# Nonuniqueness results for constant sixth order $Q$-curvature metrics on
spheres with higher dimensional singularities

João Henrique Andrade  ,  Paolo Piccione  and  Juncheng Wei  Department of
Mathematics, University of British Columbia
V6T 1Z2, Vancouver, Canada
and
Institute of Mathematics and Statistics, University of São Paulo
05508-090, São Paulo, Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>Institute of Mathematics and Statistics, University of São Paulo
05508-090, São Paulo, Brazil<EMAIL_ADDRESS>Department of Mathematics,
University of British Columbia
V6T 1Z2, Vancouver-BC, Canada<EMAIL_ADDRESS>
###### Abstract.

We prove nonuniqueness results for constant sixth order $Q$-metrics on
complete locally conformally flat $n$-dimensional Riemannian manifolds with
$n\geqslant 7$. More precisely, assuming a positive Green function exists for
the sixth order GJMS operator, our objective is two-fold. First, we use a
classical bifurcation technique to prove that there exists infinitely many
constant $Q$-curvature metrics on $\mathbb{S}^{1}\times\mathbb{S}^{n-1}$. As a
by-product, we find the sixth order Yamabe invariant on this product manifold
can be arbitrarily close to that of the round dimensional sphere, generalizing
a result of Schoen about the classical Yamabe invariant. Second, when the
underlying manifold is noncompact, we apply a bifurcation technique on
Riemannian covering to construct infinitely many complete metrics with
constant sixth order $Q$-curvature conformal to
$\mathbb{S}^{n_{1}}\times\mathbb{R}^{n_{2}}$ or
$\mathbb{S}^{n_{1}}\times\mathbb{H}^{n_{2}}$, where $n_{1}+n_{2}\geqslant 7$.
Consequently, we obtain infinitely many solutions to the singular constant
GJMS equation on round spheres $\mathbb{S}^{n}\setminus\mathbb{S}^{k}$ blowing
up along a minimal equatorial subsphere with $0\leqslant k<\frac{n-6}{2}$;
this dimension restriction is sharp in the topological sense.

###### Key words and phrases:

Bifurcation theory, Nonlinear ODEs, Sixth order equations, Prescribed
$Q$-curvature, Critical PDEs

###### 2020 Mathematics Subject Classification:

35J60, 35B09, 35J30, 35B40, 53C18, 34C23, 58J55

This work was partially supported by Fundação de Amparo à Pesquisa do Estado
de São Paulo (FAPESP), Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), and Natural Sciences and Engineering Research Council of
Canada (NSERC). J.H.A. was supported by FAPESP #2020/07566-3 and
#2021/15139-0. P.P. was supported by FAPESP #2016/23746-6 and CNPq
#313773/2021-1. J.W. was supported by Natural Sciences and Engineering
Research Council of Canada (NSERC)

###### Contents

  1. 1 Introduction
  2. 2 Notation
  3. 3 Toy model
    1. 3.1 Variational structure
    2. 3.2 Cylindrical coordinates
    3. 3.3 Delaunay metrics
    4. 3.4 Hamiltonian energy
    5. 3.5 Linearized operator
    6. 3.6 Fourier eigenmodes
    7. 3.7 Jacobi fields
    8. 3.8 Bifurcation via Morse index
  4. 4 The noncompact case
    1. 4.1 Riemannian coverings
    2. 4.2 Bifurcation via infinite tower of Riemannian coverings

##  1\. Introduction

In recent years, active research has been made on analogs of the Yamabe
problem and its singular counterpart. In each of these problems, one seeks a
representative of a conformal class with prescribed constant curvature of some
type. In the classical case, the scalar curvature functional is naturally the
first one to be studied. In more modern examples, fully nonlinear
$\sigma_{\ell}$-curvatures with $\ell\in\mathbb{N}$ and fractional higher
order $Q_{2\sigma}$-curvatures with $\sigma\in\mathbb{R}_{+}$ are also
considered.

In this manuscript, we focus on the integer case $\sigma=m\in\mathbb{N}$, more
precisely $m=3$. Let $(M^{n},g)$ with $n\geqslant 7$ be $n$-dimensional
Riemannian manifold; $g$ is called the background metric or (ambient metric),
which will be fixed throughout the paper. In consonance with the plethora of
results for the Yamabe problem, the scope of nonuniqueness in the constant
sixth order $Q$-curvature problem ought to be much more prosperous, both
geometrically and topologically. Our main purpose here is to exhibit an
extensive nonuniqueness phenomena on a broad class of Riemannian manifolds.

We set $A_{g},W_{g},B_{g}$ to be respectively the Schouten tensor, Weyl
tensor, Bach tensor given by

$\displaystyle A_{g}$
$\displaystyle:=\frac{1}{n-2}\left(\operatorname{Ric}_{g}-\frac{1}{2(n-1)}R_{g}g\right)$
(1.1) $\displaystyle W_{g}$ $\displaystyle:=\accentset{\circ}{\rm
Rm}_{g}-A_{g}\mathchoice{\mathbin{\ooalign{$\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$\cr}}{}}g$
(1.10) $\displaystyle B_{g}$
$\displaystyle:=\Delta_{g}A_{g}-\nabla_{g}^{2}\operatorname{tr}_{g}A_{g}+2\accentset{\circ}{\rm
Rm}_{g}\cdot A_{g}-(n-4)A_{g}\times
A_{g}-\|A_{g}\|^{2}g-2(\operatorname{tr}_{g}A_{g})A_{g}$ (1.11)

where these expressions are written in an abstract index-free manner (see
Section 2).

