# Convergence of the fractional Yamabe flow for a class of initial data

Hardy Chan<EMAIL_ADDRESS>Yannick Sire<EMAIL_ADDRESS>Liming Sun
<EMAIL_ADDRESS>Department of Mathematics, University of British
Columbia,
Vancouver, B.C., Canada, V6T 1Z2  Department of Mathematics, Johns Hopkins
University,
3400 N. Charles St., Baltimore, MD 21218, USA

###### Abstract

This work is a follow-up on the work of the second author with P.
Daskalopoulos and J.L. Vázquez [12]. In this latter work, we introduced the
Yamabe flow associated to the so-called fractional curvature and prove some
existence result of mild (semi-group) solutions. In the present work, we
continue this study by proving that for some class of data one can prove
actually convergence of the flow in a more general context. We build on the
approach in [27] as simplified in the book of M. Struwe [30].

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries and technical tools
  3. 3 Long time existence and convergence
  4. 4 The compact case
  5. 5 The noncompact case
  6. A Some elliptic estimates

##  1 Introduction

The resolution of the Yamabe problem, i.e. finding a metric in a given
conformal of a closed manifold with constant scalar curvature has been a
landmark in geometric analysis after the series of works [33, 31, 5, 25].
Later a parabolic proof of the previous elliptic results, was somehow
desirable and in his seminal paper Hamilton [18] introduced the so-called
Yamabe flow. Given a compact Riemannian manifold $(M,g_{0})$ of dimension
$n\geq 2$, Hamilton introduced in [18] the following evolution for a metric
$g(t)$

$\left\\{\begin{array}[]{l}\partial_{t}g(t)=-\Big{(}\text{Scal}_{g(t)}-\text{scal}_{g(t)}\Big{)}g(t)\\\
g(0)=g_{0},\end{array}\right.$ (1)

where $\text{Scal}_{g(t)}$ is the scalar curvature of $g(t)$ and

$\text{scal}_{g(t)}=\text{vol}_{g(t)}(M)^{-1}\int_{M}\text{Scal}_{g(t)}\,d\text{vol}_{g(t)}.$

This gave rise to an extensive literature, see e.g. [11, 34, 27, 6, 7].

On the other hand, in a seminal paper [16] Graham and Zworski constructed for
every $\gamma\in(0,n/2)$ a conformally covariant operator $P^{g}_{\gamma}$ on
the conformal infinity of a Poincaré–Einstein manifold. These operators appear
to be the higher-order generalizations of the conformal Laplacian. They
coincide with the GJMS operators of [15] for suitable integer values of
$\gamma$. This paved the way to define an interpolated quantity
$R^{g}_{\gamma}$ for each $\gamma\in(0,n/2)$, which is just the scalar
curvature for $\gamma=1$, and the $Q$-curvature for $\gamma=2$. This new
notion of curvature has been investigated in [24, 14, 9, 13, 21] and is called
the fractional curvature. Unfortunately, this notion of curvature (except in
the case $\gamma=\frac{1}{2}$ (see [9])), at the present knowledge, does not
carry any clear geometric meaning. Nonetheless, from the analytical point of
view, it interpolates between several well-known geometric quantities and one
can hope that their investigations will shed some light on these matters.

In the aforementioned series of papers, all the technqiues used in studying
the so-called fractional Yamabe problem are of elliptic nature. The aim of the
present article is to develop a parabolic theory. The paper is twofold. We
first collect all the necessary tools to deal with this new fractional flow.
Then we prove convergence for certain class of initial data.

We now introduce the flow under study. On a compact Poincaré–Einstein manifold
$(M,g_{0})$ let $P_{\gamma}^{g}$, where $\gamma\in(0,1)\subset(0,\frac{n}{2})$
be the conformal fractional Laplacian satisfying

$P_{\gamma}^{g_{0}}(uf)=u^{\frac{n+2\gamma}{n-2\gamma}}P_{\gamma}^{g}(f)\quad\text{
for all }{f}\in{C}^{\infty}(M),$ (2)

under the conformal change

$g=u^{\frac{4}{n-2\gamma}}g_{0}.$ (3)

In particular on $(\mathbb{R}^{n},|dx|^{2})$ we have
$P_{\gamma}^{|dx|^{2}}=(-\Delta_{\mathbb{R}^{n}})^{\gamma}$.

The volume element on $(M,g_{0})$ is denoted by $d\mu_{0}$. By replacing
$g_{0}$ by its constant multiple we may assume the $(M,g_{0})$ has unit
volume, $\mu_{0}(M)=1$. With a conformal metric (3) we write

$d\mu=d\mu_{g}=u^{\frac{2n}{n-2\gamma}}\,d\mu_{0}.$

Let
$R=R_{\gamma}^{g}=P_{\gamma}^{g}(1)=u^{-\frac{n+2\gamma}{n-2\gamma}}P_{\gamma}^{g_{0}}(u)$
be the fractional curvature. As previously mentioned, this is the scalar
curvature when $\gamma=1$ and the $Q$-curvature when $\gamma=2$. Its average
is denoted by

$s=s_{\gamma}^{g}=\int_{M}R_{\gamma}^{g}\,d\mu.$

Consider the volume-preserving fractional (note the suppressed $\gamma$)
Yamabe flow

$\begin{cases}\frac{n-2\gamma}{4}\partial_{t}g=(s-R)g,\\\
g(0)=g_{0},\end{cases}$

i.e.

$\begin{cases}\partial_{t}{u}=(s-R)u,\\\ u(0)=1.\end{cases}$ (4)

This new geometrical problem has been firstly introduced by Jin and Xiong in
[20] where the authors investigate the flow on the sphere $M=\mathbb{S}^{n}$
with the round metric, the conformally flat case. Only in this context was the
flow actually introduced, but the generalization on any compact manifold $M$
is straightforward and has been done in [12]. That the flow preserves the
volume in time is a rather important property for the global existence.

Depending on the need, the flow (4) is sometimes alternatively expressed as a
fast diffusion fractional equation, namely

$\begin{split}\frac{n-2\gamma}{n+2\gamma}\partial_{t}\left(u^{\frac{n+2\gamma}{n-2\gamma}}\right)&=-P_{\gamma}^{g_{0}}(u)+s_{\gamma}^{g}u^{\frac{n+2\gamma}{n-2\gamma}}.\end{split}$

It is convenient define the Yamabe functional

$E(u)=\frac{\displaystyle\int_{M}uP_{\gamma}^{g_{0}}{u}\,d\mu_{0}}{\left(\displaystyle\int_{M}u^{\frac{2n}{n-2\gamma}}d\mu_{0}\right)^{\frac{n-2\gamma}{n}}},$
(5)

as it appears naturally in the variational formulation throughout the paper.
Then the Yamabe constant for the class $[g]$ containing $g_{0}$ is given by

$Y_{\gamma}(M,[g])=\inf_{0\neq{u}\in{H^{\gamma}(M)}}E(u).$ (6)

A feature in all the proofs of the convergence of the Yamabe flow is the use
at some point the so-called Positive Mass Theorem, as has already been present
in [25, 6, 7]. This is associated to the Green’s function. Suppose $M$ is the
conformal infinity of a Poincaré–Einstein manifold $(X^{n+1},g_{+})$. Assume
$Y_{\gamma}(M,[g])>0$ and $\lambda_{1}(g_{+})\geq\frac{n^{2}}{2}-\gamma^{2}$.
Then for each $y\in M$, there exists a Green’s function $G(x,y)$ on
$\bar{X}\backslash\\{y\\}$ (see [21, Prosposition 1.5]). In the fractional
case, the Positive Mass Conjecture can be formulated in terms of the expansion
of Green’s function.

######  Conjecture 1.1.

Assume that $\gamma\in(0,1)$, $n>2\gamma$ and $(M,[g])$ is a Poincaré–Einstein
manifold with $Y_{\gamma}(M,[g])>0$. Fix any $y\in M$. Then there exists a
small neighborhood of $y$ in $(\bar{X},\bar{g})$, which is diffeomorphic to a
small neighborhood $\mathcal{N}\subset\mathbb{R}_{+}^{n+1}$ of 0, such that

$G(x,0)=g_{n,\gamma}|x|^{-(n-2\gamma)}+A+\psi(x)\quad\text{for
}x\in\mathcal{N}$

Here
$g_{n,\gamma}=\pi^{-n/2}2^{-2\gamma}\Gamma(\gamma)^{-1}\Gamma(\frac{n}{2}-\gamma)$
and $\psi$ is a function in $\mathcal{N}$ satisfying

$|\psi(x)|\leq C|x|^{\min\\{1,2\gamma\\}}\quad\text{ and
}\quad|\nabla\psi(x)|\leq C|x|^{\min\\{0,2\gamma-1\\}}$

for some constant $C>0$.

The Positive Mass Theorem for the operators $P^{g}_{\gamma}$ even for
$\gamma\in(0,1)$ is out of reach at the moment, for several reasons due to the
non-locality assumption of the operator and the lack of tools to treat this
case. So we naturally assume that Positive Mass holds in our main theorem as
follows.

######  Theorem 1.2.

For $\gamma\in(0,1)$, assume that $Y_{\gamma}(M,[g])>0$,
$\lambda_{1}(g_{+})\geq\frac{n^{2}}{2}-\gamma^{2}$ and, in the case
$\gamma\in(\frac{1}{2},1)$, $H=0$, where $H$ denotes the mean curvature of
$\partial_{\infty}X=M$. Assume also the Positive Mass Conjecture holds with
$A>0$. If $E$ is initially small in the sense that111Indeed, since $u(0)=1$,
the initial energy is given by
$E(u(0))=\frac{\int_{M}R(0)\,d\mu_{0}}{\mu_{0}(M)^{\frac{n-2\gamma}{n}}}=s_{0}.$

$s_{0}\leq\left[(Y_{\gamma}(M,[g]))^{\frac{n}{2\gamma}}+Y_{\gamma}(\mathbb{S}^{n})^{\frac{n}{2\gamma}}\right]^{\frac{2\gamma}{n}},$
(7)

then the flow (4) converges.

######  Remark 1.3.

For $\gamma=\frac{1}{2}$, [1] has proved the convergence of flow under more
general assumptions. As previously mentioned, the operators $P^{g}_{\gamma}$,
hence the fractional curvatures, are defined for every number (up to
resonances) between $0$ and $n/2$. However, several major difficulties arise
when one considers $\gamma>1$. First the maximum principle fails at the
elliptic level and second the parabolic theory is completely open in this
range. We leave as an open problem the investigation of these higher order
curvatures. However, we will mention in the present paper the argument working
in the larger range $\gamma>1$.

######  Remark 1.4.

In our main theorem, we didn’t specify in which sense the flow converges.
Following previous works, the flow is globally defined and Hölder continuous.
It is an open question to prove that this is actually smooth, though such a
result is expected. Implicitly, we assume the flow to be smooth in order to
use Simon’s inequality. The only proof of smoothness of the flow is in the
Euclidean setting (see [32]) and the proof does not adapt straightforwardly to
the manifold case. We postpone such result to future work.

Let us also remark that, on the other hand, singular solutions do exist, at
least for the elliptic problem. For the classical Yamabe problem, solutions
with a prescribed singular set have been constructed by Mazzeo and Pacard [22]
in 1996. This is recently extended by Ao, DelaTorre, Fontelos, González, Wei
and the first author [3] to the fractional case $\gamma\in(0,1)$. By a result
of González, Mazzeo and the second author [13], the dimension $k$ of the
singularity satisfies an inequality that includes in particular
$k<(n-2\gamma)/2$. When $\gamma=1$, such dimension restriction is sharp
according to the celebrated result of Schoen and Yau [26]. This is also known
to Chang, Hang and Yang [10] when $\gamma=2$.

Our strategy follows the one in the book of M. Struwe [30] simplifying his
original argument in virtue of the works of Brendle [6, 7]. This is based on a
series of curvature bounds which allow compactness and a recent global
compactness result [23] in the spirit of Struwe’s original one, developed by
Palatucci and Pisante (holding actually for all powers of $\gamma\in(0,n/2)$).
The nonlocality of the flow induces several difficulties that one has to
overcome using new inequalities which will be described over the paper.

##  2 Preliminaries and technical tools

In this section, we provide several tools to deal with our conformally
covariant operators of fractional orders.

We will always assume that $(M,g_{0})$ is the conformal infinity of
$(X,\bar{g}_{0})$, both equipped with appropriate metrics, and $\rho$ is the
associated defining function. Details can be found in [9].

######  Proposition 2.5 (Integration by parts).

Assume $H=0$ when $\gamma\in(\frac{1}{2},1)$. For any $v,w\in{C}^{\infty}(M)$,
we have

$\int_{M}P_{\gamma}^{g}(v)w\,d\mu=\int_{M}P_{\gamma}^{g}(w)v\,d\mu.$

###### Proof.

We recall the “improved” extension [9] for $P_{\gamma}^{g_{0}}-R_{g_{0}}$
without the zeroth order term, namely

$\begin{cases}-\operatorname{div}(\rho^{1-2\gamma}\nabla{W})=0&\text{ in
}(X,\bar{g}_{0}),\\\ W=w&\text{ on }(M,g_{0}),\\\
(P_{\gamma}^{g_{0}}-R_{g_{0}})w=-c_{\gamma}\lim\limits_{\rho\to
0}\rho^{1-2\gamma}\partial_{\rho}W&\text{ on }(M,g_{0}).\end{cases}$ (8)

where $c_{\gamma}$ is a positive constant (which can be found [9]).

