I'm reading [Tawfik - The Yamabe problem](https://www.math.mcgill.ca/gantumur/math580f12/Yamabe.pdf): the PDE is
$$
\Delta \varphi+h(x) \varphi=\lambda f(x) \varphi^{q-1}. \label{1}\tag{1}
$$
**Theorem (Yamabe)**. For $2<q<N=N=2 n /(n-2)$, there exists a $C^{\infty}$ strictly positive $\varphi\_{q}$ satisfying \eqref{1} with $\lambda=\mu\_{q}$ and $I\_{q}\left(\varphi\_{q}\right)=\mu\_{q}$.


The proof contains several steps:


1. For $2<q \leq N, \mu\_{q}$ is finite.
2. Let $\left\{\varphi\_{i}\right\}$ be a minimizing sequence such that $\int\limits\_{M} f(x) \varphi\_{i}^{q} d V=1$: then


	* $\varphi\_{i} \in H\_{1}$
	* $\varphi\_{i} \geq 0$,
	* $\lim \_{i \rightarrow \infty} I\_{q} \left(\varphi\_{i}\right)=\mu\_{q}$.
	And we have that the set of the $\varphi\_{i}$ is bounded in $H\_{1}$.
3. If $2<q<N$, there exists a non-negative function $\varphi\_{q} \in H\_{1}$ satisfying
$$
I\_{q}\left(\varphi\_{q}\right)=\mu\_{q}
$$
and
$$
\int\_{M} f(x) \varphi\_{q}^{q} d V=1.
$$
4. $\varphi\_{q}$ satisfies \eqref{1} weakly in $H\_{1}$.
5. $\varphi\_{q} \in C^{\infty}$ for $2 \leq q<N$ and the functions $\varphi\_{q}$ are uniformly bounded for $2 \leq q \leq q\_{0}<N$.


**I'm confused about this step.**


The proof is as follows:  

*Let $G(P, Q)$ be the Green's function. $\varphi\_{q}$ satisfies the integral equation
$$
\begin{aligned}
\varphi\_{q}(P)=& V^{-1} \int\_{M} \varphi\_{q}(Q) d V(Q) \\
&+\int\_{M} G(P, Q)\left[\mu\_{q} f(Q) \varphi\_{q}^{q-1}-h(Q) \varphi\_{q}\right] d V(Q)
\end{aligned}\label{2}\tag{2}
$$
We know that $\varphi\_{q} \in L^{r\_{0}}$ with $r\_{0}=N$. Since by A 6 part 3 there exists a constant $B$ such that $\lvert G(P, Q)\rvert \leq B[d(P, Q)]^{2-n}$, then according to Sobolev's lemma A1 and its corollary, $\varphi\_{q} \in L^{r\_{1}}$ for $2<q \leq q\_{0}$ with
$$
\frac{1}{r\_{1}}=\frac{n-2}{n}+\frac{q\_{0}-1}{r+0}-1=\frac{q\_{0}-1}{r\_{0}}-\frac{2}{n}
$$
and there exists a constant $A\_{1}$ such that $\left\lVert\varphi\_{q}\right\rVert\_{r\_{1}} \leq A\_{1}\left\lVert\varphi\_{q}\right\rVert\_{r\_{0}}^{q-1}$. By induction we see that $\varphi\_{q} \in L^{r\_{k}}$ with
$$
\frac{1}{r\_{k}}=\frac{q\_{0}-1}{r\_{k-1}}-\frac{2}{n}=\frac{\left(q\_{0}-1\right)^{k}}{r\_{0}}-\frac{2}{n} \frac{\left(q\_{0}-1\right)^{k}-1}{q\_{0}-2}
$$
and there exists a constant $A\_{k}$ such that $\left\lVert\varphi\_{q}\right\rVert\_{r\_{k}} \leq A\_{k}\left\lVert\varphi\_{q}\right\rVert\_{r\_{0}}^{(q-1)^{k}}$. If for $k$ large enough, $1 / r\_{k}$ is negative, then $\varphi\_{q} \in L\_{\infty}$. Indeed suppose $1 / r\_{k-1}>0$ and $1 / r\_{k}<0$. Then $$
\frac{q\_{0}-1}{ r\_{k-1}}-\frac{2}{n}<0
$$ and Hölder's inequality A4 applied to \eqref{2} yields $\left\lVert\varphi\_{q}\right\rVert\_{\infty} \leq C\left\lVert\_{q}\right\rVert\_{r\_{k-1}}^{q-1}$ where $C$ is a constant. There exists a $k$ such that
$$
\frac{1}{r\_{k}}=\left(q\_{0}-1\right)^{k}\left[\frac{1}{r\_{0}}-\frac{2}{n\left(q\_{0}-2\right)}\right]+\frac{2}{n\left(q\_{0}-2\right)}<0
$$
because $n\left(q\_{0}-2\right)<2 r\_{0}=2 N$, since $q\_{0}<N=\frac{2 n}{n-2}$. Moreover, there exists a constant $A\_{k}$ which does not depend on $q \leq q\_{0}$ such that
$$
\left\lVert\varphi\_{q}\right\rVert\_{\infty} \leq A\_{k}\left\lVert\varphi\_{q}\right\rVert\_{N}^{(q-1)^{k}}.
$$*


