A problem arising from reading a lecture on the Yamabe problem of how the Hölder inequality is used I'm reading Tawfik - The Yamabe problem: 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:

*

*For $2<q \leq N, \mu_{q}$ is finite.


*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 $H_{1}$.



*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
$$
\int_{M} f(x) \varphi_{q}^{q} d V=1.
$$


*$\varphi_{q}$ satisfies \eqref{1} weakly in $H_{1}$.


*$\varphi_{q} \in C^{\infty}$ 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: 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 $\varphi$.