First we prove that

$\int_{M}P_{\gamma}^{g_{0}}(v)w\,d\mu_{0}=\int_{M}P_{\gamma}^{g_{0}}(w)v\,d\mu_{0}.$

Indeed, denoting $V$ and $W$ to be the extension of $v$ and $w$ respectively,
we have

$\begin{split}\int_{M}\left(P_{\gamma}^{g_{0}}(v)w-P_{\gamma}^{g_{0}}(w)v\right)\,d\mu_{0}&=\int_{M}\left((P_{\gamma}^{g_{0}}-R)(v)w-(P_{\gamma}^{g_{0}}-R)(w)v\right)\,d\mu_{0}\\\
&=c_{\gamma}\lim_{\rho\to
0}\int_{M_{\rho}}\rho^{1-2\gamma}\left(W\partial_{\rho}V-V\partial_{\rho}W\right)\,d\bar{\mu}_{0}\\\
&=c_{\gamma}\int_{X}\operatorname{div}\left(\rho^{1-2\gamma}\left(W\nabla{V}-V\nabla{W}\right)\right)\,d\bar{\mu}_{0}\\\
&=0.\end{split}$

Here $M_{\rho}$ denotes the level set at level $\rho$. For a conformal metric
$g=u^{\frac{4}{n-2\gamma}}g_{0}$, we have

$\int_{M}P_{\gamma}^{g}(v)w\,d\mu=\int_{M}u^{-\frac{n+2\gamma}{n-2\gamma}}P_{\gamma}^{g_{0}}(uv)wu^{\frac{2n}{n-2\gamma}}\,d\mu_{0}=\int_{M}P_{\gamma}^{g_{0}}(uv)uw\,d\mu_{0}.$

Hence the result follows. ∎

We now compute crucial quantities involving the time-derivatives of $R$ and
$s$. These computations can be justified by a standard approximation argument.
Hereafter we also write $R(t)=R_{\gamma}^{g(t)}$, etc.

######  Lemma 2.6.

We have

  1. 1.

$\partial_{t}R(t)=\frac{n+2\gamma}{n-2\gamma}R(R-s)-P_{\gamma}^{g}(R-s)=-(P_{\gamma}^{g}-R)(R)+\frac{4\gamma}{n-2\gamma}R(R-s)$.

  2. 2.

$\partial_{t}s(t)=-2\int_{M}|R-s|^{2}\,d\mu$.

###### Proof.

  1. 1.

Using the definition of the flow, we have

$\begin{split}\partial_{t}R(t)&=-\dfrac{n+2\gamma}{n-2\gamma}\dfrac{u_{t}}{u}R+u^{-\frac{n+2\gamma}{n-2\gamma}}P_{\gamma}^{g_{0}}\left(\dfrac{u_{t}}{u}u\right)\\\
&=\dfrac{n+2\gamma}{n-2\gamma}R(R-s)-P_{\gamma}^{g}(R-s).\end{split}$

  2. 2.

Similarly we compute, using additionally Lemma 2.5,

$\begin{split}\partial_{t}s(t)&=\partial_{t}\int_{M}R\,d\mu\\\
&=\int_{M}R_{t}\,d\mu+\dfrac{2n}{n-2\gamma}\int_{M}R\dfrac{u_{t}}{u}\,d\mu\\\
&=\dfrac{n+2\gamma}{n-2\gamma}\int_{M}R(R-s)\,d\mu-\int_{M}P_{\gamma}^{g}(R-s)\,d\mu\\\
&\quad\;-\dfrac{2n}{n-2\gamma}\int_{M}R(R-s)\,d\mu\\\
&=\left(\dfrac{n+2\gamma}{n-2\gamma}-1-\dfrac{2n}{n-2\gamma}\right)\int_{M}(R-s)^{2}\,d\mu\\\
&=-2\int_{M}(R-s)^{2}\,d\mu.\end{split}$

This completes the proof. ∎

Next we show that $R(t)\geq 0$ for all $t$ provided that $R(0)>0$.
Quantitatively we have

######  Lemma 2.7.

For any $t\geq 0$, we have

$R(t)\geq{e}^{-\frac{4\gamma}{n-2\gamma}s(0)t}\min_{M}{R}(0)>0.$

###### Proof.

The extension problem for Lemma 2.6(1) reads, for $U|_{M}=e^{\frac{4\gamma
t}{n-2\gamma}}R(t)$,

$\begin{cases}\operatorname{div}(\rho^{1-2\gamma}U)=0&\text{ in
}(X,\bar{g})\\\ -c_{\gamma}\lim\limits_{\rho\to
0}\rho^{1-2\gamma}\partial_{\rho}U=-\partial_{t}U+\dfrac{4\gamma}{n-2\gamma}U(R-s+s(0))&\text{
in }(M,g_{0}).\end{cases}$

Testing this with $V$ such that
$V|_{M}=\min\left\\{U-\min_{M}U(0),0\right\\}$, we have, as long as $R(t)>0$,

$\begin{split}0&=\int_{X}-\operatorname{div}(\rho^{1-2\gamma}U)V\,d\bar{\mu}\\\
&=\int_{X}\rho^{1-2\gamma}\nabla{U}\cdot\nabla{V}\,d\bar{\mu}-\int_{M}(P_{\gamma}^{g_{0}}-R)(U)V\,d\mu\\\
&=\int_{X}\rho^{1-2\gamma}|\nabla
V|^{2}\,d\bar{\mu}+\dfrac{1}{2}\partial_{t}\int_{M}V^{2}\,d\mu-\dfrac{4\gamma}{n-2\gamma}\int_{M}U(R-s+s(0))V\,d\mu\\\
&\geq\int_{X}\rho^{1-2\gamma}|\nabla
V|^{2}\,d\bar{\mu}+\dfrac{1}{2}\partial_{t}\int_{M}V^{2}\,d\mu+\dfrac{4\gamma}{n-2\gamma}\int_{M}U|V|R\,d\mu\\\
&\geq\dfrac{1}{2}\partial_{t}\int_{M}V^{2}\,d\mu\end{split}$

where we have used the facts that $V\leq 0$ and $s\leq{s(0)}$. Integrating in
$t$, we obtain $V\equiv 0$ up to the time where $\min_{M}U\geq 0$, so that

$R(t)>e^{-\frac{4\gamma}{n-2\gamma}s(0)t}\min_{M}R(0),$

as desired. ∎

######  Proposition 2.8.

Given any $T>0$, we can find positive constants $C(T)$ such that

$C(T)^{-1}\leq u(t)\leq C(T)$

for all $0\leq t\leq T$.

###### Proof.

The function $u(t)$ satisfies

$\displaystyle\partial_{t}u=-(R-s)u\leq s(0)u$ (9)

then $u(t)\leq e^{s(0)T}$ for $0\leq t\leq T$. Since $R(t)\geq 0$ for $0\leq
t\leq T$, then

$\displaystyle P_{\gamma}^{g_{0}}u=R(t)u^{\frac{n+2\gamma}{n-2\gamma}}\geq 0.$
(10)

It follows from [8, Lemma 4.9] that $u$ satisfies a Harnack inequality such
that

$\inf_{M}u\geq C(T)\sup_{M}u,$

for some $C(T)>0$. Then the proposition is proved. ∎

For $q\geq 1$ consider the functionals

$S_{q}(g)=\int_{M}(R_{\gamma}^{g})^{q}\,d\mu_{g},\quad
F_{q}(g)=\int_{M}|R_{\gamma}^{g}-s_{\gamma}^{g}|^{q}\,d\mu_{g}.$ (11)

In particular $s_{\gamma}^{g}=S_{1}(g)$.

######  Lemma 2.9.

For $1\leq{q}<\frac{n}{2\gamma}$, we have

$F_{q+1}(g(t))\leq C(T,q,g_{0}),$ (12)

for all $0\leq t\leq T$. If the flow exists for all $t>0$, then

$\liminf_{t\to\infty}F_{q+1}(g(t))=0.$ (13)

###### Proof.

We compute, for $q\in[1,\frac{n}{2\gamma})$,

$\begin{split}&\quad\,\partial_{t}S_{q}(g)\\\
&=\int_{M}qR^{q-1}R_{t}\,d\mu+\dfrac{2n}{n-2\gamma}\int_{M}R^{q}\dfrac{u_{t}}{u}\,d\mu\\\
&=-q\int_{M}(P_{\gamma}^{g_{0}}-R)(R-s)R^{q-1}\,d\mu+q\dfrac{4\gamma}{n-2\gamma}\int_{M}R(R-s)R^{q-1}\,d\mu\\\
&\qquad-\dfrac{2n}{n-2\gamma}\int_{M}R^{q}(R-s)\,d\mu\\\
&=-q\int_{M}(P_{\gamma}^{g_{0}}-R)(R)R^{q-1}\,d\mu+\dfrac{2(2\gamma{q}-n)}{n-2\gamma}\int_{M}R^{q}(R-s)\,d\mu\\\
&\leq-\dfrac{2(n-2\gamma{q})}{n-2\gamma}\int_{M}R^{q}(R-s)\,d\mu\leq 0,\\\
\end{split}$ (14)

the last inequality following from the extension problem (8). Indeed, if $U$
is the extension for $R$, then

$\begin{split}\int_{M}(P_{\gamma}^{g_{0}}-R)(R)R^{q-1}\,d\mu&=-c_{\gamma}\lim\limits_{\rho\to
0}\rho^{1-2\gamma}\int_{M}\partial_{\rho}UR^{q-1}\,d\bar{\mu}\\\
&=c_{\gamma}\int_{X}\rho^{1-2\gamma}\nabla{U}\cdot\nabla\left(U^{q-1}\right)\,d\bar{\mu}\\\
&=\dfrac{4(q-1)}{q^{2}}c_{\gamma}\int_{X}\rho^{1-2\gamma}\left|\nabla(U^{\frac{q}{2}})\right|^{2}\,d\bar{\mu}\\\
&\geq 0.\end{split}$

Integrating (14), we have

$\begin{split}\int_{0}^{\infty}F_{q+1}(g(t))\,dt&=\int_{0}^{\infty}\int_{M}|R-s|^{q+1}\,d{\mu}dt\\\
&\leq\int_{0}^{\infty}\int_{M}(R^{q}-s^{q})(R-s)\,d{\mu}dt\\\
&\leq\dfrac{n-2\gamma}{2(n-2\gamma{q})}S_{q}(g_{0}).\end{split}$

In particular,

$\liminf_{t\to\infty}F_{q+1}(g(t))=0.$

∎

##  3 Long time existence and convergence

The short time and long time existence of $u$ has been studied by [12]. One
can use the method in [4] to show that for any $T>0$, $u\in
C^{\alpha}((0,T]\times M)$ for some $\alpha$. Here we are providing a proof
follows from Brendle’s approach [6].

######  Proposition 3.10.

For any fixed $\frac{n}{2\gamma}<p<\frac{n+2\gamma}{2\gamma}$, let
$\alpha=2\gamma-\frac{n}{p}>0$. Then for any $T>0$, there exists a constant
$C(T)$ such that

$\displaystyle|u(x_{1},t_{1})-u(x_{2},t_{2})|\leq
C(T)((t_{1}-t_{2})^{\frac{\alpha}{2}}+d(x_{1},x_{2})^{\alpha}).$

###### Proof.

Using Lemma 2.9 and Proposition 2.8, for
$\frac{n}{2\gamma}<p<\frac{n+2\gamma}{2\gamma}$,

$\displaystyle\int_{M}|P_{\gamma}^{g_{0}}u(t)|^{p}d\mu\leq C(T)$ (15)

for all $0\leq t\leq T$ and

$\displaystyle\int_{M}|\partial_{t}u|^{p}d\mu\leq C(T).$ (16)

By [17, Theorem 4], the inequality (15) implies that

$|u(x,t)-u(y,t)|\leq C(T)d(x,y)^{\alpha}$

where $\alpha=2\gamma-\frac{n}{p}$ and $t\in[0,T]$. Using (16), we obtain

$\displaystyle|u(x,t_{1})-u(x,t_{2})|$ $\displaystyle\leq$ $\displaystyle
C(t_{1}-t_{2})^{-\frac{n}{2}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|u(x,t_{1})-u(x,t_{2})|\,d\mu_{0}(y)$
$\displaystyle\leq$ $\displaystyle
C(t_{1}-t_{2})^{-\frac{n}{2}}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|u(y,t_{1})-u(y,t_{2})|\,d\mu_{0}(y)+C(T)(t_{1}-t_{2})^{\frac{\alpha}{2}}$
$\displaystyle\leq$ $\displaystyle
C(t_{1}-t_{2})^{-\frac{n-2}{2}}\sup_{t\in[t_{1},t_{2}]}\int_{B_{\sqrt{t_{1}-t_{2}}}(x)}|\partial_{t}u|\,d\mu_{0}(y)+C(T)(t_{1}-t_{2})^{\frac{\alpha}{2}}$
$\displaystyle\leq$ $\displaystyle
C(t_{1}-t_{2})^{1-\frac{n}{2p}}\sup_{t\in[t_{1},t_{2}]}\left(\int_{M}|\partial_{t}u|^{p}\,d\mu_{0}(y)\right)^{\frac{1}{p}}+C(T)(t_{1}-t_{2})^{\frac{\alpha}{2}}$
$\displaystyle\leq$ $\displaystyle C(T)(t_{1}-t_{2})^{\gamma-\frac{n}{2p}}$

for all $x\in M$ and $t_{1},t_{2}\in[0,T]$ satisfying $0<t_{1}-t_{2}<1$. Thus
the assertion is proved. ∎

Now we show that the convergence is uniform.

######  Lemma 3.11.

For any $p\in[1,\frac{n+2\gamma}{2\gamma})$, $F_{p}(g(t))\to 0$ as
$t\to+\infty$.

