# Compactness of singular solutions to the sixth order GJMS equation

João Henrique Andrade  ,  João Marcos do Ó  ,  Jesse Ratzkin  and  Juncheng
Wei  Department of Mathematics, University of British Columbia
V6T 1Z2, Vancouver-BC, Canada
and
Institute of Mathematics and Statistics, University of São Paulo
05508-090, São Paulo-SP, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>Department of Mathematics, Federal University of Paraíba
58051-900, João Pessoa-PB, Brazil<EMAIL_ADDRESS>Department of Mathematics,
Universität Würzburg
97070, Würzburg-BA, Germany<EMAIL_ADDRESS>Department of Mathematics, University of British Columbia
V6T 1Z2, Vancouver-BC, Canada<EMAIL_ADDRESS>
###### Abstract.

We study compactness properties of the set of conformally flat singular
metrics with constant, positive sixth order $Q$-curvature on a finitely
punctured sphere. Based on a recent classification of the local asymptotic
behavior near isolated singularities, we introduce a notion of necksize for
these metrics in our moduli space, which we use to characterize compactness.
More precisely, we prove that if the punctures remain separated and the
necksize at each puncture is bounded away from zero along a sequence of
metrics, then a subsequence converges with respect to the Gromov–Hausdorff
metric. Our proof relies on an upper bound estimate which is proved using
moving planes and a blow-up argument. This is combined with a lower bound
estimate which is a consequence of a removable singularity theorem. We also
introduce a homological invariant which may be of independent interest for
upcoming research.

###### Key words and phrases:

Tri-Laplacian, Critical exponent, Sixth order equations, Local asymptotic
behavior, Emden–Fowler solutions

###### 2020 Mathematics Subject Classification:

35J60, 35B09, 35J30, 35B40

This work was partially supported by São Paulo Research Foundation (FAPESP)
#2020/07566-3 and #2021/15139-0, Paraíba State Research Foundation (FAPESQ)
#3034/2021, and National Council for Scientific and Technological Development
(CNPq) #312340/2021-4 and #429285/2016-7, and Natural Sciences and Engineering
Research Council of Canada (NSERC)

##  1\. Introduction

In recent years, there has been active research into analogs of the Yamabe
problem and its singular counterpart. In each of these problems, one seeks a
representative of a conformal class with constant curvature of some type,
scalar curvature in the classical case, and some $\sigma_{k}$-curvature or one
of Branson’s $Q^{2m}$-curvatures in more modern examples. Conformal invariance
(or, more generally, covariance) often complicates these problems, leading to
singular solutions and the lack of compactness in the space of solutions. For
this reason, it is always appealing to characterize which geometric properties
in the solution space imply compactness.

In the present paper, we study the moduli space of complete, conformally flat
metrics with constant sixth order $Q^{6}$-curvature on a finitely punctured
sphere. Our main result generalizes a theorem of Pollack [25] in the scalar
curvature setting, stating that so long as the punctures remain separated and
certain geometric necksizes bounded away from zero, the the corresponding
subset of moduli space is compact in the Gromov-Hausdorff topology.

Let $n\geqslant 7$ and denote the $n$-dimensional sphere by $\mathbb{S}^{n}$.
For $N\in\mathbb{N}$ we let
$\Lambda=\\{p_{1},\dots,p_{N}\\}\subset\mathbb{S}^{n}$ be a finite subset and
seek complete metrics on $\Omega:=\mathbb{S}^{n}\backslash\Lambda$ of the form
$g=U^{{4}/{n-6}}g_{0}$, where $g_{0}$ is the standard round metric. The fact
that $g$ is complete on $\Omega$ forces $\liminf_{p\rightarrow
p_{i}}U(p)=\infty$ for each $i=1,\dots,N$. Furthermore, we prescribe the
resulting metric to have constant $Q^{6}$-curvature, which we normalize to be

$Q_{n}=Q^{6}(g_{0})=\frac{n(n^{4}-20n^{2}+64)}{2^{5}}.$ (1.1)

We define $Q^{6}(g)$ the quantity in Definition A.3 for any smooth metric.

The $Q$-curvature $Q^{6}$ behaves well under a conformal change of metric.
More precisely, the condition that $g=U^{{4}/{(n-6)}}g_{0}$ satisfies
$Q^{6}(g)=Q_{n}$ on $\Omega=\mathbb{S}^{n}\backslash\Lambda$ is equivalent to
the PDE

$P^{6}_{g_{0}}U=c_{n}U^{\frac{n+6}{n-6}}\quad\mbox{on}\quad\Omega,$
($\mathcal{Q}_{6,g_{0},N}$)

where $c_{n}=\frac{n-6}{2}{Q}_{n}$ is a normalizing constant. The operator on
the left-hand side is the sixth order GJMS operator on the sphere defined by

$P_{g_{0}}^{6}=\left(-\Delta_{g_{0}}+\frac{(n-6)(n+4)}{4}\right)\left(-\Delta_{g_{0}}+\frac{(n-4)(n+2)}{4}\right)\left(-\Delta_{g_{0}}+\frac{n(n-2)}{4}\right),$
(1.2)

and after a conformal change of metric $g=U^{{4}/{n-6}}g_{0}$, it transforms
as

$P^{6}_{g}\phi=U^{-\frac{n+6}{n-6}}P^{6}_{g_{0}}(U\phi)\quad\mbox{for
all}\quad\phi\in\mathcal{C}^{\infty}(\Omega).$ (1.3)

For more details on this subject, we refer the interested reader to [19, 10,
7, 12].

In [14] Graham, Jenne, Mason and Sparling constructed conformally covariant
differential operators $P_{g}^{2m}$ on a compact $n$-dimensional Riemannian
manifold $(M^{n},g)$ for any $m\in\mathbb{N}$ such the leading order term of
$P_{g}^{2m}$ is $(-\Delta_{g})^{m}$. One can then construct the associated
$Q$-curvature of order $2m$ by $Q_{g}^{2m}=P_{g_{0}}^{2m}(1)$. In the special
case $m=1$, one recovers the conformal Laplacian

$P_{g}^{2}=-\Delta_{g}+\frac{n-2}{4(n-1)}R_{g}\quad{\rm with}\quad
Q_{g}^{2}=\frac{n-2}{4(n-1)}R_{g},$

where $\Delta_{g}$ is the Laplace-Beltrami operator of $g$ and $R_{g}$ is its
scalar curvature. Subsequently, Grahan and Zworski [15] and Chang and González
[8] extended these definitions in the case the background metric is the round
metric on the sphere to obtain (nonlocal) operators $P_{g_{0}}^{\sigma}$ of
any order $\sigma\in(0,n/2)$ as well as the corresponding $Q$-curvatures of
order $\sigma$. Once again, the leading order part of $P_{g_{0}}^{\sigma}$ is
$(-\Delta_{g_{0}})^{\sigma}$, understood as the principal value of a singular
integral operator. We write the formulae for $P_{g}^{2}$, $P_{g}^{4}$ and
$P_{g}^{6}$ explicitly in Definitions A.2 and A.4. Nevertheless, the
expressions for $P_{g}^{\sigma}$ and $Q_{g}^{\sigma}$ for a general
$\sigma\in\mathbb{R}_{+}$ are more complicated (see for instance [11]).

We remark that the nonlinearity on the right-hand side of
($\mathcal{Q}_{6,g_{0},N}$) has critical growth with respect to the Sobolev
embedding $W^{3,2}(\mathbb{R}^{n})\hookrightarrow
L^{2^{\\#}}(\mathbb{R}^{n})$, where $2^{\\#}=\frac{2n}{n-6}$. It is well known
that this embedding is not compact, reflecting the conformal invariance of the
PDE ($\mathcal{Q}_{6,g_{0},N}$).

It will be convenient to transfer the PDE ($\mathcal{Q}_{6,g_{0},N}$) to
Euclidean space, which we can do using the standard stereographic projection
(with the north pole in $\Omega$, and thus a regular point of any of the
metrics we consider). After stereographic projection, we can write

$g_{0}=u_{\rm sph}^{\frac{4}{n-6}}\delta,\qquad u_{\rm
sph}(x)=\left(\frac{1+|x|^{2}}{2}\right)^{\frac{6-n}{2}},$

where $\delta$ is the Euclidean metric. In these coordinates our conformal
metric takes the form $g=U^{{4}/{(n-6)}}g_{0}=(U\cdot u_{\rm
sph})^{{4}/{(n-6)}}\delta$. Thus,
$u\in\mathcal{C}^{\infty}(\mathbb{R}^{n}\setminus\Gamma)$ given by $u=U\cdot
u_{\rm sph}$ is a positive singular solution to the transformed equation

$\displaystyle(-\Delta)^{3}u=c_{n}u^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\Gamma,$ ($\mathcal{Q}_{6,\delta,N}$)

where $\Delta$ is the usual flat Laplacian and $\Gamma$ is the image of the
singular set $\Lambda$ under the stereographic projection. As a notational
shorthand, we adopt the convention that $U$ refers to a conformal factor
relating the metric $g$ to the round metric, i.e. $g=U^{{4}/{(n-6)}}g_{0}$,
while $u$ refers to a conformal factor relating the metric $g$ to the
Euclidean metric, i.e. $g=u^{{4}/{(n-6)}}\delta$, with the two related by
$u=Uu_{\rm sph}$.

######  Remark 1.1.

In this Euclidean setting, the transformation law (1.3) in particular implies
the scaling law for ($\mathcal{Q}_{6,\delta,N}$), namely if $u$ solves
($\mathcal{Q}_{6,\delta,N}$) then so does
$u_{\lambda}(x):=\lambda^{\frac{n-6}{2}}u(\lambda x)$ for any $\lambda>0$.

We study the compactness properties of both the unmarked and the marked moduli
spaces of admissible constant sixth $Q$-curvature metrics. We define the
unmarked moduli space as

$\displaystyle\mathcal{M}^{6}_{N}=\left\\{g\in[g_{0}]:\mbox{$g$ is complete on
$\mathbb{S}^{n}\setminus\Lambda$ with $\\#\Lambda=N$},\;Q^{6}_{g}\equiv
Q_{n}\right\\},$ (1.4)

and the marked moduli space as

$\displaystyle\mathcal{M}^{6}_{\Lambda}=\left\\{g\in[g_{0}]:\mbox{$g$ is
complete on $\mathbb{S}^{n}\setminus\Lambda$},\;Q^{6}_{g}\equiv
Q_{n}\right\\}.\ $

Intuitively, in the unmarked moduli space we fix only the number of punctures,
whereas in the marked moduli space, we fix the punctures themselves. We place
the Gromov-Hausdorff topology on both the marked and unmarked moduli spaces.

The first step to understanding the properties of the marked moduli space
$\mathcal{M}^{6}_{N}$ is to study the conformally flat equation

$\displaystyle(-\Delta)^{3}u=c_{n}u^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathbb{B}^{*}_{R},$ ($\mathcal{P}_{6,R}$)

where $\mathbb{B}_{R}^{*}:=\\{x\in\mathbb{R}^{n}:0<|x|<R\\}$ is the punctured
ball for $R<+\infty$. Allowing $R\rightarrow+\infty$ turns
($\mathcal{P}_{6,R}$) into the following PDE on the punctured space

$\displaystyle(-\Delta)^{3}u=c_{n}u^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\\{0\\}.$ ($\mathcal{P}_{6,\infty}$)

On this subject, the classification of non-singular solutions to
($\mathcal{P}_{6,\infty}$) is provided in [29]. Later on, in [18] it is proved
that blow-up limit solutions do exist. Recently, based on a topological
shooting method, the first and last authors classified all possible solutions
to this limit equation [3].