**I'm confused why for $k$ large enough, $1 / r\_{k}$ is negative, then $\varphi\_{q} \in L\_{\infty}$ and how the Hölder inequality is applied to \eqref{2}.**


The Hölder inequality is stated as follows:  

**Proposition A4 (Hölder's inequality)**. Let $M$ be a Riemannian manifold. If $f \in$ $L^{r}(M) \cap L^{q}(M)$, $1 \leq r<q \leq \infty$, then $f \in L^{p}$ for $p \in[r, q]$ and
$$
\lVert f\rVert\_{p} \leq\lVert f\rVert\_{r}^{a}\lVert f\rVert\_{q}^{1-\alpha}
$$
with $a=\dfrac{1 / p-1 / q}{1 / r-1 / q}$.



A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used
### Application of Holder's inequality


Notice that the estimate on the Green's function means that
$$ \int |G(P,Q)|^\alpha ~dQ $$
is bounded whenever $\alpha < \frac{n}{n-2}$ (and the bound can be taken to be uniform; that is independent of $P$).
By Holder's inequality, we have
$$ \int G(P,Q) F(Q) ~dQ \leq \|G(P,-)\|\_{L^\alpha} \|F\|\_{L^\beta} $$
when $\beta^{-1} = 1 - \alpha^{-1}$. To ensure that the first term is bounded, we need $\alpha < \frac{n}{n-2}$ as described above; this means that $\beta^{-1} < 1 - \frac{n-2}{n} = \frac{2}{n}$.


So applying to equation (2) (noting that $h$ and $f$ are smooth, bounded, and can be discarded), we see that if $\varphi$ is such that both $\varphi\in L^\beta$ and $\varphi^{q-1} \in L^{\beta'}$ with $\beta, \beta' > \frac{n}{2}$, then we can apply the chain of reasoning above to conclude that $\varphi$ is uniformly bounded.


By assumption $q - 1 > 1$; if $\varphi^{q-1}\in L^{\beta'}$ then $\varphi\in L^{(q-1)\beta'}$. This means that we only need:



> 
> if $\varphi^{q-1} \in L^{\beta'}$ with $\beta' > \frac{n}2$, then the equation will guarantee that $\varphi\in L^\infty$.
> 
> 
> 


This final condition can be re-written as $\varphi \in L^\beta$ for some $\beta > \frac{n}{2} (q-1)$. So you just need to argue somehow that $\varphi$ can be taken to have this degree of integrability.


### Iteration process


Pretty much your intuition is correct. The idea is that solving equation (1), if you know that $\varphi$ is a priori in $L^\beta$, with $\beta < \frac{n}2(q-1)$, then the Sobolev inequality will tell you that
$$ \varphi \in L^{\gamma}, \quad \gamma = \frac{n \beta }{n(q-1) - 2\beta} $$
**Our hope is that starting with $\varphi\in L^\beta$, we can improve it to some $L^\gamma$ for $\gamma > \beta$. And repeating this improvement should eventually get us above the threshold $\frac{n}2(q-1)$.**


Examining this $\gamma$, we see that (under still the assumption that $\beta < \frac{n}2(q-1)$)
$$ \gamma > \beta \iff \frac{n}{n(q-1) - 2\beta} > 1 \iff 2 + \frac{2\beta}{n} > q $$
**In particular**: if we start the iteration process with the first step such that $\gamma > \beta$, then for all subsequent steps we will continue to get improvements from the process.


For the first step, we have $\beta = \frac{2n}{n-2}$ by Sobolev embedding since we pre-supposed $\varphi\in H^1$. So if we know that
$$ q < 2 + \frac{2\beta}{n} = 2 + \frac{4}{n-2} = \frac{2n}{n-2} $$
we can guarantee that during the iteration process, **starting from initial knowledge that $\varphi\in H^1$**, that at every step we will improve that integrability of $\varphi$.