###### Proof.

We use the notation $z^{p}=|z|^{p-1}z$. Using the Stroock–Varopoulos
inequality

$\int_{M}f^{p-1}P_{\gamma}^{g}{f}\,d\mu\geq\dfrac{4(p-1)}{p^{2}}\int_{M}|f|^{\frac{p}{2}}P_{\gamma}^{g}\left(|f|^{\frac{p}{2}}\right)\,d\mu$

together with the Sobolev inequality

$0<Y_{\gamma}(M,[g])=\inf_{0\neq{f}\in{C}^{\infty}(M)}\dfrac{\int_{M}fP_{\gamma}^{g}{f}\,d\mu}{\left(\int_{M}|f|^{\frac{2n}{n-2\gamma}}\,d\mu\right)^{\frac{n-2\gamma}{n}}},$

we compute

$\begin{split}\partial_{t}F_{p}(g)&=-p\int_{M}(R-s)^{p-1}P_{\gamma}^{g}(R-s)\,d\mu+p\dfrac{n+2\gamma}{n-2\gamma}\int_{M}|R-s|^{p}\,d\mu\\\
&\quad\,+2ps_{t}\int_{M}(R-s)^{p-1}\,d\mu+\dfrac{2n}{n-2\gamma}\int_{M}|R-s|^{p}\dfrac{u_{t}}{u}\,d\mu\\\
&\leq-\dfrac{4(p-1)}{pY_{\gamma}(M,[g])}F_{p^{*}}(g)^{\frac{n-2\gamma}{n}}+\dfrac{p(n+2\gamma)-2n}{n-2\gamma}F_{p+1}\\\
&\quad\,+\dfrac{p(n+2\gamma)}{n-2\gamma}sF_{p}+2pF_{2}(g)F_{p-1}(g)\end{split}$

where we denote $p^{*}=\frac{np}{n-2\gamma}$. Using Hölder’s inequality with
the conjugate exponents $\theta=\frac{n-2\gamma}{2\gamma{p}}$ and
$1-\theta=\frac{2\gamma(p+1)-n}{2\gamma{p}}$ such that
$p+1=\theta{p^{*}}+(1-\theta)p$, and Young’s inequality with
$\alpha=\frac{n}{2\gamma{p}}<1$, we have

$\begin{split}F_{2}(g)F_{p-1}(g)&\leq{F}_{p+1}(g)\\\
F_{p+1}(g)&\leq{F}_{p^{*}}(g)^{\frac{n-2\gamma}{2\gamma{p}}}F_{p}(g)^{\frac{2\gamma(p+1)-n}{2\gamma{p}}}\\\
&\leq\delta{F}_{p^{*}}^{\frac{n-2\gamma}{n}}+C(\delta)F_{p}(g)^{1+\frac{2\gamma}{2\gamma{p}-n}}\end{split}$

for any $\delta>0$. Combining with the above estimates, we have

$\begin{split}\partial_{t}F_{p}(g)&\leq{C}F_{p}(g)+CF_{p}(g)^{1+\beta},\end{split}$

with $\beta=\frac{1}{p(1-\alpha)}\frac{2\gamma}{2\gamma{p}-n}>0$. Recalling
(12), standard ODE analysis implies that

$\lim_{t\to\infty}F_{p}(g(t))=0.$

∎

Now we have proved $u(t)$ exists for $(0,\infty)$ and it is hölder in space
and time for any finite time interval. We want to study the convergence of
$u(t)$.

Let $r_{0}>0$ denote a lower bound for the injectivity radius on $(M,g_{0})$.
Fix $\varphi\in{C}_{c}^{\infty}(B_{r_{0}}(0))$ such that $\varphi=1$ on
$B_{r_{0}/2}\subset\mathbb{R}^{n}$. For $x,y\in{M}$, let
$\varphi_{y}(x)=\varphi(\exp_{y}^{-1}(x))$, where $\exp$ is the exponential
map in the metric $g_{0}$. Let us also denote, for functions $u$ and $\bar{u}$
defined on $M$ and $\mathbb{R}^{n}$ respectively,

$\operatorname{vol}(u)=\mu(M)=\int_{M}u^{\frac{2n}{n-2\gamma}}\,d\mu_{0},\qquad\operatorname{\overline{vol}}(\bar{u})=\int_{\mathbb{R}^{n}}\bar{u}^{\frac{2n}{n-2\gamma}}\,dx.$

For any sequence of time, we have the profile decomposition by [23].

######  Lemma 3.12.

For any sequence $t_{k}\to\infty$, there exists an integer $L$ and sequences
$x_{k,l}$, $\varepsilon_{k,l}$, $l=1\dots L$ such that, passing to a
subsequence if necessary,

$u(t_{k})-\sum_{l=1}^{L}{u}_{(x_{k,l},\varepsilon_{k,l})}\to
u_{\infty}\quad\text{ in }H^{\gamma}(M,g_{0}),$ (17)

where $u_{\infty}\geq 0$ solves

$\displaystyle
P^{g_{0}}_{\gamma}u_{\infty}=s_{\infty}u_{\infty}^{\frac{n+2\gamma}{n-2\gamma}}\quad\text{
on }(M,g_{0}),$ (18)

and

$u_{(x_{k,l},\varepsilon_{k,l})}(x)=\varphi_{x_{k,l}}(x)\bar{u}\left(\varepsilon_{k,l}^{-1}\exp_{x_{k,l}}^{-1}(x)\right),$

with
$\bar{u}=\bar{\alpha}_{n,\gamma}\left(1+|x|^{2}\right)^{-\frac{n-2\gamma}{2}}$,
the standard bubble solving

$\displaystyle(-\Delta_{\mathbb{R}^{n}})^{\gamma}\bar{u}=\bar{u}^{\frac{n+2\gamma}{n-2\gamma}}\quad\text{
on }\mathbb{R}^{n}.$

Moreover,

$\displaystyle\operatorname{vol}(u(t_{k}))=\operatorname{vol}(u_{\infty})+L\cdot\operatorname{\overline{vol}}(\bar{u}).$
(19)

###### Proof.

By [23], such profile decomposition holds as long as the Palais–Smale
condition is verified222The authors proved the result in $\mathbb{R}^{n}$. In
the manifold setting, the proof is almost identical.. Indeed, from Lemma 3.11,

$\displaystyle\int_{M}|P_{\gamma}^{g_{0}}u-su^{\frac{n+2\gamma}{n-2\gamma}}|^{\frac{2n}{n+2\gamma}}d\mu_{0}=\int_{M}|R-s|^{\frac{2n}{n+2\gamma}}d\mu\to
0.$

Hence the result follows. ∎

Actually (19) means

$1=\left(\frac{E(u_{\infty})}{s_{\infty}}\right)^{\frac{n}{2\gamma}}+L\left(\frac{Y_{\gamma}(\mathbb{S}^{n})}{s_{\infty}}\right)^{\frac{n}{2\gamma}},$

which is

$s_{\infty}=\left[E(u_{\infty})^{\frac{n}{2\gamma}}+LY_{\gamma}(\mathbb{S}^{n})^{\frac{n}{2\gamma}}\right]^{\frac{2\gamma}{n}}.$

Obviously $E(u_{\infty})\geq Y_{\gamma}(M,[g])$. By Lemma 2.6 and the
assumption (7),

$s_{\infty}\leq
s_{0}\leq\left[(Y_{\gamma}(M,[g]))^{\frac{n}{2\gamma}}+Y_{\gamma}(\mathbb{S}^{n})^{\frac{n}{2\gamma}}\right]^{\frac{2\gamma}{n}}.$

By the Aubin inequality (see [14])

$Y_{\gamma}(M,[g])\leq Y_{\gamma}(\mathbb{S}^{n}),$ (20)

we conclude that either

  1. 1.

$L=0$ and $u_{\infty}>0$; or

  2. 2.

$L=1$ and $u_{\infty}\equiv 0$.

######  Remark 3.13.

The Aubin inequality (20) can be proved using concentration-compactness (as
opposed to test functions) for any $\gamma\in(0,n/2)$.

Using the version of strong maximum principle, again proved in [14], one has

######  Lemma 3.14.

Either $u_{\infty}>0$ or $u_{\infty}\equiv 0$.

Thus the above cases are a dichotomy and are to be referred to as the _compact
case_ and the _noncompact case_ respectively.

The following proposition is crucial in proving the convergence, and its proof
is the content of Sections 4–5.

######  Proposition 3.15.

For any sequence $t_{k}\to\infty$ there exist constants $\delta\in(0,1)$ such
that for a subsequence there holds

$s(t_{k})-s_{\infty}\leq
CF_{\frac{2n}{n+2\gamma}}(g(t_{k}))^{\frac{n+2\gamma}{2n}(1+\delta)}.$

One consequence of this proposition is

######  Lemma 3.16.

There exist constants $\delta\in(0,1)$ and $T$ such that for all $t>T$ there
holds

$s(t)-s_{\infty}\leq CF_{2}(g(t))^{\frac{1+\delta}{2}}.$

###### Proof.

Suppose this is not true. One can find a sequence $t_{k}\to\infty$ such that

$s(t_{k})-s_{\infty}\geq F_{2}(g(t_{k}))^{\frac{1+1/k}{2}}.$

However, Proposition 3.15 can be applied to this sequence

$s(t_{k})-s_{\infty}\leq
CF_{\frac{2n}{n+2\gamma}}(g(t_{k}))^{\frac{n+2\gamma}{2n}(1+\delta)}\leq
CF_{2}(g(t_{k}))^{\frac{1+\delta}{2}},$

the last inequality following from Hölder’s inequality. Putting the two
inequalities together, we obtain

$1\leq CF_{2}(g(t_{k}))^{\frac{\delta-1/k}{2}},$

which contradicts Lemma 3.11 when $k$ is sufficiently large. ∎

######  Lemma 3.17.

We have

$\int_{0}^{\infty}F_{2}(g(t))^{\frac{1}{2}}dt<\infty.$

###### Proof.

Recall the relation

$\frac{d}{dt}(s(t)-s_{\infty})=-2F_{2}(g(t))\leq-C(s(t)-s_{\infty})^{\frac{2}{1+\delta}}$

where $\delta\in(0,1)$. This differential inequality implies

$s(t)-s_{\infty}\leq Ct^{-\frac{1+\delta}{1-\delta}}$

for some constant $C>0$ and $t$ sufficiently large. Using Hölder’s inequality,
we obtain

$\int_{T}^{2T}F_{2}(g(t))^{\frac{1}{2}}dt\leq
T^{\frac{1}{2}}\left(\int_{T}^{2T}F_{2}(g(t))dt\right)^{\frac{1}{2}}\leq\frac{T^{\frac{1}{2}}}{4}(s(T)-s(2T))^{\frac{1}{2}}\leq
CT^{-\frac{\delta}{1-\delta}}$

if $T$ sufficently large. Since $\delta\in(0,1)$, we conclude that

$\int_{1}^{\infty}F_{2}(g(t))^{\frac{1}{2}}dt\leq\sum_{k=0}^{\infty}\int_{2^{k}}^{2^{k+1}}F_{2}(g(t))^{\frac{1}{2}}dt\leq
C\sum_{k=0}^{\infty}2^{-\frac{\delta}{1-\delta}k}\leq C.$

∎

######  Proposition 3.18.

Given any $\epsilon_{0}>0$, there exists a real number $r>0$ and
$q>\frac{n}{2\gamma}$ such that

$\int_{B_{r}(x)}|R(g(t))|^{q}d\mu(t)\leq\epsilon_{0}$

for all $x\in M$ and $t\geq 0$.

###### Proof.

We can find a real number $T>0$ such that

$\int_{T}^{\infty}\left(\int_{M}u(t)^{\frac{2n}{n-2\gamma}}(R(g(t)-s(t))^{2}d\mu_{0}\right)\leq\frac{\epsilon_{0}}{n}.$

Choosing a real number $r>0$ such that

$\int_{B_{r}(x)}u(t)^{\frac{2n}{n-2\gamma}}d\mu_{0}\leq\frac{\epsilon_{0}}{2}$

for all $x\in M$ and $0\leq t\leq T$. Then for any $t\geq T$, we have

$\begin{split}&\int_{B_{r}(x)}u(t)^{\frac{2n}{n-2\gamma}}d\mu_{0}\\\
\leq&\int_{B_{r}(x)}u(T)^{\frac{2n}{n-2\gamma}}d\mu_{0}+\frac{n}{2}\int_{T}^{\infty}\left(\int_{M}u(t)^{\frac{2n}{n-2\gamma}}(R(g(t)-s(t))^{2}d\mu_{0}\right)\\\
\leq&\epsilon_{0}.\end{split}$

From Lemma 3.11, we can find $p,q\in(\frac{n}{2\gamma},\frac{n}{2\gamma}+1)$
such that $q<p$ and

$\int_{M}|R(g(t))|^{p}d\mu_{g(t)}\leq C$

for some constant $C$ independent of $t$. By the previous part of this proof,
one can find $r>0$ independent of $t$ such that

$\int_{B_{r}(x)}d\mu(t)\leq\epsilon_{0}.$

Using Hölder’s inequality, then

$\int_{B_{r}(x)}|R(g(t))|^{q}d\mu(t)\leq C\epsilon_{0}^{1-\frac{q}{p}}.$

Since $\epsilon_{0}$ can be chosen arbitrarily small, the proposition is
proved. ∎

######  Proposition 3.19.

Suppose the assumption of Thm 1.2 are satisfied, then we have the uniform
upper and lower bound of $u(t)$, that is

$\sup_{M}u(t)\leq C,\quad\inf_{M}u(t)\geq C^{-1}$

here $C$ is a positive constant independent of $t$.

###### Proof.

We will need Proposition A.31 and verify its assumption is satisfied. Since
our flow is volume preserving, then

$\int_{M}d\mu_{g(t)}=1$

and Lemma 2.6 implies that

$\int_{M}R(g(t))d\mu_{g(t)}=\int_{M}u(t)P_{\gamma}^{g_{0}}u(t)d\mu_{0}\leq
s_{0}.$

Now Proposition 3.18 means that we can find a uniform radius for any point
$x\in M$ and $t>0$. Therefore we can arrive at an uniform upper bound of $u$
by Proposition A.31. For the lower bound of $u$, it is just a consequence of
the Harnack inequality of [8].

∎

Our next goal is to prove Proposition 3.15 for the two cases.

##  4 The compact case

In this case we have $u_{\infty}>0$. We first need a spectral decomposition
with respect to weighted eigenfunctions of $P_{\gamma}^{g_{0}}$.

######  Proposition 4.20.

There exist sequences $\\{\psi_{a}\\}_{a\in\mathbb{N}}\subset C^{\infty}(M)$
and $\\{\lambda_{a}\\}_{a\in\mathbb{N}}\subset\mathbb{R}$, with
$\lambda_{a}>0$, satisfying:

(i) For all $a\in{\mathbb{N}}$,

$P_{\gamma}^{g_{0}}\psi_{a}=\lambda_{a}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}\,,\text{
in }M.$

(ii) For all $a,b\in{\mathbb{N}}$,

$\int_{M}\psi_{a}\psi_{b}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}d\mu_{0}=\begin{cases}1\,,&\text{if}\>a=b\,,\\\
0\,,&\text{if}\>a\neq b\,.\end{cases}$

(iii) The span of $\\{\psi_{a}\\}_{a\in{\mathbb{N}}}$ is dense in $L^{2}(M)$.