One can merge these classification results into the statement below

######  Theorem A.

Let $u$ be a positive solution to ($\mathcal{P}_{6,\infty}$). Assume that

  1. (a)

the origin is a removable singularity, then there exists
$x_{0}\in\mathbb{R}^{n}$ and $\varepsilon>0$ such that $u$ is radially
symmetric about $x_{0}$ and, up to a constant, is given by

$u_{x_{0},\varepsilon}(x)=\left(\frac{2\varepsilon}{1+\varepsilon^{2}|x-x_{0}|^{2}}\right)^{\frac{n-6}{2}}.$
(1.5)

These are called the $($sixth order$)$ spherical solutions $($or bubbles$)$.

  2. (b)

the origin is a non-removable singularity, then $u$ is radially symmetric
about the origin. Moreover, there exist
$\varepsilon_{0}\in(0,\varepsilon^{*}_{n}]$ and $T\in(0,T_{\varepsilon_{0}}]$
such that

$u_{\varepsilon,T}(x)=|x|^{\frac{6-n}{2}}v_{\varepsilon}(\ln|x|+T).$ (1.6)

Here $\varepsilon^{*}_{n}=K_{0}^{(n-6)/6}$, $T_{\varepsilon}\in\mathbb{R}$ is
the fundamental period of the unique $T$-periodic bounded solution $v_{T}$ to
the following sixth order IVP,

$\begin{cases}v^{(6)}-K_{4}v^{(4)}+K_{2}v^{(2)}-K_{0}v=c_{n}v^{\frac{n+6}{n-6}}\\\
v(0)=\varepsilon_{0},\ v^{(2)}(0)=\varepsilon_{2},\
v^{(4)}(0)=\varepsilon_{4},\ v^{(1)}(0)=v^{(3)}(0)=v^{(5)}(0)=0,\end{cases}$

where $K_{4},K_{2},K_{0},\varepsilon^{*}_{n}$ are dimensional constants
$\varepsilon_{0}\in(0,\varepsilon^{*}_{n}]$ $($See (2)$)$. These are called
$($sixth order$)$ Emden–Fowler solutions.

In [18], it is shown that solutions to ($\mathcal{P}_{6,R}$) with $R<+\infty$
satisfy a priori bound near the isolated singularity, which implies that they
behave like the solutions to the limit equation near the isolated singularity

######  Theorem B.

Let $u$ be a positive singular solution to ($\mathcal{P}_{6,R}$). Suppose that
$-\Delta u\geqslant 0$ and $\Delta^{2}u\geqslant 0$. Then

$u(x)=(1+\mathrm{o}(1))u_{\varepsilon,T}(|x|)\quad{\rm as}\quad x\rightarrow
0,$ (1.7)

where $u_{\varepsilon,T}$ belongs to the family (1.6).

These two results combined motivate the following definition

######  Definition 1.2.

Let $g\in\mathcal{M}_{N}$ with a singular set $\Lambda\subset\mathbb{S}^{n}$,
$\\#\Lambda=N$, and let $p_{j}\in\Lambda$. Let
$g=U^{{4}/{(n-6)}}g_{0}=u^{{4}/{(n-6)}}\delta$ where we choose stereographic
coordinates centered at $p_{j}$. By (1.7) we know
$u(x)=u_{\varepsilon_{j},T_{j}}(|x|)(1+\mathrm{o}(|x|))$ for some
$\varepsilon_{j}\in(0,\varepsilon_{n}^{*}]$. This $\varepsilon_{j}$ is the
asymptotic necksize of the metric $g$ at the puncture $p_{j}$.

Now we have conditions to state our main compactness theorem for the unmarked
moduli space

######  Theorem 1.3.

Let $N\geqslant 3$ and let $0<\delta_{1},\delta_{2}<1$ be positive real
numbers. Then the set

$\mathcal{Q}^{6}_{\delta_{1},\delta_{2}}=\left\\{g\in\mathcal{M}^{6}_{N}:\mathrm{d}_{g_{0}}(p_{j},p_{\ell})\geqslant\delta_{1}\;{\rm
for\ each}\;j\neq\ell\;{\rm
and}\;\varepsilon_{j}(g)\geqslant\delta_{2}\right\\}.$

is sequentially compact with respect to the Gromov–Hausdorff topology.

######  Remark 1.4.

Notice that as a consequence of Theorem A (a), it follows that
$\mathcal{M}_{1}=\varnothing$. Also, from Theorem A (b), we have that
$\mathcal{M}_{p_{1},p_{2}}=(0,\varepsilon_{n}^{*}]$ for any $p_{1}\neq p_{2}$,
where $\varepsilon_{n}^{*}\in(0,1)$. Moreover, it follows that
$\mathcal{M}_{2}=(0,\varepsilon_{n}^{*}]\times((\mathbb{S}^{n}\times\mathbb{S}^{n}\setminus{\rm
diag})/SO(n+1,1))$, where the group $SO(n+1,1)$ of conformal transformations
acts on each $\mathbb{S}^{n}$ factor simultaneously. These metrics are called
the Delaunay metrics. Furthermore, they all correspond to a doubly punctured
sphere and are rotationally invariant.

######  Remark 1.5.

It is worthwhile to now describe the possible degenerations of a sequence of
metrics in $\mathcal{M}_{N}^{6}$. Let
$\\{g_{k}=(U_{k})^{{4}/{n-6}}g_{0}\\}\in\mathcal{M}_{N}^{6}$ be a sequence
that leaves every compact subset. We denote the singular set of $g_{k}$ by
$\Lambda_{k}=\\{p_{1,k},\dots,p_{N,k}\\}$ and the asymptotic necksize of
$g_{k}$ at the puncture $p_{j,k}$ as $\varepsilon_{j,k}$. Then either
$\lim_{k\rightarrow\infty}\varepsilon_{j,k}=0$ for some $j$ or
$\lim_{k\rightarrow\infty}p_{j,k}=\lim_{k\rightarrow\infty}p_{j^{\prime},k}$
for some $j\neq j^{\prime}$. We sketch these two degenerations in Figure 1.
$($It is possible that both degenerations happen simultaneously.$)$ In either
case, in the limit one obtains a metric
$g_{\infty}\in\mathcal{M}_{N^{\prime}}^{6}$ for some $N^{\prime}<N$. In this
way, one can compactify the moduli space $\mathcal{M}_{N}^{6}$ by gluing
copies of $\mathcal{M}_{N^{\prime}}^{6}$ for $N^{\prime}<N$ to
$\partial\mathcal{M}_{N}^{6}$. We speculate that this compactification would
not give a smooth manifold with boundary, but rather that
$\partial\mathcal{M}_{N}^{6}$ is in general a stratified space.

$*$$(p_{1},\varepsilon_{1})$$*$$(p_{4},\varepsilon_{4})$$*$$(p_{2},\varepsilon_{2})$$*$$(p_{3},\varepsilon_{3})$$*$$*$$p_{1}\rightarrow
p_{2}$$*$$*$$*$$*$$*$$\circ$$\varepsilon_{4}\rightarrow 0$ Figure 1. The two
possible degenerations in the moduli space $\mathcal{M}_{4}^{6}$.

Let us compare our main results with the second and fourth order analogs. In
the same spirit as our main result, it was proved in [25] and [2] that the
moduli sets below

$\displaystyle\mathcal{Q}^{2}_{\delta_{1},\delta_{2}}\subset{\mathcal{M}}^{2}_{N}=\left\\{g\in[g_{0}]:\mbox{$g$
is complete}\;\mbox{and}\;R_{g}\equiv 2^{-1}(n-4)\right\\}.$ (1.8)

and

$\displaystyle\mathcal{Q}^{4}_{\delta_{1},\delta_{2}}\subset{\mathcal{M}}^{4}_{N}=\left\\{g\in[g_{0}]:\mbox{g
is complete},\;R_{g}\geqslant 0,\;\mbox{and}\;Q^{4}_{g}\equiv
2^{-3}n(n^{2}-4)\right\\}.$ (1.9)

are also sequentially compact.

Based on classifications results like Theorem A and Theorem B, much more is
known about the moduli spaces in (1.8) and (1.9). In fact, in some classical
works of Mazzeo and Pacard [21] used gluing techniques to prove that there
exists a family of solutions in (1.8). Furthermore, Mazzeo, Pollack, and
Uhlenbeck [22], this space turns out to be a finite-dimensional analytic
submanifold furnished with a natural Lagrangian structure. On the moduli space
(1.9), less is known; it is not proved yet whether this is non-empty. Some of
the authors in [1] proved that this property holds for non-degenerate
manifolds with a suitable hypothesis on the vanishing of the Weyl tensor,
However, the standard round sphere is not included in this class.

Inspired by the arguments in [25], the proof of Theorem 1.3 is divided into
three parts that we describe as follows. First, we need to introduce the so-
called sixth order geometric Pohozaev invariant, which is related to the
Hamiltonian energy of the limiting ODE [27, 24]. Second, we obtain an a priori
upper and for positive singular solutions to ($\mathcal{Q}_{6,g_{0},N}$),
estimates which are accomplished by combining a sliding method, a blow-up
argument, and a Harnack inequality. From this, we obtain uniform bound on
certain Hölder norms, which by compactness, allows us to extract a limit, up
to subsequence. Third, we use the first order asymptotic expansion for the
Green function of the sixth order GJMS operator near the pole and the fact the
necksizes are away from zero shows that this limit is non-trivial. Finally,
one can apply a removable singularity theorem to conclude the proof.

The rest of the paper is divided as follows. In Section 2, we define the
logarithmic cylindrical change of variables and we use the conformal
invariance between the punctured space and the cylinder to transform
($\mathcal{Q}_{6,g_{0},N}$) into a PDE on the cylinder. In Section 3, we
describe all singular solutions on a doubly punctured sphere. These Delaunay
metrics are especially important because they provide asymptotic models for
the metrics in $\mathcal{M}_{N}^{6}$ near a given puncture point. In Section
4, we define the sixth order Pohozaev invariants associated with
($\mathcal{Q}_{6,g_{0},N}$). In Section 5, we prove a priori upper and lower
bound estimates for positive singular solutions to
($\mathcal{Q}_{6,g_{0},N}$). In Section 6, we prove the compactness statement
in Theorem 1.3.

######  Remark 1.6.

Several of our supporting results below generalize to the Paneitz operators
and $Q$-curvatures of any order $\sigma\in(0,n/2)$, at least in the
conformally flat setting. In particular, the convexity result of Lemma 5.1 and
the upper bound of Proposition 5.2 both generalize, and may be of independent
interest. On the other hand, some parts of the proof of Theorem 1.3 do not
carry over. In particular, at this time we cannot classify all two-ended
constant $Q^{\sigma}$-curvature metrics on the sphere, which is very important
for our proof.

##  2\. Cylindrical coordinates

This section is devoted to constructing a change of variables that transforms
the local singular PDE ($\mathcal{P}_{6,R}$) problem into a nice ODE problem
with constant coefficients. This is the conformally flat problem associated
with ($\mathcal{Q}_{6,g_{0},N}$).

