal diffeomorphisms, and establish that this renormalized sequence converges weakly in $ W^{1,2} $ and $C^2_{loc}(S^n)\setminus\{P\} $ (P denotes the north pole) to a function $\psi$, which they want to show is a minimizer. However towards the end of page 56, they claim that $$ \|\psi_{s}\|_{p}^p \geq Vol(S^n)^{1-\frac{s}{p}}\|\varphi_{s}\|_{s}^p ,$$ implies $ \|\psi\|_{p} \geq 1.$ It is this claim which I have problems with. The control they have on $\psi_{s}$ near the north pole is not enough to rule out the mass "escaping out to the infinity". Indeed one can consider a family of Aubin-Talenti bubbles on the sphere which would show this claim cannot be true, unless they are using something else about $\psi_{s}$ I'm not seeing. Hope someone can clarify this issue.