(iv) We have $\lim_{a\to\infty}\lambda_{a}=\infty$.

###### Proof.

Since we are assuming $R_{g_{0}}>0$, for each $f\in L^{2}(M)$ we can define
$T(f)=u$, where $u\in H^{\gamma}(M)$ is the unique solution of

$P_{\gamma}^{g_{0}}u=fu_{\infty}^{\frac{4\gamma}{n-2\gamma}}\text{ in }\>M\,,$

It has been proved in [14] that the first eigenvalue of the operator is
positive, hence

$\int_{M}uP^{g_{0}}_{\gamma}u\,d\mu_{0}$

defines an (equivalent) norm in $H^{\gamma}(M)$ which is compactly embedded in
$L^{2}(M)$, and the operator $T:L^{2}(M)\to L^{2}(M)$ is compact. Integrating
by parts, we see that $T$ is symmetric with respect to the inner product

$(\psi_{1},\psi_{2})\mapsto\int_{M}\psi_{1}\psi_{2}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\,d\mu_{0}.$
(21)

Then the result follows from the spectral theorem for compact operators. ∎

######  Corollary 4.21.

For any $u,v\in{H^{\gamma}}(M)$, we have

$\int_{M}uP_{\gamma}^{g_{0}}{v}\,d\mu_{0}\leq\left\|{u}\right\|_{H^{\gamma}}\left\|{v}\right\|_{H^{\gamma}}.$

###### Proof.

This follows directly from the eigenfunction expansion. If
$u=\sum_{a}\mu_{a}\psi_{a}$ and $v=\sum_{b}\nu_{b}\psi_{b}$, then

$\begin{split}\int_{M}uP_{\gamma}^{g_{0}}{v}\,d\mu_{0}&=\int_{M}\sum_{a,b}\mu_{a}\psi_{a}\cdot\nu_{b}\lambda_{b}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{b}\,d\mu_{0}\\\
&=\sum_{a,b}\lambda_{b}\mu_{a}\nu_{b}\delta_{ab}\\\
&\leq\sqrt{\sum_{a}\lambda_{a}\mu_{a}^{2}}\sqrt{\sum_{a}\lambda_{a}\nu_{a}^{2}}\\\
&=\left\|{u}\right\|_{H^{\gamma}}\left\|{v}\right\|_{H^{\gamma}}.\end{split}$

∎

Our next goal is show the coercivity in $H^{\gamma}(M,g_{0})$ of the second
variation operator of the Yamabe functional at certain error $w_{k}$ (defined
below in (24)). This is the content of Proposition 4.24 and requires a
projection onto a finite dimensional subspace that we now introduce.

Let $A\subset{\mathbb{N}}$ be a finite set such that
$\lambda_{a}>\frac{n+2\gamma}{n-2\gamma}s_{\infty}$ for all $a\notin A$, and
define the projection

$\Pi(f)=\sum_{a\notin
A}\left(\int_{M}\psi_{a}fd\mu_{0}\right)\psi_{a}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}=f-\sum_{a\in
A}\left(\int_{M}\psi_{a}fd\mu_{0}\right)\psi_{a}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\,.$

Note that this definition facilitates the computations for the lemma below and
is not the canonical projection with respect to the inner product defined in
(21), which would read

$\tilde{\Pi}(f)=\sum_{a\notin{A}}\left(\int_{M}\psi_{a}fu_{\infty}^{\frac{4\gamma}{n-2\gamma}}\,d\mu_{0}\right)\psi_{a}.$

We are going to construct functions $\bar{u}_{z}$, which are perturbations of
$u_{\infty}$ in a finite dimensional subspace, and whose derivatives satisfy
nice orthogonality conditions.

######  Lemma 4.22.

There exists $\zeta>0$ with the following significance: for all
$z=(z_{1},...,z_{|A|})\in\mathbb{R}^{|A|}$ with $|z|\leq\zeta$, there exists a
smooth function $\bar{u}_{z}$ satisfying,

$\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}(\bar{u}_{z}-u_{\infty})\psi_{a}\,d\mu_{0}=z_{a}\>\>\>\>\text{for
all}\quad\>a\in A\,,$ (22)

and

$\Pi\left(P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\right)=0\,.$
(23)

Moreover, the mapping $z\mapsto\bar{u}_{z}$ is real analytic.

As a result,

###### Proof.

This is just an application of the implicit function theorem and a standard
argument to reach real analyticity. ∎

######  Lemma 4.23.

There exists $0<\delta<1$ such that

$E(\bar{u}_{z})-E(u_{\infty})\leq C\sup_{a\in
A}\left|\int_{M}\psi_{a}\left(P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\,\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\right)d\mu_{0}\right|^{1+\delta}\,,$

if $|z|$ is sufficiently small.

###### Proof.

Observe that the function $z\mapsto E(\bar{u}_{z})$ is real analytic.
According to results of Łojasiewicz (see equation (2.4) in [28, p.538]), there
exists $0<\delta<1$ such that

$|E(\bar{u}_{z})-E(u_{\infty})|\leq\sup_{a\in A}\left|\frac{\partial}{\partial
z_{a}}E(\bar{u}_{z})\right|^{1+\delta}\,,$

if $|z|$ is sufficiently small. Now we can follow the lines in [6, Lemma 6.5]
to calculate the partial derivative of the function $z\mapsto E(\bar{u}_{z})$,

$\begin{split}\frac{\partial}{\partial
z_{a}}E(\bar{u}_{z})&=2\dfrac{\displaystyle\int_{M}\left(P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\right)\tilde{\psi}_{a,z}\,d\mu_{0}}{\left(\displaystyle\int_{M}\bar{u}_{z}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{n}}}\\\
&\quad\;-2\left(\dfrac{\displaystyle\int_{M}\bar{u}_{z}P_{\gamma}^{g_{0}}\bar{u}_{z}\,d\mu_{0}}{\displaystyle\int_{M}\bar{u}_{z}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}}-s_{\infty}\right)\dfrac{\displaystyle\int_{M}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\tilde{\psi}_{a,z}\,d\mu_{0}}{\left(\displaystyle\int_{M}\bar{u}_{z}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{n}}},\end{split}$

where $\tilde{\psi}_{a,z}:=\frac{\partial}{\partial z_{a}}\bar{u}_{z}$ for
$a\in A$. According to (22) and (23), we know that $\tilde{\psi}_{a,z}$
satisfies

$\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\tilde{\psi}_{a,z}\psi_{b}\,d\mu_{0}=\begin{cases}1\quad&a=b,\\\
0\quad&a\neq b,\end{cases}$

for $b\in A$ and
$\Pi\left(P_{\gamma}^{g_{0}}\tilde{\psi}_{a,z}-s_{\infty}\bar{u}_{z}^{\frac{4\gamma}{n-2\gamma}}\tilde{\psi}_{a,z}\right)=0$.
Moreover, (23) implies

$P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}=\sum_{b\in{A}}\left(\int_{M}\left(P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\right)\psi_{b}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\,d\mu_{0}\right)\psi_{b}.$

We therefore obtain

$\begin{split}\frac{\partial}{\partial
z_{a}}E(\bar{u}_{z})&=2\dfrac{\displaystyle\int_{M}\left(P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\right)\psi_{a}\,d\mu_{0}}{\left(\displaystyle\int_{M}\bar{u}_{z}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{n}}}\\\
&\quad\;+2\sum_{b\in
A}\dfrac{\displaystyle\int_{M}\left(P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\right)\psi_{b}\,d\mu_{0}\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\bar{u}_{z}\psi_{b}\,d\mu_{0}}{\displaystyle\int_{M}\bar{u}_{z}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}}\\\
&\quad\qquad\qquad\cdot\dfrac{\displaystyle\int_{M}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\tilde{\psi}_{a,z}\,d\mu_{0}}{\left(\displaystyle\int_{M}\bar{u}_{z}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{n}}},\end{split}$

for all $a\in A$. Then the bounds for $u_{\infty}$ and $\bar{u}_{z}$ yield

$\sup_{a\in A}\left|\frac{\partial}{\partial z_{a}}E(\bar{u}_{z})\right|\leq
C\sup_{a\in
A}\left|\int_{M}\psi_{a}\left(P_{\gamma}^{g_{0}}\bar{u}_{z}-s_{\infty}\bar{u}_{z}^{\frac{n+2\gamma}{n-2\gamma}}\right)d\mu_{0}\right|.$

From this, the lemma follows. ∎

For any $k\in\mathbb{N}$, we consider the best approximation in
$H^{\gamma}(M)$ of $u_{k}=u(t_{k})$ among the family
$\left\\{\bar{u}_{z}\right\\}$. More precisely, we choose $z_{k}$ with
$|z_{k}|\leq\zeta$ such that

$\displaystyle\int_{M}(\bar{u}_{z_{k}}-u_{k})P_{\gamma}^{g_{0}}(\bar{u}_{z_{k}}-u_{k})\,d\mu_{0}=\min_{|z|\leq\zeta}\int_{M}(\bar{u}_{z}-u_{k})P_{\gamma}^{g_{0}}(\bar{u}_{z}-u_{k})\,d\mu_{0}$

By (17), we have $u_{k}\to u_{\infty}$ in $H^{\gamma}(M)$. As
$\bar{u}_{0}=u_{\infty}$, this implies that $z_{k}\to 0$ as $k\to\infty$. One
can decompose

$u_{k}=\bar{u}_{z_{k}}+w_{k},$ (24)

such that

$\left\|{w_{k}}\right\|_{H^{\gamma}}\to 0\quad\text{ as }{k\to\infty}.$

It also follows from the variational properties of $\bar{u}_{z_{k}}$ that

$\int_{M}P_{\gamma}^{g_{0}}(w_{k})\tilde{\psi}_{a,z_{k}}\,d\mu_{0}=0,$

for any $a\in A$. Again noticing the fact that $z_{k}\to 0$ as $k\to\infty$
one can deduce, via Corollary 4.21, that

$\begin{split}\lambda_{a}\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}\,w_{k}\,d\mu_{0}&=\int_{M}w_{k}P_{\gamma}^{g_{0}}\psi_{a}\,d\mu_{0}\\\
&=\int_{M}(\psi_{a}-\tilde{\psi}_{a,z_{k}})P_{\gamma}^{g_{0}}{w_{k}}\,d\mu_{0}\\\
&\leq\left\|{\psi_{a}-\tilde{\psi}_{a,z_{k}}}\right\|_{H^{\gamma}}\left\|{w_{k}}\right\|_{H^{\gamma}}\\\
&=o(1)\|w_{k}\|_{H^{\gamma}}\end{split}$ (25)

for any $a\in A$.

######  Proposition 4.24.

There exists $c>0$ such that

$\displaystyle\frac{n+2\gamma}{n-2\gamma}s_{\infty}\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\,d\mu_{0}\leq(1-c)\int_{M}w_{k}P_{\gamma}^{g_{0}}w_{k}\,d\mu_{0},$

for all $k$ sufficiently large.

###### Proof.

Suppose this were not true. Then there would be a subsequence, still denoted
$w_{k}$, such that we may rescale them to $\tilde{w}_{k}$ satisfying

$1=\int_{M}\tilde{w}_{k}P_{\gamma}^{g_{0}}\tilde{w}_{k}\,d\mu_{0}\leq\liminf_{k\to\infty}\frac{n+2\gamma}{n-2\gamma}s_{\infty}\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\,d\mu_{0}.$

Then $\tilde{w}_{k}$ is bounded in $H^{\gamma}$ and consequently
$\tilde{w}_{k}\rightharpoonup\tilde{w}$ weakly in $H^{\gamma}$ for some
$\tilde{w}$. The above inequality implies in particular that

$1\leq\frac{n+2\gamma}{n-2\gamma}s_{\infty}\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\tilde{w}^{2}\,d\mu_{0},$

so that $\tilde{w}\not\equiv 0$. On the other hand,

$\int_{M}\tilde{w}P_{\gamma}^{g_{0}}\tilde{w}\,d\mu_{0}\leq\frac{n+2\gamma}{n-2\gamma}s_{\infty}\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\tilde{w}^{2}\,d\mu_{0},$

or

$\sum_{a\in\mathbb{N}}\lambda_{a}\left(\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}\tilde{w}\,d\mu_{0}\right)^{2}\leq\sum_{a\in\mathbb{N}}\dfrac{n+2\gamma}{n-2\gamma}s_{\infty}\left(\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}\tilde{w}\,d\mu_{0}\right)^{2}.$

However, (25) shows that

$\lambda_{a}\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}\tilde{w}\,d\mu_{0}=0,$

for any $a\in A$, from which we arrive at a contradiction by the choice of
$A$. ∎

We now estimate $w_{k}$ quantitatively.

######  Lemma 4.25.

There exist constants $C>0$ and $k_{0}$ such that for $k\geq k_{0}$ there
holds

$\|w_{k}\|_{H^{\gamma}}\leq
C\left(\int_{M}\left|R(t_{k})-s_{\infty}\right|^{\frac{2n}{n+2\gamma}}\,d\mu_{g(t_{k})}\right)^{\frac{n+2\gamma}{2n}}.$

We need some elementary inequalities, see also [6, (137), (156)].