######  Definition 2.1.

We define the sixth order autonomous Emden-Fowler change of variables as
follows. Let $R>0$ and $T=-\ln R$ and
$\mathcal{C}_{T}=(T,\infty)\times\mathbb{S}^{n-1}$. We then define

$\mathfrak{F}:\mathcal{C}^{\infty}(B_{R}^{*})\rightarrow\mathcal{C}^{\infty}(\mathcal{C}_{T}),\qquad\mathfrak{F}(u)(t,\theta)=e^{-\gamma_{n}t}u(e^{-t}\theta)=v(t,\theta),$
(2.1)

where $\gamma_{n}=\frac{n-6}{2}$.

It is easy to show the inverse transform is given by

$\mathfrak{F}^{-1}:\mathcal{C}^{\infty}(\mathcal{C}_{T})\rightarrow\mathcal{C}^{\infty}(B_{R}^{*}),\qquad\mathfrak{F}^{-1}(v)(x)=|x|^{-\gamma_{n}}v(-\ln|x|,x/|x|)=u(x).$

Using $\mathfrak{F}$ and performing a lengthy computation we arrive at the
following sixth order nonlinear PDE on $\mathcal{C}_{T}$:

$-P^{6}_{\rm cyl}v=c_{n}v^{\frac{n+6}{n-6}}\quad{\rm
on}\quad{\mathcal{C}}_{T}.$ ($\mathcal{C}_{T}$)

Here $P^{6}_{\rm cyl}$ is the sixth order GJMS operator associated to the
cylindrical metric $g_{\rm cyl}=dt^{2}+d\theta^{2}$ on
$\mathbb{R}\times\mathbb{S}^{n-1}$, and it is given by

$\displaystyle P^{6}_{\rm cyl}:=P^{6}_{\rm rad}+P^{6}_{\rm ang},$

where

$\displaystyle P^{6}_{\rm
rad}:=\partial_{t}^{(6)}-K_{4}\partial_{t}^{(4)}+K_{2}\partial_{t}^{(2)}-K_{0}$

and

$\displaystyle P^{6}_{\rm
ang}:=2\partial^{(4)}_{t}\Delta_{\theta}-J_{3}\partial^{(3)}_{t}\Delta_{\theta}+J_{2}\partial^{(2)}_{t}\Delta_{\theta}-J_{1}\partial_{t}\Delta_{\theta}+J_{0}\Delta_{\theta}+3\partial^{(2)}_{t}\Delta^{2}_{\theta}-L_{0}\Delta^{2}_{\theta}+\Delta^{3}_{\theta}$

with

$\displaystyle K_{0}=2^{-8}(n-6)^{2}(n-2)^{2}(n+2)^{2}$ $\displaystyle
K_{2}=2^{-4}(3n^{4}-24n^{3}+72n^{2}-96n+304)$ $\displaystyle
K_{4}=2^{-2}(3n^{2}-12n+44)$ $\displaystyle
J_{0}=2^{-3}(3n^{4}-18n^{3}-192n^{2}+1864n-3952)$ (2.2) $\displaystyle
J_{1}=2^{-1}(3n^{3}+3n^{2}-244n+620)$ $\displaystyle J_{2}=2n^{2}+13n-68$
$\displaystyle J_{3}=2(n+1)$ $\displaystyle L_{0}=2^{-2}(3n^{2}-12n-20)$

dimensional constants.

######  Remark 2.2.

The following decomposition holds

$P^{6}_{\rm rad}=L_{\lambda_{1}}\circ L_{\lambda_{2}}\circ L_{\lambda_{3}},$

where $L_{\lambda_{j}}:=-\partial_{t}^{2}+\lambda_{j}$ for $j=1,2,3$ with

$\lambda_{1}=\frac{n-6}{2},\quad\lambda_{2}=\frac{n-2}{2},\quad{\rm
and}\quad\lambda_{3}=\frac{n+2}{2}.$

We refer the reader to [3, Proposition 2.7] for the proof.

##  3\. Spherical and Delaunay metrics

In this section, we present some particular model metrics on the moduli space.
Let $p_{1},p_{2}\in\mathbb{S}^{n}$, which without loss of generality can be
chosen such that $p_{1}=\mathbf{e}_{n}$ is the north pole and $p_{2}=-p_{1}$
is the south pole. The conformal factor
$U:\mathbb{S}^{n}\backslash\\{p_{1},p_{2}\\}\rightarrow(0,\infty)$ determines
a metric $g\in\mathcal{M}_{p_{1},p_{2}}$ and after composing with a
stereographic projection it corresponds to a singular solution to
($\mathcal{P}_{6,\infty}$)

Applying the cylindrical transform (2.1) to this PDE in turn yields

$-P^{6}_{\rm cyl}v=c_{n}v^{\frac{n+6}{n-6}}\quad{\rm
on}\quad{\mathcal{C}}_{\infty}:=\mathbb{R}\times\mathbb{S}^{n}.$

Next, using those solutions to ($\mathcal{Q}_{6,g_{0},N}$) are radially
symmetric with respect to the origin, ($\mathcal{C}_{T}$) reduces to a sixth
order ODE problem

$-v^{(6)}+K_{4}v^{(4)}-K_{2}v^{(2)}+K_{0}v=c_{n}v^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathbb{R}.$ ($\mathcal{O}_{6,\infty}$)

From this last formulation, we quickly compute the cylindrical solution

$v_{\rm
cyl}(t)=\left(\frac{K_{0}}{c_{n}}\right)^{\frac{12}{n-6}}=\left(\frac{K_{0}}{c_{n}}\right)^{\frac{6}{\gamma_{n}}}=\varepsilon_{n}^{*}>0,$

which is the only constant solution. Transforming back from the cylinder to
$\mathbb{R}^{n}\backslash\\{0\\}$ we see

$u_{\rm cyl}(x)=\mathfrak{F}^{-1}(v_{\rm
cyl})=\left(\frac{K_{0}}{c_{n}}\right)^{\frac{12}{n-6}}|x|^{-\gamma_{n}},\quad
g_{\rm cyl}=u_{\rm cyl}^{\frac{4}{n-6}}\delta.$

We have already encountered the spherical solution, given by

$u_{\rm sph}(x)=\left(\frac{1+|x|^{2}}{2}\right)^{-\gamma_{n}}\quad{\rm
and}\quad g_{\rm sph}=u_{\rm sph}^{4/(n-6)}\delta,$ (3.1)

which is the particular case of (1.5) with $\epsilon=1$ and $x_{0}=0$.
Applying the Emden-Fowler change of variables to $u_{\rm sph}$ we obtain

$v_{\rm sph}(t,\theta)=\mathfrak{F}(u_{\rm sph})(t,\theta)=(\cosh
t)^{-\gamma_{n}}.$

In this setting, Theorem A classifies all positive solutions
$v_{\varepsilon_{0}}\in\mathcal{C}^{6}(\mathbb{R})$ to
($\mathcal{O}_{6,\infty}$) in terms of the necksize
$\varepsilon_{0}\in(0,\varepsilon^{*}_{n}]$, where
$\varepsilon_{0}=\min_{\mathbb{R}}v\in(0,\varepsilon^{*}_{n}]$. Varying the
parameter $\varepsilon$ from its maximal value of $\varepsilon_{n}^{*}$ to
$0$, we see that the Delaunay solutions in Theorem A (b) interpolate between
the cylindrical solution $v_{\rm cyl}$ and the spherical solution $v_{\rm
sph}$. We denote the minimal period of $v_{\varepsilon}$ by $T_{\varepsilon}$.

######  Definition 3.1.

For each $\varepsilon\in(0,\varepsilon_{n}^{*}]$ the Delaunay metric of
necksize $\varepsilon$ is

$g_{\varepsilon}=v_{\varepsilon}^{\frac{4}{n-6}}(\mathrm{d}t^{2}+\mathrm{d}\theta^{2})=u_{\varepsilon}^{\frac{4}{n-6}}\delta,$

where $u_{\varepsilon}=\mathfrak{F}^{-1}(v_{\varepsilon})$. Observe that we
have equivalently defined $g_{\varepsilon}$ as a metric on
$\mathcal{C}_{-\infty}$, using $v_{\varepsilon}$ as the conformal factor, and
on $\mathbb{R}^{n}\backslash\\{0\\}$, using
$u_{\varepsilon}=\mathfrak{F}^{-1}(v_{\varepsilon})$ as the conformal factor.

We can reformulate the expansion (1.7) to read

######  Proposition 3.2.

Let $g\in\mathcal{M}^{6}_{N}$ with the singular set $\Lambda$ and let
$p\in\Lambda$. Then there exists a Delaunay solution $u_{\varepsilon}$ such
that in stereographic coordinates centered at $p$ the asymptotic expansion

$g=((1+\mathrm{o}(|x|))u_{\varepsilon,R}(x))^{\frac{4}{n-6}}\delta,\qquad
u_{\varepsilon,R}(x)=u_{\varepsilon}(Rx).$

We can restate this asymptotic expansion as

$\displaystyle g$ $\displaystyle=$
$\displaystyle((1+\mathrm{o}(|x|))\mathfrak{F}^{-1}(v_{\varepsilon}(\cdot+T))(x))^{\frac{4}{n-6}}\delta=((1+\mathrm{o}(e^{-t}))v_{\varepsilon}(t+T))^{\frac{4}{n-6}}(\mathrm{d}t^{2}+\mathrm{d}\theta^{2}).$

In other words, any admissible metric is asymptotic to a translated Delaunay
metric near a puncture. In the formulae above $R$ and $T$ are related by
$R=-\ln T$.

##  4\. Pohozaev invariants

We now turn to a discussion of the existence and specific form of a family of
homological integral invariants of solutions of equation
($\mathcal{Q}_{6,g_{0},N}$). These homological invariants were discovered in
their simplest form by S. Pohozaev [24], and generalized by R. Schoen [27] for
the Riemannian setting.

As a starting point, we define the energy $\mathcal{H}_{\rm cyl}$ by

$\mathcal{H}_{\rm cyl}(v):=\mathcal{H}_{\rm rad}(v)+\mathcal{H}_{\rm
ang}(v)+F(v),$ (4.1)

where

$\mathcal{H}_{\rm
rad}(v):=\frac{1}{2}{v^{(3)}}^{2}+\frac{K_{4}}{2}{v^{(2)}}^{2}+\frac{K_{2}}{2}{v^{(1)}}^{2}-\frac{K_{0}}{2}v^{2}+v^{(5)}v^{(1)}-v^{(4)}v^{(2)}-{K_{4}}v^{(3)}v^{(1)},$

is the radial part,

$\displaystyle\mathcal{H}_{\rm ang}(v)$
$\displaystyle:=-J_{4}\left(\partial_{t}^{(3)}\nabla_{\theta}v\partial_{t}\nabla_{\theta}v-|\partial_{t}^{(2)}\nabla_{\theta}v|^{2}\right)-\frac{J_{2}}{2}|\partial_{t}^{(2)}\nabla_{\theta}v|^{2}-\frac{J_{1}}{2}|\partial_{t}^{(2)}\nabla_{\theta}v|^{2}$
$\displaystyle-\frac{J_{0}}{2}|\nabla_{\theta}v|^{2}+\frac{L_{2}}{2}|\partial_{t}^{(2)}\Delta_{\theta}v|^{2}+\frac{L_{0}}{2}|\partial_{t}^{(2)}\Delta_{\theta}v|^{2}+\frac{1}{2}|\Delta_{\theta}v|^{2}.$

is the angular part, and

$F(v):=\frac{c_{n}(n-6)}{2n}|v|^{\frac{2n}{n-6}}$

is the nonlinear term.