######  Lemma 4.26.

Let $a,b>0$. For $p>0$, we have

$|a^{p}-b^{p}|\leq{C}|a-b|^{p}+Ca^{p-1}|a-b|.$

For $p>1$,

$|a^{p}-b^{p}-pa^{p-1}(a-b)|\leq{C}a^{\max\left\\{p-2,0\right\\}}|a-b|^{\min\left\\{p,2\right\\}}+C|a-b|^{p}.$

Moreover, for $p>2$,

$\left|a^{p}-b^{p}-pa^{p-1}(a-b)+\dfrac{p(p-1)}{2}b^{p-2}(a-b)^{2}\right|\\\
\leq{C}a^{\max\left\\{p-3,0\right\\}}|a-b|^{\min\left\\{p,3\right\\}}+C|a-b|^{p}.$

###### Proof.

Let $h=a-b$. The first one follows directly from

$|a^{p}-(a-h)^{p}|\leq\begin{cases}Ca^{p-1}|h|&\text{ for
}|h|\leq\frac{a}{2},\\\ C|h|^{p}&\text{ for }|h|\geq\frac{a}{2}.\end{cases}$

The second estimate is similar when $p\geq 2$, where we expand to the second
order,

$|a^{p}-(a-h)^{p}-pa^{p-1}h|\leq\begin{cases}Ca^{p-2}|h|^{2}&\text{ for
}|h|\leq\frac{a}{2},\\\ C|h|^{p}&\text{ for }|h|\geq\frac{a}{2}.\end{cases}$

When $p<2$, in the regime $|h|\leq\frac{a}{2}$ we simply bound
$a^{p-2}\leq{C}|h|^{p-2}$.

The same argument applied to the last estimate reads

$\left|a^{p}-(a-h)^{p}-pa^{p-1}h+\dfrac{p(p-1)}{2}(a-h)^{p-2}h^{2}\right|\leq\begin{cases}Ca^{p-3}|h|^{3}&\text{
for }|h|\leq\frac{a}{2},\\\ C|h|^{p}&\text{ for
}|h|\geq\frac{a}{2},\end{cases}$

hence the result. ∎

###### Proof of Lemma 4.25.

Denote $\hat{w}_{k}$ to be the projection of $w_{k}$ onto the subspace
$\\{\psi_{a}\mid a\not\in A\\}$,

$\hat{w}_{k}=\sum_{a\not\in
A}\left(\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}w_{k}\,d\mu_{0}\right)\psi_{a},$

so that

$\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}\hat{w}_{k}\,d\mu_{0}=0$

for any $a\in A$. Moreover, (25) shows that

$\begin{split}\int_{M}(\hat{w}_{k}-w_{k})P_{\gamma}^{g_{0}}(\hat{w}_{k}-w_{k})d\mu_{0}&=\sum_{a\in
A}\lambda_{a}\left(\int_{M}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\psi_{a}w_{k}d\mu_{0}\right)^{2}\\\
&=o(1)\|w_{k}\|_{H^{\gamma}}^{2}.\end{split}$

In other words,
$\|\hat{w}_{k}-w_{k}\|_{H^{\gamma}}=o(1)\|w_{k}\|_{H^{\gamma}}$. With the
decomposition $u_{k}=\bar{u}_{z_{k}}+w_{k}$, we calculate the linearization

$\begin{split}&\quad\;\left(R(t_{k})-s_{\infty}\right)u_{k}^{\frac{n+2\gamma}{n-2\gamma}}\\\
&=P_{\gamma}^{g_{0}}{u_{k}}-s_{\infty}u_{k}^{\frac{n+2\gamma}{n-2\gamma}}\\\
&=P_{\gamma}^{g_{0}}\bar{u}_{z_{k}}-s_{\infty}\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}+P_{\gamma}^{g_{0}}w_{k}-s_{\infty}u_{k}^{\frac{n+2\gamma}{n-2\gamma}}+s_{\infty}\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}\\\
&=P_{\gamma}^{g_{0}}\bar{u}_{z_{k}}-s_{\infty}\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}+P_{\gamma}^{g_{0}}w_{k}-\frac{n+2\gamma}{n-2\gamma}s_{\infty}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}w_{k}+I_{k},\end{split}$
(26)

where

$\begin{split}I_{k}&=s_{\infty}\left(\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}-u_{k}^{\frac{n+2\gamma}{n-2\gamma}}+\frac{n+2\gamma}{n-2\gamma}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}w_{k}\right)\\\
&=\frac{n+2\gamma}{n-2\gamma}s_{\infty}\left(\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}-u_{\infty}^{\frac{4\gamma}{n-2\gamma}}\right)w_{k}\\\
&\quad\;+s_{\infty}\left(\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}-u_{k}^{\frac{n+2\gamma}{n-2\gamma}}+\frac{n+2\gamma}{n-2\gamma}\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}w_{k}\right).\end{split}$

Using Lemma 4.26,

$\begin{split}|I_{k}|&\leq{C}u_{\infty}^{\frac{6\gamma-n}{n-2\gamma}}|\bar{u}_{z_{k}}-u_{\infty}||w_{k}|+C|\bar{u}_{z_{k}}-u_{\infty}|^{\frac{4\gamma}{n-2\gamma}}|w_{k}|\\\
&\quad\;+C\bar{u}_{z_{k}}^{\max\left\\{0,\frac{4\gamma}{n-2\gamma}-1\right\\}}|w_{k}|^{\min\left\\{\frac{n+2\gamma}{n-2\gamma},2\right\\}}+C|w_{k}|^{\frac{n+2\gamma}{n-2\gamma}}\end{split}$

By the Sobolev embedding $H^{\gamma}\hookrightarrow L^{\frac{2n}{n-2\gamma}}$
and the smallness of $\bar{u}_{z_{k}}-u_{\infty}$ and $w_{k}$, we conclude

$\int_{M}|I_{k}\hat{w}_{k}|\,d\mu_{0}\leq\int_{M}|I_{k}||w_{k}|\,d\mu_{0}\leq
o(1)\|w_{k}\|_{H^{\gamma}}^{2}$

as $k\to\infty$. (Note that we need $u_{\infty}\geq{c}>0$ in case
$6\gamma<n$.) Notice that the projection $\hat{w_{k}}$ satisfies

$\int_{M}\left(P_{\gamma}^{g_{0}}\bar{u}_{z_{k}}-s_{\infty}\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}\right)\hat{w}_{k}\,d\mu_{0}=0$

because of (23). Now, using Proposition 4.24,

$\begin{split}c\|w_{k}\|^{2}_{H^{\gamma}}&\leq\int_{M}\left(P_{\gamma}^{g_{0}}w_{k}-\frac{n+2\gamma}{n-2\gamma}s_{\infty}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}w_{k}\right)w_{k}\,d\mu_{0}\\\
&=\int_{M}\left(P_{\gamma}^{g_{0}}{w}_{k}-\frac{n+2\gamma}{n-2\gamma}s_{\infty}u_{\infty}^{\frac{4\gamma}{n-2\gamma}}w_{k}\right)\hat{w}_{k}\,d\mu_{0}+o(1)\|w_{k}\|^{2}_{H^{\gamma}}\\\
&=\int_{M}\left(\left(R(t_{k})-s_{\infty}\right)u_{k}^{\frac{n+2\gamma}{n-2\gamma}}\hat{w}_{k}-I_{k}\hat{w}_{k}\right)\,d\mu_{0}+o(1)\|w_{k}\|^{2}_{H^{\gamma}}\\\
&\leq{C}\left(\int_{M}\left|R(t_{k})-s_{\infty}\right|^{\frac{2n}{n+2\gamma}}\,d\mu_{g(t_{k})}\right)^{\frac{n+2\gamma}{2n}}\|w_{k}\|_{H^{\gamma}}+o(1)\|w_{k}\|^{2}_{H^{\gamma}}\end{split}$

and the claim follows by making $k$ large enough. ∎

A related computation completes the estimate in Lemma 4.23.

######  Lemma 4.27.

There exist $C>0$ and $k_{0}$ such that for $k\geq k_{0}$ there holds

$E(\bar{u}_{z_{k}})-E(u_{\infty})\leq{C}\left(\int_{M}\left|R(t_{k})-s_{\infty}\right|^{\frac{2n}{n+2\gamma}}\,d\mu_{g(t_{k})}\right)^{\frac{n+2\gamma}{2n}(1+\delta)}.$

###### Proof.

Recalling (26),

$P_{\gamma}^{g_{0}}\bar{u}_{z_{k}}-s_{\infty}\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}=\left(R(t_{k})-s_{\infty}\right)u_{k}^{\frac{n+2\gamma}{n-2\gamma}}-P_{\gamma}^{g_{0}}w_{k}-s_{\infty}\left(\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}-u_{k}^{\frac{n+2\gamma}{n-2\gamma}}\right),$

it suffices to bound the projection of each term on the right hand side onto
the finite dimensional subspace spanned by $\psi_{a}$, where $a\in{A}$. We
have

$\int_{M}P_{\gamma}^{g_{0}}(w_{k})\psi_{a}\,d\mu_{0}\leq
C\|w_{k}\|_{H^{\gamma}}$

by Corollary 4.21. Also,

$\begin{split}\int_{M}s_{\infty}\left(\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}-u_{k}^{\frac{n+2\gamma}{n-2\gamma}}\right)\psi_{a}\,d\mu_{0}&\leq{C}\int_{M}\left(u_{k}+|w_{k}|\right)^{\frac{4\gamma}{n-2\gamma}}|w_{k}\psi_{a}|\,d\mu_{0}\\\
&\leq C\|w_{k}\|_{H^{\gamma}}.\end{split}$

Using Lemma 4.25, we get

$\sup_{a\in
A}\left|\int_{M}\left(P_{\gamma}^{g_{0}}\bar{u}_{z_{k}}-s_{\infty}\bar{u}_{z_{k}}^{\frac{n+2\gamma}{n-2\gamma}}\right)\psi_{a}\,d\mu_{0}\right|\\\
\leq{C}\left(\int_{M}\left|R(t_{k})-s_{\infty}\right|^{\frac{2n}{n+2\gamma}}\,d\mu_{g(t_{k})}\right)^{\frac{n+2\gamma}{2n}}.$

Our claim follows from Lemma 4.23. ∎

We can finally turn to the proof of Proposition 3.15 in the compact case.

###### Proof of Proposition 3.15.

By the conformal relation (2), we compute

$\begin{split}E(u_{k})&=\int_{M}(\bar{u}_{z_{k}}+w_{k})P_{\gamma}^{g_{0}}(\bar{u}_{z_{k}}+w_{k})\,d\mu_{0}\\\
&=\int_{M}\bar{u}_{z_{k}}P_{\gamma}^{g_{0}}\bar{u}_{z_{k}}\,d\mu_{0}+2\int_{M}R(t_{k})u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}\,d\mu_{0}-\int_{M}w_{k}P_{\gamma}^{g_{0}}w_{k}\,d\mu_{0}\\\
&=s_{\infty}+2\int_{M}\left(R(t_{k})-s_{\infty}\right)u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}\,d\mu_{0}\\\
&\quad\;-\int_{M}\left(w_{k}P_{\gamma}^{g_{0}}{w_{k}}-\frac{n+2\gamma}{n-2\gamma}s_{\infty}\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\right)\,d\mu_{0}+J_{k}\end{split}$

where

$\begin{split}J_{k}&=(E(\bar{u}_{z_{k}})-s_{\infty})\left(\int_{M}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}d\mu_{0}\right)^{\frac{n-2\gamma}{n}}+s_{\infty}\left(\left(\int_{M}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}d\mu_{0}\right)^{\frac{n-2\gamma}{n}}-1\right)\\\
&\quad\;+s_{\infty}\int_{M}\left(2u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}-\frac{n+2\gamma}{n-2\gamma}\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\right)d\mu_{0}.\end{split}$

Since $x\mapsto x^{\frac{n-2\gamma}{n}}$ is a concave function,

$\begin{split}\left(\int_{M}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{n}}-1&\leq\frac{n-2\gamma}{n}\left(\int_{M}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}-1\right)\\\
&=\frac{n-2\gamma}{n}\int_{M}\left(\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}-u_{k}^{\frac{2n}{n-2\gamma}}\right)\,d\mu_{0}.\end{split}$

Then we can estimate the error term as

$\begin{split}J_{k}&\leq(E(\bar{u}_{z_{k}})-s_{\infty})\left(\int_{M}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}d\mu_{0}\right)^{\frac{n-2\gamma}{n}}\\\
&\quad\;-s_{\infty}\int_{M}\Bigg{(}\frac{n-2\gamma}{n}u_{k}^{\frac{2n}{n-2\gamma}}-\frac{n-2\gamma}{n}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}\\\
&\qquad\qquad\qquad-2u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}+\frac{n+2\gamma}{n-2\gamma}\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\Bigg{)}d\mu_{0}.\end{split}$

Recalling $u_{k}=\bar{u}_{z_{k}}+w_{k}$, the last integrand is a multiple of

$\left(\bar{u}_{z_{k}}+w_{k}\right)^{\frac{n-2\gamma}{n}}-\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}-\dfrac{2n}{n-2\gamma}\left(\bar{u}_{z_{k}}+w_{k}\right)^{\frac{2n}{n-2\gamma}}w_{k}+\dfrac{1}{2}\cdot\dfrac{2n}{n-2\gamma}\cdot\dfrac{n+2\gamma}{n-2\gamma}\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}$

hence the pointwise estimate in Lemma 4.26 applies to yield

$\begin{split}&\quad\;\int_{M}\left|\frac{n-2\gamma}{n}u_{k}^{\frac{2n}{n-2\gamma}}-\frac{n-2\gamma}{n}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}-2u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}+\frac{n+2\gamma}{n-2\gamma}\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\right|\,d\mu_{0}\\\
&\leq{C}\int_{M}\bar{u}_{z_{k}}^{\max\left\\{0,\frac{6\gamma-n}{n-2\gamma}\right\\}}|w_{k}|^{\min\left\\{\frac{2n}{n-2\gamma},3\right\\}}\,d\mu_{0}+C\int_{M}|w_{k}|^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\\\
&\leq{C}\left(\int_{M}|w_{k}|^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{2n}\min\left\\{\frac{2n}{n-2\gamma},3\right\\}}\\\
&\leq{C}\|w_{k}\|_{H^{\gamma}}^{\min\left\\{\frac{2n}{n-2\gamma},3\right\\}}.\end{split}$

Now the results of Lemma 4.25 and 4.27 imply

$\begin{split}J_{k}&\leq(E(\bar{u}_{z_{k}})-s_{\infty})\left(\int_{M}\bar{u}_{z_{k}}^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{n}}+C\|w_{k}\|_{H^{\gamma}}^{\min\left\\{\frac{2n}{n-2\gamma},3\right\\}}\\\
&\leq{C}\left(\int_{M}\left|R(t_{k})-s_{\infty}\right|^{\frac{2n}{n+2\gamma}}\,d\mu_{g(t_{k})}\right)^{\frac{n+2\gamma}{2n}(1+\delta)}\\\
&\leq{C}F_{\frac{2n}{n+2\gamma}}(g(t_{k}))^{\frac{n+2\gamma}{2n}(1+\delta)}+C(s(t_{k})-s_{\infty})^{1+\delta}.\end{split}$

It follows from Hölder’s inequality, Proposition 4.24 and Lemma 4.25 that the
remaining terms are also bounded by

$\begin{split}&\quad\;2\int_{M}\left(R(t_{k})-s_{\infty}\right)u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}\,d\mu_{0}\\\
&\qquad\qquad-\int_{M}\left(w_{k}P_{\gamma}^{g_{0}}{w_{k}}-\frac{n+2\gamma}{n-2\gamma}s_{\infty}\bar{u}_{z_{k}}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\right)\,d\mu_{0}\\\
&\leq{C}\left(\int_{M}\left|R(t_{k})-s_{\infty}\right|^{\frac{2n}{n+2\gamma}}\,d\mu_{g(t_{k})}\right)^{\frac{n+2\gamma}{2n}}\|w_{k}\|_{L^{\frac{2n}{n-2\gamma}}}-c\|w_{k}\|^{2}_{H^{\gamma}}\\\
&\leq{C}\left(\int_{M}|R(t_{k})-s_{\infty}|^{\frac{2n}{n+2\gamma}}d\mu_{g(t_{k})}\right)^{\frac{n+2\gamma}{n}}\\\
&\leq{C}F_{\frac{2n}{n+2\gamma}}(g(t_{k}))^{\frac{n+2\gamma}{2n}(1+\delta)}+C(s(t_{k})-s_{\infty})^{1+\delta}.\end{split}$

Combining our expansion of $E(u_{k})$ and the previous estimates, we get

$s(t_{k})-s_{\infty}=E(u_{k})-s_{\infty}\leq
CF_{\frac{2n}{n+2\gamma}}(g(t_{k}))^{\frac{n+2\gamma}{2n}(1+\delta)}+C(s(t_{k})-s_{\infty})^{1+\delta}.$

Since $s(t_{k})\to s_{\infty}$ as $k\to\infty$ and $\delta\in(0,1)$, then

$s(t_{k})-s_{\infty}\leq
CF_{\frac{2n}{n+2\gamma}}(g(t_{k}))^{\frac{n+2\gamma}{2n}(1+\delta)},$

as desired. ∎

##  5 The noncompact case

In this case we have $u_{\infty}=0$. Following [21] with the assumption of
Positive Mass Theorem for our operators, there is a test function $u$ such
that

$E(u)=\frac{\displaystyle\int_{M}uP_{\gamma}^{g_{0}}{u}\,d\mu_{0}}{\left(\displaystyle\int_{M}u^{\frac{2n}{n-2\gamma}}d\mu_{0}\right)^{\frac{n-2\gamma}{n}}}<Y_{\gamma}(\mathbb{S}^{n}).$

Such $u$ is found through the rescaling and relocation of standard bubble
$\bar{u}$, possibly truncated or perturbed. By specifying the relocation and
rescaling parameters $(x_{0},\varepsilon)$ of such test function, we use the
notation $u_{(x_{0},\varepsilon)}$ for a more precise purpose. Near $x_{0}$,
$u_{(x_{0},\varepsilon)}(x)$ is comparable to

$\bar{\alpha}_{n,\gamma}s_{\infty}^{-\frac{n-2\gamma}{4\gamma}}\varepsilon^{-\frac{n-2\gamma}{2}}\bar{u}\left(\varepsilon^{-1}\exp_{x_{0}}^{-1}(x)\right),$

where $\bar{\alpha}_{n,\gamma}$ can be found at [21, 1-23].

From the profile decomposition, we know that $u_{k}=u(t_{k})$ approaches some
$u_{(x_{k},\varepsilon_{k})}$ in $H^{\gamma}$. We prefer to use the best
approximation in the following sense,

$\int_{M}\left(u_{k}-\alpha_{k}u_{(x_{k},\varepsilon_{k})}\right)P_{\gamma}^{g_{0}}(u_{k}-\alpha_{k}u_{(x_{k},\varepsilon_{k})})\,d\mu_{0}\\\
=\min_{\alpha>0,x\in M,\varepsilon>0}\int_{M}\left(u_{k}-\alpha
u_{(x,\varepsilon)}\right)P_{\gamma}^{g_{0}}\left(u_{k}-\alpha
u_{(x,\varepsilon)}\right)\,d\mu_{0}.$

Then

$u_{k}=\alpha_{k}u_{(x_{k},\varepsilon_{k})}+w_{k}=:v_{k}+w_{k}$

with some suitable $x_{k},\varepsilon_{k}$ and
$\alpha_{k}\to\operatorname{const}>0$. Then we have the following lemma from
the variation of three parameters $\alpha,\varepsilon$, and $x$ respectively:

######  Lemma 5.28.

As $k\to\infty$, there hold

  1. 1.

$\displaystyle\int_{M}v_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}\,d\mu_{0}=o(1)\|w_{k}\|_{H^{\gamma}}$;

  2. 2.

$\displaystyle\int_{M}v_{k}^{\frac{n+2\gamma}{n-2\gamma}}\frac{\varepsilon_{k}^{2}-d(x,x_{k})^{2}}{\varepsilon_{k}^{2}+d(x,x_{k})^{2}}w_{k}\,d\mu_{0}=o(1)\|w_{k}\|_{H^{\gamma}}$;

  3. 3.

$\displaystyle\int_{M}v_{k}^{\frac{n+2\gamma}{n-2\gamma}}\frac{\varepsilon_{k}\exp_{x_{k}}^{-1}(x)}{\varepsilon_{k}^{2}+d(x,x_{k})^{2}}w_{k}\,d\mu_{0}=o(1)\|w_{k}\|_{H^{\gamma}}$.

###### Proof.

By the choice of $\alpha_{k}$, we get

$\int_{M}w_{k}P_{\gamma}^{g_{0}}u_{(x_{k},\varepsilon_{k})}\,d\mu_{0}=0.$

Moreover, one can expand

$\alpha_{k}P_{\gamma}^{g_{0}}{u}_{(x_{k},\varepsilon_{k})}=P_{\gamma}^{g_{0}}(u_{k}-w_{k})=R(t_{k})u_{k}^{\frac{n+2\gamma}{n-2\gamma}}-P_{\gamma}^{g_{0}}w_{k}=s_{\infty}v_{k}^{\frac{n+2\gamma}{n-2\gamma}}+I_{k}-P_{\gamma}^{g_{0}}w_{k}$

where

$I_{k}=\left(R(t_{k})-s_{\infty}\right)u_{k}^{\frac{n+2\gamma}{n-2\gamma}}+s_{\infty}\left(u_{k}^{\frac{n+2\gamma}{n-2\gamma}}-v_{k}^{\frac{n+2\gamma}{n-2\gamma}}\right)\to
0\quad\text{ in }L^{\frac{2n}{n+2\gamma}}.$

Then

$s_{\infty}\int_{M}w_{k}P_{\gamma}^{g_{0}}v_{k}^{\frac{n+2\gamma}{n-2\gamma}}\,d\mu_{0}=o(1)\|w_{k}\|_{L^{\frac{2n}{n-2\gamma}}}+\|w_{k}\|_{H^{\gamma}}^{2}=o(1)\|w_{k}\|_{H^{\gamma}},$

establishing Claim (1). Claim (2) and Claim (3) can be proved similarly. ∎

######  Lemma 5.29.

There exist constants $c>0$ and $k_{0}$ such that for $k\geq k_{0}$ there
holds

$\frac{n+2\gamma}{n-2\gamma}s_{\infty}\int_{M}v_{k}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\,d\mu_{0}\leq(1-c)\int_{M}w_{k}P_{\gamma}^{g_{0}}w_{k}\,d\mu_{0}.$

###### Proof.

Suppose it were not true. Then one would be able to extract a sequence of
rescaled $\tilde{w}_{k}=a_{k}w_{k}$ such that

$1=\int_{M}\tilde{w}_{k}P_{\gamma}^{g_{0}}\tilde{w}_{k}\,d\mu_{0}\leq\liminf_{k\to\infty}\frac{n+2\gamma}{n-2\gamma}s_{\infty}\int_{M}v_{k}^{\frac{4\gamma}{n-2\gamma}}\tilde{w}_{k}^{2}\,d\mu_{0}$

Define

$\hat{w}_{k}(x)=\varepsilon_{k}^{\frac{n-2\gamma}{2}}\tilde{w}_{k}(\exp_{x_{k}}(\varepsilon_{k}\xi)):B_{R/\epsilon_{k}}(0)\subset
T_{x_{k}}M\to\mathbb{R}$

for some $R<\imath_{0}$, the injectivity radius of $(M,g_{0})$. Then
$\hat{w}_{k}$ is bounded in $H^{\gamma}(B_{R/\varepsilon_{k}}(0))$ and
consequently $\hat{w}_{k}\rightharpoonup\hat{w}$ weakly in
$H^{\gamma}_{\textnormal{loc}}(\mathbb{R}^{n})$ for some $\hat{w}$ satisfying

$\int_{\mathbb{R}^{n}}\frac{\hat{w}(\xi)^{2}}{(1+|\xi|^{2})^{2\gamma}}\,d\xi>0$

and

$\displaystyle\int_{\mathbb{R}^{n}}\hat{w}(\xi)(-\Delta_{\mathbb{R}^{n}})^{\gamma}\hat{w}(\xi)\,d\xi\leq\alpha_{n,\gamma}^{\frac{4\gamma}{n-2\gamma}}\frac{n+2\gamma}{n-2\gamma}\int_{\mathbb{R}^{n}}\frac{\hat{w}(\xi)^{2}}{(1+|\xi|^{2})^{2\gamma}}\,d\xi.$
(27)

However, it follows from Lemma 5.28 that

$\begin{split}\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|\xi|^{2}}\right)^{\frac{n+2\gamma}{2}}\hat{w}(\xi)\,d\xi&=0,\\\
\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|\xi|^{2}}\right)^{\frac{n+2\gamma}{2}}\frac{1-|\xi|^{2}}{1+|\xi|^{2}}\hat{w}(\xi)\,d\xi&=0,\\\
\int_{\mathbb{R}^{n}}\left(\frac{1}{1+|\xi|^{2}}\right)^{\frac{n+2\gamma}{2}}\frac{\xi}{1+|\xi|^{2}}\hat{w}(\xi)\,d\xi&=0.\end{split}$
(28)

We want to prove the above three equalities and (27) together imply
$\hat{w}(\xi)=0$, which will clearly give us a contradiction. To this end, it
is better to work on sphere $\mathbb{S}^{n}$. Denote by $\Sigma$ the
stereographic projection of the sphere $\mathbb{S}^{n}$ onto $\mathbb{R}^{n}$
with respect to the north pole. More precisely,

$\begin{split}\forall\,x=&\,(x_{1},\dots,x_{n+1})\in\mathbb{S}^{n},\quad\Sigma(x)=\xi=(\xi_{1},\dots,\xi_{n})\in\mathbb{R}^{n}\\\
&\text{where }\xi_{i}=\frac{x_{i}}{1-x_{n+1}}.\end{split}$

It is known that the standard metric of $\mathbb{S}^{n}$ and $\mathbb{R}^{n}$
are related by

$g_{\mathbb{S}^{n}}=\frac{4}{(1+|\xi|^{2})^{2}}|d\xi|^{2}=\rho(\xi)^{\frac{4}{n-2\gamma}}|d\xi|^{2},\quad\rho(\xi)=\left(\frac{2}{1+|\xi|^{2}}\right)^{\frac{n-2\gamma}{2}}.$

For any $\hat{w}(\xi)\in H^{\gamma}(\mathbb{R}^{n})$, we define a function $v$
on $\mathbb{S}^{n}$ by $v(x)=(\rho^{-1}\hat{w})(\xi)$, $\xi=\Sigma(x)$. The
conformal property reads

$\displaystyle(-\Delta_{\mathbb{R}^{n}})^{\gamma}\hat{w}$
$\displaystyle=\rho^{\frac{n+2\gamma}{n-2\gamma}}P^{g_{\mathbb{S}^{n}}}_{\gamma}(v).$

Consequently,

$\displaystyle\int_{\mathbb{R}^{n}}\hat{w}(-\Delta_{\mathbb{R}^{n}})^{\gamma}\hat{w}\,d\xi$
$\displaystyle=\int_{\mathbb{S}^{n}}vP^{g_{\mathbb{S}^{n}}}_{\gamma}(v)\,d\mu_{\mathbb{S}^{n}},$
(29)
$\displaystyle\int_{\mathbb{R}^{n}}\rho^{\frac{4\gamma}{n-2\gamma}}\hat{w}^{2}(\xi)\,d\xi$
$\displaystyle=\int_{\mathbb{S}^{n}}v^{2}(x)\,d\mu_{g_{\mathbb{S}^{n}}}.$ (30)

The spectrum of $P^{g_{\mathbb{S}^{n}}}_{\gamma}$ is known; for example, see
[20]. Namely, for any $k\geq 0$

$P^{g_{\mathbb{S}^{n}}}_{\gamma}(Y^{(k)})=\frac{\Gamma(k+\frac{n}{2}+\gamma)}{\Gamma(k+\frac{n}{2}-\gamma)}Y^{(k)},$

where $Y^{(k)}$ are spherical harmonics of degree $k\geq 0$ and $\Gamma$ is
the Gamma function. The three equalities in (28) mean exactly that $v$ is
orthogonal to any $Y^{(0)}$ and $Y^{(1)}$. Therefore

$\displaystyle\int_{\mathbb{S}^{n}}vP^{g_{\mathbb{S}^{n}}}_{\gamma}(v)\,d\mu_{\mathbb{S}^{n}}\geq\frac{\Gamma(2+\frac{n}{2}+\gamma)}{\Gamma(2+\frac{n}{2}-\gamma)}\int_{\mathbb{S}^{n}}v^{2}(x)\,d\mu_{g_{\mathbb{S}^{n}}}.$