Evaluating a derivative, one can easily verify $\mathcal{H}_{\rm cyl}(v)$ is
constant for any solution $v$ of the PDE ($\mathcal{C}_{T}$). We further
observe that the last term $F$ in (4.1) is homogeneous of degree
$\frac{2n}{n-6}$ while the remaining terms are all homogeneous of degree $2$.

######  Definition 4.1.

Let $v\in\mathcal{C}^{6}(\mathcal{C}_{T})$ be a positive solution to
($\mathcal{C}_{T}$). We define its cylindrical Pohozaev invariant as

$\mathcal{P}_{\rm
cyl}(v):=\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
cyl}(v)\mathrm{d}\theta$

for any $t>T$. Observe that this integral does not, in fact, depend on $t$.

In light of the cylindrical transformation from Definition 2.1, we can define
this invariant in spherical coordinates

######  Definition 4.2.

Let $u\in\mathcal{C}^{6}(\mathbb{B}^{*}_{R})$ be a positive solution to
($\mathcal{P}_{6,R}$). We define its spherical Pohozaev invariant as

$\mathcal{P}_{\rm sph}(u):=(\mathcal{P}_{\rm
cyl}\circ\mathfrak{F}^{-1})(u)=\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
cyl}(\mathfrak{F}^{-1}(u))\mathrm{d}\theta.$

Finally, in terms of conformal metrics, we have the following definition of an
invariant associated with metrics in the moduli space.

######  Definition 4.3.

Let $g\in\mathcal{M}^{6}_{N}$ and $p_{j}\in\Lambda$. We define its radial
$($or dilational$)$ Pohozaev invariant at the puncture $p_{j}$ as follows.
Choose stereographic coordinates sending $p_{j}$ to the origin and write
$g=u^{\frac{4}{n-6}}\delta$ in these coordinates. Then define

$\mathcal{P}_{\rm rad}(g,p_{j}):=\mathcal{P}_{\rm
sph}(u)=\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
cyl}(\mathfrak{F}^{-1}(u))\mathrm{d}\theta.$

The most important result of this section states that bounding the radial
Pohozaev invariants away from zero is equivalent to bounding the necksizes of
the Delaunay asymptotes away from zero.

######  Proposition 4.4.

Let $g\in\mathcal{M}^{6}_{N}$ and $p_{j}\in\Lambda$. Then
$\mathcal{P}_{\textrm{rad}}(g,p_{j})$ is well-defined, negative and depends
only on the necksize $\varepsilon_{j}$ of the Delaunay asymptote at
$p_{j}\in\Lambda$. Moreover, decreasing $\varepsilon_{j}$ will increase
$\mathcal{P}_{\textrm{rad}}(g,p_{j})$ and if $\varepsilon_{j}\searrow 0$ then
$\mathcal{P}_{\textrm{rad}}(g,p_{j})\nearrow 0$.

###### Proof.

By construction, the integral defining $\mathcal{P}_{\rm rad}(g,p_{j})$ does
not depend on which sphere $\\{t\\}\times\mathbb{S}^{n-1}$ we choose, so long
as $t$ is sufficiently large, and therefore $\mathcal{P}_{\rm rad}$ is well-
defined. By the asymptotics in Theorem B we know that the conformal factor is
asymptotic to a Delaunay solution $u_{\varepsilon}$, and so letting
$t\rightarrow\infty$ we see

$\mathcal{P}_{\rm
rad}(g,p_{j})=\lim_{t\rightarrow\infty}\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
cyl}(\mathfrak{F}^{-1}(u))\mathrm{d}\theta=\lim_{t\rightarrow\infty}\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
cyl}(v_{\varepsilon})\mathrm{d}\theta<0.$

The remaining properties follow directly from energy ordering of the Delaunay
solutions as described in [3, Lemma 4.14]. ∎

######  Remark 4.5.

One often finds integral invariants in geometric variational problems. For
more details on a class of general higher order conformally invariant locally
conserved tensors, we cite [13]. These invariants arise from the conformal
invariance of ($\mathcal{Q}_{6,g_{0},N}$), by Noether’s famous conservation
theorem.

For our later applications we will need a slight refinement of Proposition
4.4.

######  Proposition 4.6.

Let $v\in\mathcal{C}^{6}(\mathcal{C}_{T})$ be a positive solution to the
following rescaled equation

$-P_{\rm cyl}^{6}v=Av^{\frac{n+6}{n-6}}$

for some constant $A$ and let

$\mathcal{H}_{\rm cyl}^{A}(v)=\mathcal{H}_{\rm rad}(v)+\mathcal{H}_{\rm
ang}(v)+\frac{(n-6)A}{2n}|v|^{\frac{2n}{n-6}}.$

Then

$\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
cyl}^{A}(v)\mathrm{d}\theta$

is independent of $t$.

###### Proof.

The proposition follows from taking the derivative with respect to $t$ and
integrating by parts. ∎

##  5\. Uniform estimates

This section is devoted to proving uniform upper and lower estimates near the
singular set for positive singular solutions to ($\mathcal{Q}_{6,g_{0},N}$).

We begin by quoting a superharmonicity result of Ngô and Ye [23]. We also
remark a similar superharmonicity result for a related integral equation Ao et
al. [4].

######  Proposition A.

Let $u\in\mathcal{C}^{\infty}(\mathbb{R}^{n}\setminus\Gamma)$ be a positive
solution to ($\mathcal{Q}_{6,\delta,N}$). Then additionally $-\Delta
u\geqslant 0$ and $\Delta^{2}u\geqslant 0$ in $\mathbb{R}^{n}\setminus\Gamma$.

###### Proof.

Following [23, Proposition 1.5] we see that $u$ is both weakly superharmonic
and weakly superbiharmonic in $\mathbb{R}^{n}$. In other words, for a smooth
test function $\phi$ compactly supported in $\mathbb{R}^{n}\setminus\Gamma$,
we have

$\int_{\mathbb{R}^{n}}u(-\Delta)\phi\mathrm{d}x\geqslant 0\quad{\rm
and}\quad\int_{\mathbb{R}^{n}}u(-\Delta)^{2}\phi\mathrm{d}x\geqslant 0.$

Standard elliptic regularity then implies $u$ is superharmonic and
superbiharmonic where it is smooth, namely in $\mathbb{R}^{n}\setminus\Gamma$.
∎

The first step is a sixth order version of the convexity result [26,
Proposition 1], which is proved using the Alexandrov’s moving planes (see also
[9, Theorem 4.1] for a fourth order version).

######  Lemma 5.1.

Let $g=U^{{4}/{(n-6)}}g_{0}$ be a complete metric on
$\Omega=\mathbb{S}^{n}\backslash\Lambda$ which is conformal to the round
metric, such that $Q_{g}^{6}$ is a positive constant. Then the boundary of any
$($spherically$)$ round ball in $\Omega$ has a non-negative definite second
fundamental form with respect to $g$.

###### Proof.

We let $\mathcal{B}$ be a geodesic ball with respect to the round metric such
that $\overline{\mathcal{B}}\subset\Omega$ and choose a stereographic
projection that sends $\mathcal{B}$ to the half-space
$\\{x\in\mathbb{R}^{n}:x_{1}<0\\}$. As before, we denote the image of the
singular set $\Lambda$ under this stereographic projection by $\Gamma$. With
respect to these stereographic coordinates the metric takes the form
$g=u^{{4}/{(n-6)}}\delta$ where
$u\in\mathcal{C}^{\infty}(\mathbb{R}^{n}\setminus\Gamma)$ satisfies
($\mathcal{Q}_{6,\delta,N}$), namely
$u:\mathbb{R}^{n}\setminus\Gamma\rightarrow(0,\infty)$ satisfy

$(-\Delta)^{3}u=c_{n}u^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\Gamma.$

Furthermore, the boundary of our round ball is
$\partial\mathcal{B}=\\{x\in\mathbb{R}^{n}:x_{1}=0\\}$ and is oriented by the
inward unit normal $\eta_{g}=u^{-2/(n-6)}\partial_{x_{1}}$. It follows that
the second fundamental form $II$ and mean curvature $H$ of
$\partial\mathcal{B}$ are given by

$II_{ij}=-\langle\nabla_{\partial
x_{j}}\eta,\partial_{x_{i}}\rangle=\frac{2}{n-6}\delta_{ij}u^{\frac{8-n}{n-6}}\partial_{x_{1}}u,\quad
H=\frac{2n}{n-6}u^{\frac{8-n}{n-6}}\partial_{x_{1}}u.$

Therefore, (weak) convexity of $\partial\mathcal{B}$ follows once we show
$\partial_{x_{1}}u\geqslant 0$ along the hyperplane $\\{x_{1}=0\\}$.

By Proposition A, we have

$-\Delta u\geqslant 0\quad{\rm and}\quad(-\Delta)^{2}u\geqslant 0\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\Gamma.$

We now rewrite ($\mathcal{Q}_{6,\delta,N}$) as a second order system, letting

$\mathrm{u}_{0}=u,\quad\mathrm{u}_{1}=-\Delta u\quad{\rm
and}\quad\mathrm{u}_{2}=(-\Delta)^{2}u.$

so that we obtain
$\mathrm{u}_{i}:\mathbb{R}^{n}\setminus\Gamma\rightarrow(0,\infty)$ for
$i=0,1,2$ satisfy

$\left\\{\begin{array}[]{rcl}-\Delta\mathrm{u}_{0}&=&\mathrm{u}_{1}\geqslant
0\\\ -\Delta\mathrm{u}_{1}&=&\mathrm{u}_{2}\geqslant 0\\\
-\Delta\mathrm{u}_{2}&=&c_{n}u_{0}^{\frac{n+6}{n-6}}\geqslant
0.\end{array}\right.$ (5.1)

It follows from [28, Theorem 2.7] that the Newtonian capacity of the singular
set vanishes, i.e. $\operatorname{cap}(\Gamma)=0$. As a consequence, one can
find $a_{0}>0$ and $a_{j}\in\mathbb{R}$ for $j=1,\dots,n$ such that

$\displaystyle\begin{cases}\mathrm{u}_{0}(x)&=a_{0}|x|^{6-n}+\sum_{j=1}^{n}{a_{j}x_{j}}{|x|^{4-n}}+\mathcal{O}\left({|x|^{4-n}}\right)\\\
\mathrm{\partial}_{x_{i}}u_{0}(x)&=-(n-6)a_{0}x_{i}|x|^{4-n}+\mathcal{O}\left(|x|^{4-n}\right)\\\
\mathrm{\partial}^{2}_{x_{i}x_{j}}u_{0}(x)&=\mathcal{O}\left({|x|^{4-n}}\right),\end{cases}$
(5.2)

which, by differentiating further, yields

$\displaystyle\begin{cases}\mathrm{u}_{1}(x)&=b_{0}|x|^{4-n}+\sum_{j=1}^{n}{b_{j}x_{j}}{|x|^{2-n}}+\mathcal{O}\left({|x|^{2-n}}\right)\\\
\partial_{x_{i}}\mathrm{u}_{1}(x)&=-(n-4)b_{0}x_{i}|x|^{-n}+\mathcal{O}\left({|x|^{2-n}}\right)\\\
\partial^{(2)}_{x_{i}x_{j}}\mathrm{u}_{1}(x)&=\mathcal{O}\left({|x|^{2-n}}\right)\end{cases}$
(5.3)

and

$\displaystyle\begin{cases}\mathrm{u}_{2}(x)&=c_{0}|x|^{2-n}+\sum_{j=1}^{n}{c_{j}x_{j}}{|x|^{-n}}+\mathcal{O}\left({|x|^{-n}}\right)\\\
\partial_{x_{i}}\mathrm{u}_{2}(x)&=-(n-2)c_{0}x_{i}|x|^{-n}+\mathcal{O}\left({|x|^{2-n}}\right)\\\
\partial^{(2)}_{x_{i}x_{j}}\mathrm{u}_{2}(x)&=\mathcal{O}\left({|x|^{-n}}\right)\end{cases}$
(5.4)

as $|x|\rightarrow 0$, where $b_{0},c_{0}>0$ and $b_{j},c_{j}\in\mathbb{R}$
for $j=1,\dots,n$.