Combining the above fact with (27), (29) and (30), we shall obtain

$\displaystyle\frac{\Gamma(2+\frac{n}{2}+\gamma)}{\Gamma(2+\frac{n}{2}-\gamma)}\int_{\mathbb{R}^{n}}\rho^{\frac{4\gamma}{n-2\gamma}}\hat{w}^{2}(\xi)\,d\xi$
$\displaystyle\leq\alpha_{n,\gamma}^{\frac{4\gamma}{n-2\gamma}}\frac{n+2\gamma}{n-2\gamma}\int_{\mathbb{R}^{n}}\frac{\hat{w}^{2}(\xi)}{(1+|\xi|^{2})^{2\gamma}}\,d\xi$
$\displaystyle=\alpha_{n,\gamma}^{\frac{4\gamma}{n-2\gamma}}\frac{n+2\gamma}{n-2\gamma}2^{-2\gamma}\int_{\mathbb{R}^{n}}\rho(x)^{\frac{4\gamma}{n-2\gamma}}\hat{w}^{2}(\xi)\,d\xi.$

Retrieving $\alpha_{n,\gamma}$ from [21] gives

$\alpha_{n,\gamma}=2^{\frac{n-2\gamma}{2}}\left(\frac{\Gamma(\frac{n}{2}+\gamma)}{\Gamma(\frac{n}{2}-\gamma)}\right)^{\frac{n-2\gamma}{4\gamma}}.$

It is not difficult to see

$\frac{\Gamma(2+\frac{n}{2}+\gamma)}{\Gamma(2+\frac{n}{2}-\gamma)}>\alpha_{n,\gamma}^{\frac{4\gamma}{n-2\gamma}}\frac{n+2\gamma}{n-2\gamma}2^{-2\gamma}$

for $\gamma\in(0,1)$ and $n>2\gamma$. Thus we conclude

$\int_{\mathbb{R}^{n}}\rho^{\frac{4\gamma}{n-2\gamma}}\hat{w}^{2}(\xi)\,d\xi=0,$

implying that $\hat{w}(\xi)\equiv 0$, a contradiction. ∎

With the above estimate we now give the proof of Proposition 3.15 in the
noncompact case.

###### Proof of Proposition 3.15.

$\begin{split}E(u_{k})&=\int_{M}(v_{k}+w_{k})P_{\gamma}^{g_{0}}(v_{k}+w_{k})\,d\mu_{0}\\\
&=\int_{M}v_{k}P_{\gamma}^{g_{0}}v_{k}\,d\mu_{0}+2\int_{M}R(t_{k})u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}\,d\mu_{0}-\int_{M}w_{k}P_{\gamma}^{g_{0}}w_{k}\,d\mu_{0}\\\
&=s_{\infty}+2\int_{M}\left(R(t_{k})-s_{\infty}\right)u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}\,d\mu_{0}\\\
&\quad\;-\int_{M}\left(w_{k}P_{\gamma}^{g_{0}}{w_{k}}-\frac{n+2\gamma}{n-2\gamma}s_{\infty}v_{k}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\right)\,d\mu_{0}+J_{k},\end{split}$

where

$\begin{split}J_{k}&=(E(v_{k})-s_{\infty})\left(\int_{M}v_{k}^{\frac{2n}{n-2\gamma}}d\mu_{0}\right)^{\frac{n-2\gamma}{n}}+s_{\infty}\left(\left(\int_{M}v_{k}^{\frac{2n}{n-2\gamma}}d\mu_{0}\right)^{\frac{n-2\gamma}{n}}-1\right)\\\
&\quad\;+s_{\infty}\int_{M}\left(2u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}-\frac{n+2\gamma}{n-2\gamma}v_{k}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\right)d\mu_{0}.\end{split}$

Since $E(v_{k})<s_{\infty}$, we have

$J_{k}\leq
s_{\infty}\int_{M}\Bigg{(}-\frac{n-2\gamma}{n}u_{k}^{\frac{2n}{n-2\gamma}}+\frac{n-2\gamma}{n}v_{k}^{\frac{2n}{n-2\gamma}}\\\
+2u_{k}^{\frac{n+2\gamma}{n-2\gamma}}w_{k}-\frac{n+2\gamma}{n-2\gamma}v_{k}^{\frac{4\gamma}{n-2\gamma}}w_{k}^{2}\Bigg{)}\,d\mu_{0}.$

Similar to the case when $u_{\infty}>0$, one can get the estimate

$J_{k}\leq
C\|w_{k}\|_{H^{\gamma}}^{\min\left\\{\frac{2n}{n-2\gamma},3\right\\}}.$

Lemma 5.29 yields that

$\|w_{k}\|^{2}_{H^{\gamma}}\leq
C\int_{M}\left(w_{k}P_{\gamma}^{g_{0}}{w_{k}}-\frac{n+2\gamma}{n-2\gamma}s_{\infty}v_{k}^{\frac{4}{n-2\gamma}}w_{k}^{2}\right)\,d\mu_{0}.$

for $k\geq k_{0}$. The rest of proof follows from almost same lines as the
compact case $u_{\infty}>0$. ∎

##  Appendix A Some elliptic estimates

Here we prove a Moser Harnack inequality; similar results can be found at [2,
Appendix A] and [14, Theorem 3.4]. For a fixed boundary point
$(p_{0},0)\in\partial{X}$, we consider local coordinates
$(x,\rho)\in\mathbb{R}^{n}\times\mathbb{R}$ and use the notation

$B_{r}^{+}=\left\\{(x,\rho)\in\bar{X}:\rho>0,\,d_{\bar{g}}((x,\rho),p_{0})<r\right\\},$
$\Gamma_{r}^{0}=\left\\{(x,0)\in M:d_{g_{0}}(x,p_{0})<r\right\\},$
$\Gamma_{r}^{+}=\left\\{(x,\rho)\in\bar{X}:\rho\geq
0,\,d_{\bar{g}}((x,\rho),p_{0})=r\right\\}.$

######  Proposition A.30.

Let $U$ be a nonnegative weak solution to

$\left\\{\begin{array}[]{@{\;}r@{\;}ll}\operatorname{div}(\rho^{1-2\gamma}\nabla{U})+E(\rho)U&=0\hfill\quad\text{
in }{B}_{2r}^{+},\\\ -\lim\limits_{\rho\to
0}\rho^{1-2\gamma}\partial_{\rho}U&=f(x)\hfill\quad\text{ on
}{\Gamma}_{2r}^{0}.\end{array}\right.$

where $|E(\rho)|\leq C\rho^{1-2\gamma}$. Then for each $\bar{p}>1$ and
$q>\frac{n}{2\gamma}$,

$\sup_{B_{r}^{+}}U+\sup_{\Gamma_{r}^{0}}U\\\
\leq{C}_{\bar{p},q}\left[r^{-\frac{n+2-2\gamma}{\bar{p}}}\left\|{U}\right\|_{L^{\bar{p}}(B_{2r}^{+},\rho^{1-2\gamma})}+r^{-\frac{n}{\bar{p}}}\left\|{U}\right\|_{L^{\bar{p}}(\Gamma_{2r}^{0})}+r^{2\gamma-\frac{n}{q}}||f||_{L^{q}(\Gamma_{2r}^{0})}\right]$

for some $C_{\bar{p},q}>0$ depending on $\bar{p}$ and $q$.

###### Proof.

The Moser iteration process is by now a very standard approach. We will just
sketch the main steps. Details can be found in [2] and [14]. Since we are just
using the local information, we will prove the Harnack inequality in the
Euclidean case and use $y>0$ as the extension variable.

After scaling we can assume $r=1$. Let $\ell=\|f\|_{L^{q}(\Gamma_{2}^{0})}$
and $0\leq\eta\in C^{1}_{c}(B_{2}^{+})$. We will work with the case $\ell>0$,
for otherwise we may let an arbitrary positive $\ell$ tend to zero. Set
$\bar{U}=U+\ell$ and, for simplicity, $a=1-2\gamma$. Firstly by multiplying
the equation by $\eta^{2}\bar{U}^{\beta}$ for some $\beta>0$ and integrating
by parts, we have

$\begin{split}&\quad\;2\int_{B_{2}^{+}}y^{a}\eta\bar{U}^{\beta}\nabla\eta\nabla\bar{U}\,dxdy+\beta\int_{B_{2}^{+}}y^{a}\eta^{2}\bar{U}^{\beta-1}|\nabla\bar{U}|^{2}\,dxdy+\int_{\Gamma_{2}^{0}}\eta^{2}\bar{U}^{\beta}f(x)\,dx\\\
&=\int_{B^{+}_{2}}E(y)\eta^{2}\bar{U}^{\beta+1}\,dxdy.\end{split}$

Using Hölder’s inequality to handle the cross term, we simplify it using
Young’s inequality as

$\begin{split}\int_{B_{2}^{+}}y^{a}\eta^{2}\bar{U}^{\beta-1}|\nabla\bar{U}|^{2}\,dxdy&\leq\frac{C}{\beta^{2}}\int_{B_{2}^{+}}y^{a}|\nabla\eta|\bar{U}^{\beta+1}\,dxdy+\frac{C}{\beta}\int_{\Gamma_{2}^{0}}\eta^{2}\frac{|f|}{\ell}\bar{U}^{\beta+1}\,dx\\\
&\quad\;+\frac{C}{\beta}\int_{B_{2}^{+}}y^{a}\eta^{2}\bar{U}^{\beta+1}\,dx.\end{split}$

Define $w=\bar{U}^{\frac{1+\beta}{2}}$ and insert it to the above equation.
One gets

$\begin{split}&\quad\;\int_{B_{2}^{+}}y^{a}|\nabla(\eta w)|^{2}\,dxdy\\\ &\leq
C\frac{(\beta+1)^{2}}{\beta^{2}}\int_{B_{2}^{+}}y^{a}(|\nabla\eta|^{2}+\eta^{2})w^{2}\,dxdy+C\frac{(\beta+1)^{2}}{\beta}\int_{\Gamma_{2}^{0}}\eta^{2}w^{2}\frac{|f|}{\ell}\,dx\\\
&=:I_{1}+I_{2}.\end{split}$ (31)

For the left hand side above, one uses the trace Sobolev and weighted Sobolev
embedding (see [14, Corollary 5.3 and Proposition 3.3]) to obtain

$C\int_{B_{2}^{+}}y^{a}|\nabla(\eta
w)|^{2}\,dxdy\geq\left(\int_{\Gamma_{2}^{0}}(\eta
w)^{\frac{2n}{n-2\gamma}}\,dx\right)^{\frac{n-2\gamma}{n}}+\left(\int_{B_{2}^{+}}y^{a}(\eta
w)^{k}\,dxdy\right)^{\frac{2}{k}},$ (32)

where $C>0$ is some constant and $k\in(1,2(n+1)/n)$.

Next we estimate $I_{2}$ in (LABEL:eq:A-4). We have

$\begin{split}\int_{\Gamma_{2}^{0}}\eta^{2}w^{2}\frac{|f|}{k}dx&\leq\left\|\frac{|f|}{k}\right\|_{L^{q}(\Gamma_{2}^{0})}\|\eta
w\|_{L^{2q/(q-1)}(\Gamma_{2}^{0})}^{2}\\\ &\leq\epsilon\|\eta
w\|_{L^{2n/(n-2\gamma)}(\Gamma_{2}^{0})}^{2}+\epsilon^{-\frac{n}{2\gamma
q-n}}\|\eta w\|_{L^{2}(\Gamma_{2}^{0})}^{2}.\end{split}$ (33)

Choosing $\epsilon$ small enough, the first term of the right hand side can be
absorbed in to left hand side of (32). Plugging (33) and (32) back into
(LABEL:eq:A-4), one gets

$\begin{split}&\quad\;\left(\int_{\Gamma_{2}^{0}}(\eta{w})^{\frac{2n}{n-2\gamma}}\,dx\right)^{\frac{n-2\gamma}{n}}+\left(\int_{B_{2}^{+}}y^{a}(\eta{w})^{k}\,dxdy\right)^{\frac{2}{k}}\\\
&\leq C(1+\beta)^{\frac{4\gamma q}{2\gamma
q-n}}\left[\int_{B_{2}^{+}}y^{a}(|\nabla\eta|^{2}+\eta^{2})w^{2}\,dxdy+\int_{\Gamma_{2}^{0}}(\eta
w)^{2}\,dx\right].\end{split}$ (34)

For any $1\leq r_{1}\leq r_{2}\leq 2$, we choose $\eta$ as a cut-off function
satisfying $0\leq\eta\leq 1$, $\eta\leq 2/(r_{2}-r_{1})$ and $\eta=1$ in
$B_{r_{1}}^{+}$ and $\eta=0$ on $B_{2}^{+}\backslash B_{r_{2}}^{+}$. With this
$\eta$ in (LABEL:eq:A-10), we obtain, in terms of $\bar{U}$,

$\begin{split}&\quad\;\left(\int_{\Gamma_{r_{1}}^{0}}\bar{U}^{\frac{(\beta+1)n}{n-2\gamma}}\,dx\right)^{\frac{n-2\gamma}{n}}+\left(\int_{B_{r_{1}}^{+}}y^{a}\bar{U}^{(\beta+1)k}\,dxdy\right)^{\frac{1}{k}}\\\
&\leq C\frac{(1+\beta)^{\frac{4\gamma q}{2\gamma
q-n}}}{(r_{2}-r_{1})^{2}}\left(\int_{\Gamma_{r_{2}}^{0}}\bar{U}^{\beta+1}\,dx+\int_{B_{r_{2}}^{+}}y^{a}\bar{U}^{\beta+1}\,dxdy\right).\end{split}$
(35)