We are now ready to set up the method of moving planes applied to the triple
of functions $(\mathrm{u}_{0},\mathrm{u}_{1},\mathrm{u}_{2})$. For any
$\lambda\in\mathbb{R}$, we let
$\Sigma_{\lambda}=\\{x\in\mathbb{R}^{n}:x_{1}>\lambda\\}$ and
$T_{\lambda}=\partial\Sigma_{\lambda}=\\{x\in\mathbb{R}^{n}:x_{1}=\lambda\\}$.
We also set $\Sigma_{\lambda}^{\prime}=\Sigma_{\lambda}\setminus\Gamma$. For
any $x\in\Sigma_{\lambda}^{\prime}$, we let

$x^{\lambda}=(2\lambda-x_{1},x_{2},\dots,x_{n})$

be the reflection of $x$ across the hyperplane
$T_{\lambda}=\\{x_{1}=\lambda\\}$. Finally, our goal in moving planes is to
show that for any $\lambda\leqslant 0$ and $i=0,1,2$, we have

$\mathrm{w}_{i}^{\lambda}(x)>0\quad{\rm for}\quad i=0,1,2,$ (5.5)

where
$\mathrm{w}_{i}^{\lambda}:\Sigma_{\lambda}^{\prime}\rightarrow\mathbb{R}$ is
given by

$\mathrm{w}_{i}^{\lambda}(x)=\mathrm{u}_{i}(x)-\mathrm{u}_{i}(x^{\lambda}).$

Once we establish (5.5), letting $\lambda\nearrow 0$ the first inequality
implies $\mathrm{u}_{x_{1}}\geqslant 0$ on $T_{0}=\partial\mathcal{B}$,
completing our proof.

Observe that the expansion (5.4) implies $\mathrm{u}_{2}$ is not identically
zero. Thus, using the strong maximum principle and the last equation in (5.1),
we see that $\mathrm{u}_{2}>0$ on $\mathbb{R}^{n}\backslash\Gamma$. Working
backwards, the inequality $\mathrm{u}_{2}>0$ and the same reasoning implies
$\mathrm{u}_{1}>0$ on $\mathbb{R}^{n}\backslash\Gamma$, which then in turn
gives us $\mathrm{u}_{0}>0$ on $\mathbb{R}^{n}\backslash\Gamma$.

The singular set $\Gamma$ is compact, so there exists $R_{0}>0$ such that
$\Gamma\subset\mathbb{B}_{R_{0}}(0)$. We use the extended maximum principle
[20, Theorem 3.4] to conclude there exists $\delta>0$, depending on $R>R_{0}$,
such that

$\left.\mathrm{u}_{0}\right|_{\mathbb{B}_{R}(0)\backslash\Gamma}\geqslant\delta,\quad\left.\mathrm{u}_{1}\right|_{\mathbb{B}_{R}(0)\backslash\Gamma}\geqslant\delta,\quad{\rm
and}\quad\left.\mathrm{u}_{2}\right|_{\mathbb{B}_{R}(0)\backslash\Gamma}\geqslant\delta.$
(5.6)

Combining our expansion (5.4) with Lemma 2.3 of [5] there exists $R_{1}>0$ and
$\lambda_{1}\leqslant\lambda_{0}$ such that for each $\lambda<\lambda_{1}$ we
have

$\mathrm{w}_{0}^{\lambda}(x)>0,\quad\mathrm{w}_{1}^{\lambda}(x)>0,\quad{\rm
and}\quad\mathrm{w}_{2}^{\lambda}(x)>0\quad\textrm{ for }\quad
x\in\Sigma_{\lambda}\quad\textrm{ and }\quad|x|>R.$

Using this inequality together with (5.6) then implies that there exists
$\lambda_{2}\leqslant\lambda_{1}$ such that

$\mathrm{w}_{0}^{\lambda}(x)>0,\quad\mathrm{w}_{1}^{\lambda}(x)>0,\quad{\rm
and}\quad\mathrm{w}_{2}^{\lambda}(x)>0\quad\textrm{ on
}\quad\Sigma_{\lambda}^{\prime}\quad{\rm for\
each}\quad\lambda\leqslant\lambda_{2}.$

By construction

$\Delta\mathrm{w}_{2}^{\lambda}(x)=c_{n}\left(\mathrm{u}_{0}(x^{\lambda})^{\frac{n+6}{n-6}}-\mathrm{u}_{0}(x)^{\frac{n+6}{n-6}}\right)<0\quad\textrm{
on }\quad\Sigma_{\lambda}^{\prime}\quad{\rm for\
each}\quad\lambda\leqslant\lambda_{2}.$ (5.7)

On the other hand, the asymptotic expansion (5.4) implies

$\mathrm{w}_{2}^{\lambda}(x)\rightarrow 0\quad\textrm{ as
}\quad|x|\rightarrow\infty.$ (5.8)

Putting together (5.7), (5.8) and
$\left.\mathrm{w}_{2}^{\lambda}\right|_{T_{\lambda}}=0$, we see by the maximum
principle that $\mathrm{w}_{2}^{\lambda}(x)\geqslant 0$ for each
$x\in\Sigma_{\lambda}^{\prime}$ and $\lambda\leqslant\lambda_{2}$. However, by
the completeness of the metric $g$ on $\Omega$ we know that
$\mathrm{w}_{2}^{\lambda}$ is not identically zero on
$\Sigma_{\lambda}^{\prime}$, so again the maximum principle actually implies
$\mathrm{w}_{2}^{\lambda}(x)>0$ for each $x\in\Sigma_{\lambda}^{\prime}$ and
$\lambda\leqslant\lambda_{2}$. Once again, analogous arguments imply
$\mathrm{w}_{1}^{\lambda}>0$ and $\mathrm{w}_{0}^{\lambda}>0$ on
$\Sigma_{\lambda}^{\prime}$ for each $\lambda\leqslant\lambda_{2}$.

At this point we define

$\lambda^{*}=\sup\\{\lambda\leqslant 0:\mathrm{w}_{i}^{\mu}(x)>0\textrm{ for
each }\mu\leqslant\lambda\textrm{ and }i=0,1,2\\}$

and prove that $\lambda^{*}=0$. Following our definitions we have

$\Delta\mathrm{w}_{0}^{\lambda}(x)=-\Delta\mathrm{u}_{0}(x)+\Delta\mathrm{u}_{0}(x^{\lambda})<0$

for each $x\in\Sigma_{\lambda}^{\prime}$ and $\lambda<\lambda^{*}$, and so
$\Delta\mathrm{w}_{0}^{\lambda^{*}}\leqslant 0$ on
$\Sigma_{\lambda^{*}}^{\prime}$. By similar arguments we also have

$\Delta\mathrm{w}_{1}^{\lambda^{*}}\leqslant
0,\quad\Delta\mathrm{w}_{2}^{\lambda^{*}}\leqslant 0\textrm{ on
}\Sigma_{\lambda^{*}}^{\prime}.$

Now suppose $\lambda^{*}<0$ and let
$x^{*}\in\overline{\Sigma_{\lambda^{*}}^{\prime}}$ such that
$\mathrm{w}_{i}^{\lambda^{*}}(x^{*})=0$ for some $i=0,1,2$. If
$x^{\lambda^{*}}\in\Sigma_{\lambda^{*}}^{\prime}$ is an interior point then
the maximum principle implies $\mathrm{w}_{i}^{\lambda^{*}}\equiv 0$, which in
turn means $\mathrm{u}_{i}$ is symmetric about the hyperplane
$T_{\lambda^{*}}$. This is impossible because the singular set $\Gamma$ lies
to one side of $T_{\lambda^{*}}$. On the other hand, if $x^{*}\in
T_{\lambda^{*}}$ then by the Hopf boundary lemma (together with the fact that
$\mathrm{w}_{i}^{\lambda^{*}}$ may not be constant in
$\Sigma_{\lambda^{*}}^{\prime}$) we have

$0<\partial_{x_{1}}\mathrm{w}_{i}^{\lambda^{*}}(x^{*})=2\partial_{x_{1}}\mathrm{u}_{i}(x^{*}).$
(5.9)

However, the asymptotic expansions (5.2), (5.3) and (5.4) combined with
$\lambda^{*}<0$ tells us

$\mathrm{u}_{i}(x)-\mathrm{u}_{i}(x^{\lambda^{*}})\geq\delta_{3}\textrm{ for
}|x|>R_{2}\textrm{ and }x_{1}=\lambda^{*}$ (5.10)

for some positive numbers $\delta_{3}$ and $R_{2}$. Combining (5.9) and (5.10)
implies the inequality $\mathrm{w}_{i}^{\lambda}$ continues to hold for some
small value $\lambda<\lambda^{*}$, contradicting the definition of
$\lambda^{*}$. ∎

First, we prove the upper bound estimate. Our proof borrows from Pollack’s
proof of the corresponding upper bound in the scalar curvature case.

######  Proposition 5.2.

Let $u\in\mathcal{C}^{\infty}(\Omega)$ be a positive singular solution to
($\mathcal{Q}_{6,g_{0},N}$). There exists $C_{1}>0$ depending only on $n$ and
$d$ satisfying

$u(x)\leqslant C_{1}\mathrm{d}_{g_{0}}(x,\Lambda)^{-\gamma_{n}}.$

###### Proof.

Let $p_{0}\not\in\Lambda$, and $\rho>0$ such that
$\mathcal{B}_{\rho}(p_{0})\subset\Omega$, where $\mathcal{B}_{\rho}(p_{0})$ is
a geodesic ball with respect to the round metric. We define the auxiliary
function $\psi_{\rho}:\mathcal{B}_{\rho}(p_{0})\rightarrow\mathbb{R}$ given by

$\psi_{\rho}(x)=(\rho-\mathrm{d}_{g_{0}}(x,x_{0}))^{\gamma_{n}}u(x).$

Notice that choosing $\rho=\frac{1}{2}\mathrm{d}_{g_{0}}(x_{0},\Lambda)$, it
follows

$\psi_{\rho}(x_{0})=\rho^{\gamma_{n}}u(x_{0})=2^{-\gamma_{n}}\mathrm{d}_{g_{0}}(x_{0},\Lambda)^{\gamma_{n}}u(x_{0}).$

We claim that there exists $C>0$ depending only on $n$ such that
$\psi_{\rho}(x)\leqslant C$ for all admissible choices of $\lambda$, $u$,
$x_{0}$, and $\rho$. We suppose by contradiction that one can find sequences
$\\{\Lambda_{k}\\}_{k\in\mathbb{N}}$, $\\{u_{k}\\}_{k\in\mathbb{N}}$,
$\\{p_{0,k}\\}_{k\in\mathbb{N}}$, and $\\{\rho_{k}\\}_{k\in\mathbb{N}}$ of
admissible parameters satisfying

$M_{k}=\psi_{\rho}(p_{1,k})=\sup_{x\in\mathcal{B}_{\rho_{k}}(p_{0,k})}\psi_{\rho}(x)\rightarrow+\infty.$

Also, we observe
$\left.\psi_{\rho}\right|_{\partial\mathcal{B}_{\rho_{k}}(p_{0,k})}=0$, so
$p_{1,k}\in{\rm int}(\mathcal{B}_{\rho_{k}}(p_{0,k}))$. Next, by taking
$r_{k}=\rho_{k}-\mathrm{d}_{g_{0}}(p_{1,k},p_{0,k})$, and defining be geodesic
normal coordinates centered at $p_{1,k}$, denoted by $y$, we set

$\lambda_{k}=2u_{k}(p_{1,k})^{-\gamma_{n}},\quad
R_{k}={r_{k}}{\lambda_{k}^{-1}}=2^{-1}{r_{k}}(u_{k}(p_{1,k}))^{-\gamma_{n}}=2^{-1}M_{k}^{1/\gamma_{n}}.$

We now construct a blow-up sequence
$\\{w_{k}\\}_{k\in\mathbb{N}}\subset\mathcal{C}^{6,\alpha}(\mathbb{B}_{R_{k}})$
for some $\alpha\in(0,1)$ by
$w_{k}:\mathbb{B}_{R_{k}}(0)\rightarrow\mathbb{R}$ is such that

$w_{k}(y)=\lambda_{k}^{\gamma_{n}}u_{k}(\lambda y)\quad{\rm for\ all}\quad
k\in\mathbb{N}.$

Whence, using the conformal invariance in Remark 1.1, one can verify that the
function $w_{k}\in\mathcal{C}^{6,\alpha}(\mathbb{B}_{R_{k}})$ satisfies

$P^{6}_{\lambda g_{k}}w_{k}=c_{n}w_{k}^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathbb{B}_{R_{k}}.$

Moreover, by construction, one has

$2^{\gamma_{n}}=w_{k}(0)=\sup_{\mathbb{B}_{R_{k}}(0)}w_{k}(x)\quad{\rm for\
all}\quad k\in\mathbb{N},$

which, by Arzela–Ascoli theorem, means there exists subsequence that converges
uniformly on compacts.

In addition, it is not hard to check that the rescaled metrics $\lambda g_{0}$
converge to the classical Euclidean metric $\delta$ as $k\rightarrow\infty$.
Therefore, by taking the limit of the blow-up sequence, we obtain a positive
function $w_{\infty}\in\mathcal{C}^{6,\alpha}(\mathbb{R}^{n})$ satisfying
$w_{\infty}(0)=\sup w_{\infty}=2^{\gamma_{n}}$ and

$(-\Delta)^{3}w_{\infty}=c_{n}w_{\infty}^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathbb{R}^{n}.$

By the classification theorem in Theorem A (a), we must have

$w_{\infty}(x):={2}^{-\gamma_{n}}\left({1+|x|^{2}}\right)^{-\gamma_{n}}={2}^{-\gamma_{n}}u_{\rm
sph}(x).$

Thus each solution $u_{k}$ has a bubble for $k\gg 1$ sufficiently large. In
other terms, a small neighborhood of $p_{1,k}$ is close (in
$\mathcal{C}^{6,\alpha}$-norm) to the round metric, and hence has a concave
boundary, for $k\gg 1$ sufficiently large.

We verify this by computing the mean curvature of a geodesic sphere
explicitly. Using $g_{0}={4}{(1+|x|^{2})^{-2}}\delta$, a direct computation
shows the mean curvature of a hypersurface is given by
$H_{\Sigma}=-\operatorname{tr}_{g}\langle\nabla_{\partial\ell}\nu_{\Sigma},\partial_{m}\rangle$,
where $\nu_{\Sigma}$ is the unit inward normal vector of $\Sigma$.

A geodesic sphere centered at $p=0$ coincides with a Euclidean round sphere
centered at the origin (with a different radius), and so

$\nu_{\Sigma}=-\left(\frac{1+|x|^{2}}{2|x|}\right)x^{\ell}\partial_{x_{\ell}}.$

A straightforward computation yields

$H_{\Sigma}=-2n|x|(1+|x|^{2})+\frac{n-1+n|x|^{2}}{|x|},$

which is negative when $|x|>3$. Additionally, since

$\lim_{k\rightarrow\infty}\|w_{k}-w_{\infty}\|_{\mathcal{C}^{6,\alpha}(B_{{3R_{k}}/{4}}(0))}=0,$

it holds that $\partial B_{{3R_{k}}/{4}}(0)$ is also mean concave with respect
to the metric $\hat{g}_{k}\in{\rm Met}^{\infty}(B_{{3R_{k}}/{4}}(0))$ defined
as $\hat{g}_{k}=w_{k}^{{4}/{(n-6)}}\delta_{\ell m}$, which in turn implies
$\partial\mathbb{B}_{{3|p_{1,k}|}/{8}}(p_{1,k})$ is mean concave with respect
to the metric $g_{k}\in{\rm Met}^{\infty}(\Omega)$ given by
$\hat{g}_{k}=u_{k}^{{4}/{(n-6)}}\delta_{\ell m}$. This is contradiction with
Lemma 5.1, which proves the claim. ∎

Second, we obtain a lower bound estimate.

######  Proposition 5.3.

Let $u\in\mathcal{C}^{\infty}(\Omega)$ be a positive singular solution to
($\mathcal{Q}_{6,g_{0},N}$). There exists $C_{2}>0$ depending only on $u$
satisfying

$C_{2}\min_{j\in I_{N}}\mathrm{d}_{g_{0}}(x,p_{j})^{-\gamma_{n}}\leqslant
u(x).$

###### Proof.

Indeed, notice that by applying [17, Theorem 1.3] in cylindrical coordinates
$v=\mathfrak{F}(u)$, we obtain that $\mathcal{P}_{\rm cyl}(v)\leqslant 0$ with
equality if and only if

$\liminf_{t\rightarrow\infty}v(t,\theta)=\limsup_{t\rightarrow\infty}v(t,\theta)=\lim_{t\rightarrow\infty}v(t,\theta)=0.$

Otherwise, if $\mathcal{P}_{\rm cyl}(v)<0$, there exists $C_{2}>0$, which
depends on the solution $v$, such that $v(t,\theta)\geqslant C_{2}$. This
proves the proposition. ∎

Third, we have a version of Harnack inequality for our setting, which will be
important in the proof of our main result.

######  Proposition 5.4.

Let $\Omega\subset\mathbb{R}^{n}$ and $u\in\mathcal{C}^{\infty}(\Omega)$.
Assume that $-\Delta u\geqslant 0$, $\Delta^{2}u\geqslant 0$, and

$(-\Delta)^{3}u=f(u),$

where $f$ is either linear or superlinear and $f(0)=0$. Then, there exists
$\rho_{0}>0$ such that for $\rho\in(0,\rho_{0}]$ and $C_{3}>0$ depending only
on $\Omega$, $f$, and $\rho$, it holds

$\sup_{\mathcal{B}_{\rho}(0)}u\leqslant C_{3}\inf_{\mathcal{B}_{\rho}(0)}u.$

###### Proof.

The proof is a straightforward adaptation of [6, Theorem 3.6]. ∎

##  6\. Compactness result

In this section, we prove the main result of the manuscript.

Before proceeding to the proof, we need to obtain the existence of a positive
Green function for the sixth order GJMS of the round sphere with a prescribed
asymptotic rate near a pole given by the fundamental solution to the flat tri-
Laplacian.

######  Proposition 6.1.

Let $p\in\Lambda\subset\mathbb{(}\mathbb{S}^{n},g_{0})$ be a point on the
standard round sphere. There exists a Green function with pole at $p$, denoted
by $G_{p}:\mathbb{S}^{n}\setminus\\{p\\}\rightarrow(0,\infty)$, that satisfies

$P_{g_{0}}G_{p}=\delta_{p},$

where $P_{g_{0}}$ is the sixth order GJMS operator of the round metric given
by (1.2) and $\delta_{p}$ is the Dirac function concentrated at $p$.
Furthermore, there exists $C_{n}>0$ depending only on $n$ such that

$G_{p}(x)=C_{n}\mathrm{d}_{g_{0}}(x,p)^{6-n}+\mathcal{O}(1)$ (6.1)

in conformal normal coordinates.

###### Proof.

This is a direct application of [10, Proposition 2.1] for the standard round
sphere $(\mathbb{S}^{n},g_{0})$. ∎

###### Proof of Theorem 1.3.

Let
$\\{g_{k}\\}_{k\in\mathbb{N}}=\\{(U_{k})^{{4}/{n-6}}g_{0}\\}\subset\mathcal{M}_{N}^{6}$
be a sequence of admissible metrics, each of which is a complete, conformally
flat metric on $\Omega_{k}=\mathbb{S}^{n}\backslash\Lambda_{k}$ with
$Q^{6}_{g_{k}}\equiv Q_{n}=\frac{n(n^{4}-20n^{2}+64)}{32}$. We denote the
punctures of $g_{k}$ by

$\Lambda_{k}:={\rm
sing}(U_{k})=\\{p_{1,k},\dots,p_{N,k}\\}\subset\mathbb{S}^{n}.$

The proof will be divided into a sequence of steps.

The first step will simplify our later analysis since it allows us to assume
the singular points are fixed.

Step 1. After passing to a subsequence, we may assume that for $k\gg 1$
sufficiently large each $U_{k}$ is non-singular on the set
$K_{1}:=\mathbb{S}^{n}\setminus(\cup_{i=1}^{N}\mathcal{B}_{\delta_{1}/2}(p_{j,i}))$.

Indeed, for $0<\delta_{1}$ small enough, the set

$(\mathbb{S}^{n})^{N}\setminus\left\\{(q_{1},\dots,q_{k})\in(\mathbb{S}^{n})^{N}:\mathrm{d}_{g_{0}}(q_{j},q_{\ell})\geqslant\delta_{1}\textrm{
for each }j\neq\ell\right\\}$

is compact and contains each singular set $\Lambda_{k}$ for all
$k\in\mathbb{N}$. Thus, there exits
$\\{p_{1,\infty},\dots,p_{N,\infty}\\}\subset\mathbb{S}^{n}$, and a convergent
subsequence such that $p_{j,k}\rightarrow p_{j,\infty}$ as
$k\rightarrow+\infty$, proving Step 1.

To set notation, we define the compact sets

$K_{\ell}:=\mathbb{S}^{n}\setminus\left(\cup_{j=1}^{N}\mathcal{B}_{2^{-\ell}\delta_{1}}(p_{j,\infty})\right)\quad{\rm
for\ each}\quad\ell\in\mathbb{N}$

Notice that by construction the family $\\{K_{\ell}\\}_{\ell\in\mathbb{N}}$ is
a compact exhaustion of the limit singular set

$\Omega_{\infty}:=\mathbb{S}^{n}\setminus\Lambda_{\infty},\quad{\rm
where}\quad\Lambda_{\infty}:=\\{p_{1,\infty}.\dots,p_{k,\infty}\\}.$

Furthermore, by the convergence $p_{j,k}\rightarrow p_{j,\infty}$ as
$k\rightarrow+\infty$, for each fixed $\ell\in\mathbb{N}$ there exists
$k_{0}\gg 1$ such that $k\geqslant k_{0}$ implies $U_{k}$ is smooth in
$K_{\ell}$.

The second step is based on the uniform upper bound and states that we can
extract a limit.