If we set

$\Phi(p,r)=\left(\int_{\Gamma_{r}^{0}}\bar{U}^{p}\,dx\right)^{\frac{1}{p}}+\left(\int_{B_{r}^{+}}y^{a}\bar{U}^{p}\,dxdy\right)^{\frac{1}{p}}$

and $\theta=\min\\{\frac{n}{n-2\gamma},k\\}>1$, then (LABEL:eq:A-11) becomes

$\Phi(\theta(\beta+1),r_{1})\leq\left(\frac{C(1+\beta)^{\frac{2\gamma
q}{2\gamma q-n}}}{r_{2}-r_{1}}\right)^{\frac{2}{\beta+1}}\Phi(\beta+1,r_{2}).$

Now we can iterate the above inequality by setting $R_{m}=1+1/2^{m}$ and
$\theta_{m}=\theta^{m}\bar{p}$. Then

$\Phi(\theta_{m},1)\leq\Phi(\theta_{m},R_{m})\leq(c_{1}\theta)^{c_{2}\sum_{i=0}^{m-1}i/\theta^{i}}\Phi(\bar{p},2)\leq
C\Phi(\bar{p},2)$

for some constant $C$, because the series $\sum_{i=0}^{\infty}i/\theta^{i}$ is
convergent. Finally, since

$\lim_{p\to\infty}\Phi(p,1)=\sup_{\Gamma_{1}^{0}}\bar{U}+\sup_{B_{1}^{+}}\bar{U},$

we have

$\sup_{\Gamma_{1}^{0}}U+\sup_{B_{1}^{+}}U\leq
C\left[||U||_{L^{\bar{p}}(B_{2}^{+},y^{a})}+||U||_{L^{\bar{p}}(\Gamma_{2}^{0})}+||f||_{L^{q}(\Gamma_{2}^{0})}\right].$

Rescaling back to $B_{2r}^{+}$, we conclude the proof of theorem. ∎

######  Proposition A.31.

Suppose $(M^{n},g_{0})$ is the comformal infinity of a Poincaré–Einstein
manifold with $n>2\gamma$. For each $q>\frac{n}{2\gamma}$ we can find positive
constants $\eta_{0}=\eta_{0}(M,g_{0},q,C_{1})$ and $C=C(M,g_{0},q,C_{1})$ with
the following significance: if $g=u^{\frac{4}{n-2\gamma}}g_{0}$ is a conformal
metric and $R=P_{\gamma}^{g}(1)$ satisfying

$\int_{M}u^{\frac{2n}{n-2\gamma}}\,d\mu_{0}+\int_{M}uP_{\gamma}^{g_{0}}u\,d\mu_{0}\leq
C_{1}\quad\text{ and
}\quad\int_{\Gamma_{2r}^{0}(x)}|R|^{q}\,d\mu_{g}\leq\eta_{0}$ (36)

for $x\in M$, then we have

$u(x)\leq C.$

Before we prove this proposition, we collect some useful estimates.

######  Lemma A.32.

Let $x\in{M}$. Under the same assumptions as in Proposition A.31, there hold

$r^{-2}\int_{B_{2r}^{+}(x)}\rho^{1-2\gamma}U^{2}\,d\mu_{\bar{g}_{0}}\leq{C_{2}}$
(37)

and

$r^{-n}\int_{\Gamma_{2r}^{0}(x)}d\mu_{g}\leq{C_{2}},$

where $C_{2}$ depends only on $C_{1}$.

###### Proof.

For the first assertion, using Hölder’s inequality,

$\int_{B_{2r}^{+}}\rho^{1-2\gamma}U^{2}\,d\mu_{\bar{g}_{0}}\leq{C}r^{2}\left(\int_{B_{2r}^{+}}\rho^{1-2\gamma}U^{\frac{2(n+2-2\gamma)}{n-2\gamma}}\,d\mu_{\bar{g}_{0}}\right)^{\frac{n-2\gamma}{n+2-2\gamma}}.$

It follows from the weighted Sobolev embedding in [19, Theorem 2] and the
weighted Poincaré–Hardy inequalities (see [29]) that

$\begin{split}&\quad\;\left(\int_{B_{2r}^{+}}\rho^{1-2\gamma}U^{\frac{2(n+2-2\gamma)}{n-2\gamma}}\,d\mu_{\bar{g}_{0}}\right)^{\frac{n-2\gamma}{n+2-2\gamma}}\\\
&\leq
C\int_{B_{2r}^{+}}\rho^{1-2\gamma}|\nabla{U}|_{\bar{g}_{0}}^{2}\,d\mu_{\bar{g}_{0}}+C\int_{B_{2r}^{+}}\rho^{-1-2\gamma}U^{2}\,d\mu_{\bar{g}_{0}}\\\
&\leq{C}\int_{X}\rho^{1-2\gamma}|\nabla{U}|_{\bar{g}_{0}}^{2}\,d\mu_{\bar{g}_{0}}\\\
&\leq C\int_{M}uP_{\gamma}^{g_{0}}u\,d\mu_{0}+C\int_{M}u^{2}\,d\mu_{0}\leq
CC_{1},\end{split}$

where $C$ is large enough constant that depends only on $(X,\bar{g}_{0})$.

The second estimate is immediate. ∎

###### Proof of Proposition A.31.

The proof is similar to [1, Proposition A.3] where the author deals with the
$\gamma=\frac{1}{2}$ case. The key step is to obtain [6, (187)] in our
setting. This is the consequence of Proposition A.30, which, we stress again,
holds on the manifold $(M^{n},g_{0})$. Let $U$ be the extension of $u$ to $X$,
which satisfies

$\left\\{\begin{array}[]{@{\;}r@{\;}ll}-\operatorname{div}(\rho^{1-2\gamma}\nabla{U})+E_{g_{0}}(\rho)U&=0&\text{
in }(X,\bar{g}_{0}),\\\ U&=u&\text{ on }(M,g_{0}),\\\
-c_{\gamma}\lim\limits_{\rho\to
0}\rho^{1-2\gamma}\partial_{\rho}U&=P_{\gamma}^{g_{0}}u&\text{ on
}(M,g_{0}).\end{array}\right.$ (38)

It follows from Proposition A.30 that for any center on $M$ and any small
radius $r>0$,

$\begin{split}\sup_{\Gamma_{r}^{0}}U+\sup_{B_{r}^{+}}U&\leq
Cr^{-\frac{n+2-2\gamma}{2}}\left(\int_{B_{2r}^{+}}\rho^{1-2\gamma}U^{2}\,d\mu_{\bar{g}_{0}}\right)^{\frac{1}{2}}\\\
&\quad\;+Cr^{-\frac{n-2\gamma}{2}}\left(\int_{\Gamma_{2r}^{0}}u^{\frac{2n}{n-2\gamma}}\,d\mu_{0}\right)^{\frac{n-2\gamma}{2n}}\\\
&\quad\;+Cr^{2\gamma-\frac{n}{q}}\left(\int_{\Gamma_{2r}^{0}}|P_{\gamma}^{g_{0}}{u}|^{q}\,d\mu_{0}\right)^{\frac{1}{q}}.\end{split}$
(39)

Notice that Lemma A.32 and our assumption (36) imply

$\begin{split}r^{\frac{n-2\gamma}{2}}\sup_{B_{r}^{+}(x)}U&\leq
C_{2}r^{\frac{n-2\gamma}{2}}+Cr^{\frac{n+2\gamma}{2}-\frac{n}{q}}\left(\int_{\Gamma_{2r}^{0}(x)}|P_{\gamma}^{g_{0}}{u}|^{q}\,d\mu_{0}\right)^{\frac{1}{q}}.\end{split}$
(40)

Now let us suppose $r_{0}$ is a real number such that $r_{0}<r$ and

$(r-s)^{\frac{n-2\gamma}{2}}\sup_{B^{+}_{s}(x)}U\leq(r-r_{0})^{\frac{n-2\gamma}{2}}\sup_{B_{r_{0}}^{+}(x)}U$

for all $s<r$. Moreover, we can find $x_{0}\in B_{r_{0}}^{+}(x)$ such that

$\sup_{B_{r_{0}}^{+}(x)}U=U(x_{0}).$

We can assume $r$ is small that $d(x_{0},M)$ is achieved by a unique point
$x_{0}^{*}$ on $M$. By the definition of $r_{0}$ and $x_{0}$, we have

$\sup_{B^{+}_{\frac{r-r_{0}}{2}}(x^{*}_{0})}U\leq\sup_{B^{+}_{\frac{r+r_{0}}{2}}(x)}U\leq
2^{\frac{n-2\gamma}{2}}U(x_{0}).$

We want to show that (40) implies the existence of a fixed constant
$K=K(C_{2})$ such that for _all_ $s\leq\frac{r-r_{0}}{2}$,

$s^{\frac{n-2\gamma}{2}}U(x_{0})\leq
K+K(s^{\frac{n-2\gamma}{2}}U(x_{0}))^{\frac{n+2\gamma}{n-2\gamma}-\frac{2n}{n-2\gamma}\frac{1}{q}}\left(\int_{B_{r}^{+}(x)}|R_{g}|^{q}\,d\mu_{g}\right)^{\frac{1}{q}}.$
(41)

To that end we distinguish two cases according to the size of $s$. If
$0<s<\min\\{2d(x_{0},M),\frac{r-r_{0}}{2}\\}$, then the interior Harnack
inequality yields

$s^{\frac{n-2\gamma}{2}}U(x_{0})\leq
Cs^{-1}\left(\int_{B_{s}}\rho^{1-2\gamma}U^{2}\,d\mu_{\bar{g}_{0}}\right)^{\frac{1}{2}}\leq
K,$

for a ball $B_{s}\subset X$, using estimates similar to (37). On the other
hand, if $\min\\{2d(x_{0},M),\frac{r-r_{0}}{2}\\}\leq s\leq\frac{r-r_{0}}{2}$,
then $x_{0}\in B_{s}^{+}(x_{0}^{*})$ and
$\Gamma_{2s}^{0}(x_{0}^{*})\subset\Gamma_{2r}^{0}(x)$. We get from (40) that

$\begin{split}s^{\frac{n-2\gamma}{2}}U(x_{0})&\leq
C_{2}s^{\frac{n-2\gamma}{2}}+Cs^{\frac{n+2\gamma}{2}-\frac{n}{q}}\left(\int_{\Gamma_{2r}^{0}(x)}|P_{\gamma}^{g_{0}}u|^{q}\,d\mu_{0}\right)^{1/q}\\\
&\leq
Ks^{\frac{n-2\gamma}{2}}+Ks^{\frac{n+2\gamma}{2}-\frac{n}{q}}\left(\int_{\Gamma_{2r}^{0}(x)}u^{\frac{n+2\gamma}{n-2\gamma}q-\frac{2n}{n-2\gamma}}|R|^{q}\,d\mu_{g}\right)^{1/q},\end{split}$
(42)

so again we get (41). Therefore (41) holds for any $s\leq\frac{r-r_{0}}{2}$.

Now we choose $\eta_{0}>0$ such that

$(2K)^{\frac{n+2\gamma}{n-2\gamma}-\frac{2n}{n-2\gamma}\frac{1}{q}}\eta_{0}^{\frac{1}{q}}\leq\frac{1}{2}.$

We claim that

$\left(\frac{r-r_{0}}{2}\right)^{\frac{n-2\gamma}{2}}U(x_{0})\leq 2K.$

Indeed, if, on the contrary,
$2K\leq(\frac{r-r_{0}}{2})^{\frac{n-2\gamma}{2}}U(x_{0})$, then we let
$s=(\frac{2K}{U(x_{0})})^{\frac{2}{n-2\gamma}}\leq\frac{r-r_{0}}{2}$ in (41),
which yields

$\begin{split}2K&\leq{K}+K(2K)^{\frac{n+2\gamma}{n-2\gamma}-\frac{2n}{n-2\gamma}\frac{1}{q}}\left(\int_{\Gamma_{2r}^{0}(x)}|R_{g}|^{q}\,d\mu_{g}\right)^{\frac{1}{q}}\\\
&\leq
K+K(2K)^{\frac{n+2\gamma}{n-2\gamma}-\frac{2n}{n-2\gamma}\frac{1}{q}}\eta_{0}^{\frac{1}{q}}.\end{split}$

Clearly this contradicts the choice of $\eta_{0}$. Thus we must have

$\left(\frac{r-r_{0}}{2}\right)^{\frac{n-2\gamma}{2}}U(x_{0})\leq 2K.$

Using (42) with $s$ replaced by $\frac{r-r_{0}}{2}$, we obtain

$\begin{split}&\quad\;\left(\frac{r-r_{0}}{2}\right)^{\frac{n-2\gamma}{2}}U(x_{0})\\\
&\leq{K}\left(\frac{r-r_{0}}{2}\right)^{\frac{n-2\gamma}{2}}\\\
&\quad\;+K(2K)^{\frac{4\gamma}{n-2\gamma}-\frac{2n}{n-2\gamma}\frac{1}{q}}\left(\int_{\Gamma_{2r}^{0}(x)}|R_{g}|^{q}\,d\mu_{g}\right)^{\frac{1}{q}}\left(\frac{r-r_{0}}{2}\right)^{\frac{n-2\gamma}{2}}U(x_{0}).\end{split}$

Since
$\left(\int_{\Gamma_{2r}^{0}(x)}|R_{g}|^{q}\,d\mu_{g}\right)^{\frac{1}{q}}\leq\eta_{0}^{\frac{1}{q}}$
and
$(2K)^{\frac{n+2\gamma}{n-2\gamma}-\frac{2n}{n-2\gamma}\frac{1}{q}}\eta_{0}^{\frac{1}{q}}\leq\frac{1}{2}$,
then

$\displaystyle\left(\frac{r-r_{0}}{2}\right)^{\frac{n-2\gamma}{2}}U(x_{0})$
$\displaystyle\leq 2K\left(\frac{r-r_{0}}{2}\right)^{\frac{n-2\gamma}{2}}\leq
2Kr^{\frac{n-2\gamma}{2}}.$

Thus we conclude that

$\displaystyle r^{\frac{n-2\gamma}{2}}U(x)$
$\displaystyle\leq(r-r_{0})^{\frac{n-2\gamma}{2}}U(x_{0})$ $\displaystyle\leq
2^{\frac{n+2-2\gamma}{2}}Kr^{\frac{n-2\gamma}{2}},$

as desired. ∎

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