Step 2. The exists $U_{\infty}\in\mathcal{C}^{\infty}(\Omega_{\infty})$
solving ($\mathcal{Q}_{6,g_{0},N}$) such that

$\lim_{k\rightarrow+\infty}\|U_{\infty}-U_{k}\|_{\mathcal{C}_{\rm
loc}^{\infty}(\Omega_{\infty})}=0.$ (6.2)

In fact, using the upper bound in Proposition 5.2, one has that for each
compact subset $K\subset\Omega_{\infty}$, there exists $\alpha\in(0,1)$ and
$C_{1}>0$ depending only on $n$, $\Omega$, and $\alpha$ such that

$\|U_{k}\|_{\mathcal{C}{6,\alpha}(K)}\leqslant C_{1}\quad{\rm for\ all}\quad
k\in\mathbb{N}.$

Therefore, as a consequence of the Arzela–Ascoli theorem, one can find a limit
$U_{\infty}\in\mathcal{C}^{6,\alpha}(K)$ a convergent subsequence, which we
again denote the same, such that

$\lim_{k\rightarrow+\infty}\|U_{\infty}-U_{k}\|_{\mathcal{C}_{\rm
loc}^{6,\alpha}(\Omega_{\infty})}=0.$

Furthermore, by applying standard elliptic regularity, we directly obtain that
(6.2) holds, and so Step 2 is proved.

The next step is to show that this limit is non-trivial.

Step 3. $U_{\infty}>0$ on $\Omega_{\infty}$.

If this step were false, there would exist $p_{*}\in\Omega_{\infty}$ such that

$0=U_{\infty}(p_{*})=\lim_{k\rightarrow+\infty}U_{k}(p_{*}).$

For each $k\in\mathbb{N}$, we define $\varepsilon_{k}=U_{k}(p_{*})$ and the
rescaled function $\widehat{U}_{k}\in\mathcal{C}^{\infty}(\Omega_{k})$ given
by

$\widehat{U}_{k}(x)={\varepsilon_{k}}^{-1}U_{k}(x)\quad{\rm for\ all}\quad
k\in\mathbb{N}.$

As a consequence of Remark 1.1, it follows

$P_{g_{0}}\widehat{U}_{k}=\varepsilon_{k}^{\frac{12}{n-6}}c_{n}\widehat{U}_{k}^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\Omega_{k}\quad{\rm for\ all}\quad k\in\mathbb{N}.$

In addition, by construction, the sequence
$\\{\widehat{U}_{k}\\}_{k\in\mathbb{N}}$ satisfy the normalization

$\widehat{U}_{k}(p_{*})=1\quad{\rm for\ all}\quad k\in\mathbb{N}.$ (6.3)

By the Harnack inequality of Lemma 5.4 there exists a positive constant
$C_{1}$ depending only on $n$ and $\ell$ such that

$\sup_{K_{\ell}}|u_{\rm sph}\widehat{U}_{k}|\leqslant C_{1}.$ (6.4)

However, there is another positive constant $C_{2}$, again depending only on
$n$ and $\ell$, such that

$C_{2}\leqslant u_{\rm sph}\leqslant 2^{\gamma_{n}}.$ (6.5)

Combining (6.4) and (6.5) there exists a uniform constant $C_{3}$ such that

$\sup_{K_{\ell}}\widehat{U}_{k}\leq C_{3},$

and so by the Arzela-Ascoli theorem we may pass to a subsequence
$\widehat{U}_{k}$ that converges uniformly on compact subsets of
$\Omega_{\infty}$ to a smooth function $\widehat{U}_{\infty}$.

This limit function
$\widehat{U}_{\infty}:\Omega_{\infty}\rightarrow\mathbb{R}$ satisfies

$P_{g_{0}}\widehat{U}_{\infty}=0\quad{\rm in}\quad\Omega_{\infty}$

and so it has the form

$\widehat{U}_{\infty}=\sum_{j=1}^{N}\beta_{j}G_{p_{j,\infty}}$

for some collection of real numbers $\beta_{1},\dots,\beta_{N}$. The
normalization (6.3) implies one of the coefficients $\beta_{j_{0}}$ is
positive, so after possibly relabeling the punctures we may assume
$\beta_{1}>0$.

We now choose a stereographic projection sending $p_{1,\infty}$ to the origin
and perform the Emden-Fowler change of coordinates in Definition 2.1, which
yield the functions

$v_{k}:=\mathfrak{F}(u_{\rm sph}U_{k})\quad{\rm
and}\quad\widehat{v}_{k}:=\mathfrak{F}(u_{\rm sph}\widehat{U}_{k})$

and their respective limits

$v_{\infty}:=\mathfrak{F}(u_{\rm sph}U_{\infty})\quad{\rm
and}\quad\widehat{v}_{\infty}:=\mathfrak{F}(u_{\rm sph}\widehat{U}_{\infty}).$

The expansion (6.1) implies

$\displaystyle\widehat{v}_{\infty}(t,\theta)=e^{-\gamma_{n}t}(\cosh
t)^{-\gamma_{n}}(C_{n}e^{-\gamma_{n}t}+\mathcal{O}(1))=C_{n}+\mathcal{O}(e^{(6-n)t})\quad{\rm
as}\quad t\rightarrow+\infty.$ (6.6)

Also, observe that $\widehat{v}_{k}\in\mathcal{C}^{6}(\mathcal{C}_{T})$
satisfies the PDE

$P^{6}_{\rm
cyl}\widehat{v}_{k}=\varepsilon_{k}^{\frac{12}{n-6}}c_{n}\widehat{v}_{k}^{\frac{n+6}{n-6}}\quad{\rm
in}\quad\mathcal{C}_{T_{k}}\quad{\rm for\ all}\quad k\in\mathbb{N},$

which we combine with (6.6) and Proposition 4.6 and the convergence
$\widehat{v}_{k}\rightarrow\widehat{v}_{\infty}$ to see that for $t$
sufficiently large

$\displaystyle\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
cyl}^{\varepsilon_{k}^{\frac{12}{n-6}}c_{n}}\mathrm{d}\theta$ $\displaystyle=$
$\displaystyle\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
rad}(\widehat{v}_{k})+\mathcal{H}_{\rm
ang}(\widehat{v}_{k})+\frac{n-6}{2n}\varepsilon^{\frac{12}{n-6}}c_{n}|\widehat{v}_{k}|^{\frac{2n}{n-6}}\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle-\widetilde{C}_{n}\beta_{1}^{2}+\mathcal{O}(e^{(6-n)t}).$

for some $\tilde{C}_{n}>0$. On the other hand, by our construction we have

$\displaystyle\mathcal{P}_{\rm cyl}(v_{k})$ $\displaystyle=$
$\displaystyle\int_{\\{t\\}\times\mathbb{S}^{n-1}}\mathcal{H}_{\rm
rad}(v_{k})+\mathcal{H}_{\rm ang}(v_{k})+F(v_{k})\mathrm{d}\theta$
$\displaystyle=$
$\displaystyle\int_{\\{t\\}\times\mathbb{S}^{n-1}}\varepsilon_{k}^{2}\left(\mathcal{H}_{\rm
rad}(\widehat{v}_{k})+\mathcal{H}_{\rm
ang}(\widehat{v}_{k})\right)+\varepsilon_{k}^{\frac{2n}{n-6}}F(\widehat{v}_{k})\mathrm{d}\theta\rightarrow
0$

From (6) and (6), we find

$\lim_{k\rightarrow+\infty}\mathcal{P}_{\rm rad}(g_{k},p_{1,k})=0,$

which, together with Proposition 4.4, implies
$\lim_{k\rightarrow+\infty}\varepsilon_{1}(g_{k})=0.$ This contradicts the
hypothesis that the necksizes are bounded away from zero, that is,
$\varepsilon_{j}(g_{k})>\delta_{1}$ for some $0<\delta_{1}\ll 1$.

At last, we can complete our argument

Step 4. The metric $g_{\infty}=U_{\infty}^{\frac{4}{n-6}}g_{0}$ is a complete
metric on $\Omega_{\infty}$.

Indeed, suppose by contradiction that is $g_{\infty}$ is incomplete. Then
there exists an index $j\in\\{1,\dots,N\\}$ such that $\liminf_{x\rightarrow
p_{j,\infty}}{U}_{\infty}(x)<\infty$. In this case, the removable singularity
result in Proposition 5.3 implies

$\mathcal{P}_{\rm rad}(g_{\infty},p_{j,\infty})=0.$

However, by construction

$0=\mathcal{P}_{\textrm{rad}}(g_{\infty},p_{j,\infty})=\lim_{k\rightarrow+\infty}\mathcal{P}_{\rm
rad}(g_{k},p_{j,k})\geqslant\delta_{2},$

which, by Proposition 4.4 implies $\varepsilon_{j}(g_{k})\geqslant\delta_{2}$,
which is contradiction with the fact
$g_{k}\in\mathcal{Q}^{6}_{\delta_{1},\delta_{2}}$.

By putting all these steps together, our main theorem is proved. ∎

##  Appendix A Higher order curvature tensors

Let $(M^{n},g)$ is a Riemannian manifold with $n\geqslant 2$. In what follows,
we will always be using Einstein’s summation convection. In a local coordinate
frame, denoted by $\\{\partial_{i}\\}_{i=1}^{n}$, we let ${\rm
Rm}_{g}\in\mathfrak{T}^{3}_{1}(M)$ be the Riemannian curvature tensor,
$\accentset{\circ}{\rm Rm}_{g}\in\mathfrak{T}^{4}_{0}(M)$ be covariant Riemann
curvature tensor, and the Ricci curvature tensor ${\rm Ric}_{g}={\rm
tr}_{g}\accentset{\circ}{\rm Rm}_{g}\in\mathfrak{T}^{2}_{0}(M)$, which can be
expressed as ${\rm Ric}_{jk}=\accentset{\circ}{\rm
Rm}_{ijk}^{i}=g^{i\ell}\accentset{\circ}{\rm Rm}_{ijk\ell}$. We also consider
the scalar curvature $R_{g}={\rm tr}_{g}{\rm
Ric}_{g}\in\mathfrak{T}^{0}_{0}(M)$, defined by $R=g^{ij}{\rm Ric}_{ij},$
where $\mathfrak{T}^{r}_{s}(M)$ stands for the set of $(r,s)$-type tensor over
$M$ with $\mathfrak{T}^{0}_{0}(M)=\mathcal{C}^{\infty}(M)$ and ${\rm
tr}_{g}:\mathfrak{T}^{r}_{s}(M)\rightarrow\mathfrak{T}^{r-2}_{s}(M)$. Also for
the Laplace–Beltrami operator, we simply denote
$\Delta_{g}:=g^{ij}\nabla_{i}\nabla_{j}$, where $\nabla_{g}$ the Levi–Civita
connection associated to $g$.

It is also convenient to define some operations involving two tensors.

######  Definition A.1.

First, let us introduce the cross product $\times:{\rm Sym}_{2}(M)\times{\rm
Sym}_{2}(M)\rightarrow{\rm Sym}_{2}(M)$ is given by

$(h_{1}\times
h_{2})_{ij}:=g^{k\ell}h_{1,ik}h_{2,j\ell}=h_{1,i}^{\ell}h_{2,\ell j}.$

Second, we define a dot product $\times:{\rm Sym}_{2}(M)\times{\rm
Sym}_{2}(M)\rightarrow\mathbb{R}$, given by

$h_{1}\cdot h_{2}:={\rm tr}_{g}(h_{1}\times
h_{2})=g^{ij}g^{k\ell}h_{1}^{ik}h_{2,j\ell}=h_{1}^{jk}h_{2,jk}.$

Third, we also recall the Kulkarni–Nomizu product
$\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}:{\rm
Sym}_{2}(M)\times{\rm Sym}_{2}(M)\rightarrow\mathfrak{T}^{4}_{0}(M)$

$(h_{1}\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}h_{2})_{ijk\ell}:=h_{1,i\ell}h_{2,jk}+h_{1,jk}h_{2,i\ell}-h_{1,ik}h_{2,j\ell}-h_{1,j\ell}h_{2,ik}.$

At last, we consider $\cdot:{\rm Sym}_{2}(M)\rightarrow{\rm Sym}_{2}(M)$ and
$\delta_{g}:{\rm Sym}_{2}(M)\rightarrow\mathbb{R}$,

$(\accentset{\circ}{\rm Rm}\cdot
h)_{jk}:=R_{ijk\ell}h^{i\ell}\quad\mbox{and}\quad\left(\delta_{g}h\right)_{i}:=-\left(\operatorname{div}_{g}h\right)_{i}=-\nabla^{j}h_{ij},$

where the latter one is the $L^{2}$-formal adjoint of Lie derivative (up to
scalar multiple).

######  Definition A.2.

Let us define the Schouten tensor, Weyl tensor, Bach tensor, and nameless
tensor, respectively, by

$\displaystyle A_{g}$
$\displaystyle:=\frac{1}{n-2}\left(\operatorname{Ric}_{g}-\frac{1}{2(n-1)}R_{g}g\right)$
$\displaystyle W_{g}$ $\displaystyle:=\accentset{\circ}{\rm
Rm}_{g}-A_{g}\mathchoice{\mathbin{\ooalign{$\displaystyle\bigcirc$\cr$\displaystyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\textstyle\bigcirc$\cr$\textstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptstyle\bigcirc$\cr$\scriptstyle\wedge$\cr}}{}}{\mathbin{\ooalign{$\scriptscriptstyle\bigcirc$\cr$\scriptscriptstyle\wedge$\cr}}{}}g$
$\displaystyle B_{g}$
$\displaystyle:=\Delta_{g}A_{g}-\nabla_{g}^{2}\operatorname{tr}_{g}A_{g}+2\accentset{\circ}{\rm
Rm}_{g}\cdot A_{g}-(n-4)A_{g}\times
A_{g}-|A_{g}|^{2}g-2(\operatorname{tr}_{g}A_{g})A_{g},A$

where these expressions are written in an abstract index-free manner.

From this, we introduce the following tensors

$\displaystyle T^{2}_{g}$ $\displaystyle:=(n-2)\sigma_{1}(A_{g})g-8A_{g},$
$\displaystyle T^{4}_{g}$
$\displaystyle:=-\frac{3n^{2}-12n-4}{4}\sigma_{1}(A_{g})^{2}g+4(n-4)|A|_{g}^{2}g+8(n-2)\sigma_{1}(A_{g})A_{g}$
$\displaystyle+(n-6)\Delta_{g}\sigma_{1}(A_{g})g+48A_{g}^{2}-\frac{16}{n-4}B_{g},$
$\displaystyle T^{6}_{g}$
$\displaystyle:=-\frac{1}{8}\sigma_{3}(A_{g})-\frac{1}{24(n-4)}\langle
B_{g},A_{g}\rangle_{g},$

where $\sigma_{k}$ is the $k$-th elementary symmetric function for each
$k\in\mathbb{N}$.

Based on this notation, we introduce the concept of higher order curvatures as
follows

######  Definition A.3.

For any $g\in{\rm Met}^{\infty}(\Omega)$, let us define the $N$th order
$Q$-curvature $Q_{g}^{N}$ for $N=2,4,6$, respectively, by

$\displaystyle Q^{2}_{g}$ $\displaystyle:=R_{g}$ $\displaystyle Q^{4}_{g}$
$\displaystyle:=-\frac{1}{2(n-1)}\Delta
R_{g}-\frac{2}{(n-2)^{2}}|\operatorname{Ric}_{g}|^{2}+\frac{n^{3}-4n^{2}+16n-16}{8(n-1)^{2}(n-2)^{2}}R_{g}^{2},$
$\displaystyle Q_{g}^{6}$
$\displaystyle:=-3!2^{6}T^{6}_{g}-\frac{n+2}{2}\Delta_{g}(\sigma_{1}\left(A_{g}\right)^{2})+4\Delta_{g}|A|_{g}^{2}-8\delta\left(A_{g}\mathrm{d}\sigma_{1}\left(A_{g}\right)\right)+\Delta_{g}^{2}\sigma_{1}\left(A_{g}\right)$
$\displaystyle-\frac{n-6}{2}\sigma_{1}\left(A_{g}\right)\Delta_{g}\sigma_{1}\left(A_{g}\right)-4(n-6)\sigma_{1}\left(A_{g}\right)|A|_{g}^{2}+\frac{(n-6)(n+6)}{4}\sigma_{1}\left(A_{g}\right)^{3}.$

Associated with these curvatures, we have the following conformally invariant
operators

######  Definition A.4.

For any $g\in{\rm Met}^{\infty}(\Omega)$, let us define the $N$th order GJMS
operator $P_{g}^{N}$ for $N=2,4,6$, respectively, by

$\displaystyle P^{2}_{g}:=-\Delta_{g}+\frac{n-2}{2}R_{g}$ $\displaystyle
P^{4}_{g}:=\Delta_{g}^{2}-\mathrm{div}\left(\frac{(n-2)^{2}+4}{2(n-1)(n-2)}R_{g}g-\frac{4}{n-2}\operatorname{Ric}_{g}\right)\mathrm{d}+\frac{n-4}{2}Q^{4}_{g}$
$\displaystyle P_{g}^{6}:=-\Delta_{g}^{3}-\Delta_{g}\delta
T_{2}\mathrm{d}-\delta
T_{2}\mathrm{d}\Delta_{g}-\frac{n-2}{2}\Delta_{g}\left(\sigma_{1}\left(A_{g}\right)\Delta_{g}\right)-\delta
T_{4}\mathrm{d}+\frac{n-6}{2}Q_{g}^{6}.$

When $N=2$, the operator $P_{g}^{2}=L_{g}$ is the so-called conformal
Laplacian.

##  Appendix B Modica estimates

In this appendix, we discuss possible pointwise estimates for positive smooth
solutions to ($\mathcal{P}_{6,\infty}$). These estimates have strong geometric
implications in terms of the associated conformally flat metric.

In [16, Theorem 1.4], it is proved that positive smooth solutions to

$\Delta^{2}u=\frac{n(n-4)(n^{2}-4)}{16}u^{\frac{n+4}{n-4}}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\\{0\\}.$

satisfies the following pointwise inequality

$-\Delta u-\frac{4}{n-2}\frac{|\nabla
u|^{2}}{u}\geqslant\sqrt{\frac{n-4}{n}}u^{\frac{n}{n-4}}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\\{0\\}.$

This implies in particular that the scalar curvature $Q_{g}^{2}$ of the
conformally flat metric $g=u^{4/(n-4)}\delta$ is positive. This type of result
is known in the literature as Modica-type estimates.

In our situation, we start by writing the metric $g\in[g_{0}]$ as
$g=(u^{\frac{n-2}{n-6}})^{\frac{4}{n-2}}\delta$, we see

$Q_{g}^{2}=-\frac{4(n-1)}{n-2}u^{\frac{-(n+2)}{n-6}}\Delta\left(u^{\frac{n-2}{n-6}}\right)=-\frac{4(n-1)}{n-6}u^{-\frac{n-2}{n-6}}\left(\Delta
u+\frac{4}{n-6}\frac{|\nabla u|^{2}}{u}\right).$ (B.1)

and

$-\Delta\left(u^{\frac{n-2}{n-6}}\right)=-\Delta u-\frac{4}{n-6}\frac{|\nabla
u|^{2}}{u}.$

From this, we conclude that $Q_{g}^{2}\geqslant 0$ implies $-\Delta u\geqslant
0$, and in fact is a stronger condition. Similarly, writing
$g=(u^{\frac{n-4}{n-6}})^{\frac{4}{n-4}}\delta$, it follows

$Q_{g}^{4}=\frac{2}{n-4}u^{-\frac{n+4}{n-6}}\Delta^{2}\left(u^{\frac{n-4}{n-6}}\right).$
(B.2)

Furthermore, a long computation shows

$\displaystyle(-\Delta)^{2}\left(u^{\frac{n-4}{n-6}}\right)$ $\displaystyle=$
$\displaystyle\frac{n-4}{n-6}u^{\frac{2}{n-6}}\Delta^{2}u+\frac{8(n-4)}{(n-6)^{2}}u^{\frac{8-n}{n-6}}\langle\nabla
u,\nabla\Delta u\rangle$
$\displaystyle+\frac{4(n-4)}{(n-6)^{2}}u^{\frac{8-n}{n-6}}|D^{2}u|^{2}+\frac{8(n-4)(8-n)}{(n-6)^{3}}u^{\frac{-2(n-7)}{n-6}}D^{2}u(\nabla
u,\nabla u)$
$\displaystyle+\frac{4(n-4)(8-n)}{(n-6)^{3}}u^{\frac{-2(n-7)}{n-6}}|\nabla
u|^{2}\Delta u+\frac{2(n-7)(n-8)}{(n-6)^{4}}u^{\frac{20-3n}{n-6}}|\nabla
u|^{4},$

where

$|D^{2}u|^{2}=\sum_{i,j=1}^{n}u_{x_{i}x_{j}}^{2}\quad{\rm and}\quad
D^{2}u(\nabla u,\nabla u)=\sum_{i,j=1}^{n}u_{x_{i}x_{j}}u_{x_{i}}u_{x_{j}}.$

Hence, the conditions $Q_{g}^{2}\geqslant 0$ and $Q_{g}^{4}\geqslant 0$ are
not enough to guarantee that $\Delta^{2}u\geqslant 0$ directly.

Based on this, it is natural to ask whether the following result holds.

######  Conjecture B.1.

Let $u\in\mathcal{C}^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ be a positive
solutions to ($\mathcal{P}_{6,\infty}$). Then, the conformally flat metric
given by $g=u^{{4}/{(n-6)}}\delta$ satisfies the following pointwise estimate

$Q_{2}(u)\geqslant\sqrt{\frac{n-6}{n}}u^{\frac{n}{n-6}}\quad{\rm and}\quad
Q_{4}(u)\geqslant\sqrt{\frac{n-6}{n}}u^{\frac{n}{n-6}}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\\{0\\}$

where

$Q_{2}(u):=-\Delta u-\frac{4}{n-6}\frac{|\nabla u|^{2}}{u}.$

and

$\displaystyle Q_{4}(u)$
$\displaystyle:=\Delta^{2}u-\frac{8}{(n-6)}u^{\frac{8-n}{2}}\langle\nabla
u,\nabla\Delta
u\rangle-\frac{4}{(n-6)}u^{\frac{8-n}{2}}|D^{2}u|^{2}-\frac{8(8-n)}{(n-6)^{2}}u^{{7-n}}D^{2}u(\nabla
u,\nabla u)$ $\displaystyle-\frac{4(8-n)}{(n-6)^{2}}u^{{7-n}}|\nabla
u|^{2}\Delta u-\frac{2(n-7)(n-8)}{(n-6)^{3}(n-4)}u^{\frac{20-3n}{2}}|\nabla
u|^{4}.$

In particular, it follows that the curvatures $Q^{2}_{g}$ and $Q^{4}_{g}$
associated with the conformally flat metric $g=u^{4/(n-6)}\delta$ are both
positive.

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