We define the concept of higher order curvatures as follows

$\displaystyle Q_{2}(g)$ $\displaystyle:=\frac{n-2}{4(n-1)}R(g)$
$\displaystyle Q_{4}(g)$
$\displaystyle:=-\Delta_{g}\sigma_{1}(A_{g})+4\sigma_{2}(A_{g})-\frac{n-4}{2}\sigma_{1}(A_{g})^{2}$
(1.12) $\displaystyle Q_{6}(g)$
$\displaystyle:=-3!2^{6}v_{6}(g)-\frac{n+2}{2}\Delta_{g}(\sigma_{1}(A_{g})^{2})+4\Delta_{g}\|A_{g}\|^{2}-8\delta_{g}(A_{g}\\#\mathrm{d}\sigma_{1}(A_{g}))+\Delta_{g}^{2}\sigma_{1}(A_{g})$
$\displaystyle-\frac{n-6}{2}\sigma_{1}(A_{g})\Delta_{g}\sigma_{1}(A_{g})-4(n-6)\sigma_{1}(A_{g})\|A_{g}\|^{2}+\frac{(n-6)(n+6)}{4}\sigma_{1}(A_{g})^{3}.$

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

$\displaystyle P_{2}(g):=-\Delta_{g}+\frac{n-2}{2}R(g)$ $\displaystyle
P_{4}(g):=\Delta_{g}^{2}-\delta_{g}T_{2}(g)\\#\mathrm{d}+\frac{n-4}{2}Q_{4}(g)$
(1.13) $\displaystyle
P_{6}(g):=-\Delta_{g}^{3}-\Delta_{g}\delta_{g}T_{2}(g)\\#\mathrm{d}-\delta_{g}T_{2}(g)\\#\mathrm{d}\Delta_{g}-\frac{n-2}{2}\Delta_{g}\left(\sigma_{1}(A_{g})\Delta_{g}\right)-\delta_{g}T_{4}(g)\\#\mathrm{d}+\frac{n-6}{2}Q_{6}(g),$

where

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

with $\sigma_{\ell}$ is the $\ell$-th elementary symmetric function for each
$\ell\in\mathbb{N}$. Notice that $P_{2}(g)=L(g)$ is the so-called conformal
Laplacian, and $P_{6}(g),P_{6}(g)$ are their conformally invariant powers.

This class of higher order operators on manifolds was first introduced in the
works of Graham, Jenne, Manson, and Sparling [25], which is construction is
based on earlier constructions of conformally invariant fourth order operators
by the works of Paneitz [42] and Branson [13], that is, for $m=2$. Given
$m\in\mathbb{N}$ and a compact $n$-dimensional Riemannian manifold $(M^{n},g)$
with $n>2m$, the so-called GJMS operators are conformally covariant
differential operators, denoted by $P_{2m}(g)$, satisfying that the leading
order term of $P_{2m}(g)$ is $(-\Delta_{g})^{m}$. The construction of these
operators are based on the seminal work of Fefferman and Graham [21] on
ambient metrics for any $m\in\mathbb{N}$. From this point of view, one can
then construct the associated $Q$-curvature of order $2m$ by
$Q_{2m}(g)=P_{2m}(g)(1)$. When $m=1$, one recovers the conformal Laplacian

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

where $\Delta_{g}$ is the Laplace–Beltrami operator of $g$ and $R_{g}$ is its
scalar curvature. For the standard round sphere, Grahan and Zworski [26] and
Chang and González [17] extended these definitions to obtain (nonlocal)
operators $P_{2\sigma}(g)$ of any order $\sigma\in\mathbb{R}_{+}$ with
$n>2\sigma$ as well as the corresponding $\sigma$-fractional order
$Q$-curvatures. Once again, the leading order part of $P_{2\sigma}(g)$ is
$(-\Delta_{g})^{\sigma}$, understood as the principal value of a singular
integral operator. Although we have explicit formulas for $P_{2}(g)$,
$P_{4}(g)$ and $P_{6}(g)$, the expressions for $P_{2\sigma}(g)$ and
$Q_{2\sigma}(g)$ for a general $\sigma\in\mathbb{R}_{+}$ are far more
complicated (see, for instance [19]). For more details, we refer the
interested reader to [33, 18, 15, 22].

In general, the higher order $Q_{2m}$-curvatures defined in (1) for $m=1,2,3$
transform nicely under a conformal change, namely for any conformal metric
$\bar{g}\in[g]$, one has

$Q_{2m}({\bar{g}})=\frac{2}{n-2m}u^{-\frac{n+2m}{n-2m}}P_{2m}(g)u,$ (1.14)

where $P_{2m}(g):\mathcal{C}^{\infty}(M)\rightarrow\mathcal{C}^{\infty}(M)$ is
the GJMS operator defined in (1) and

$[g]:=\left\\{\bar{g}\in{\rm Met}^{\infty}(M):\bar{g}=u^{{4}/{(n-2m)}}g\ {\rm
for\ some}\ u\in\mathcal{C}_{+}^{\infty}(M)\right\\}$

is the conformal class of $g$, where $u\in\mathcal{C}_{+}^{\infty}(M)$ if and
only if $u\in\mathcal{C}^{\infty}(M)$ and $u>0$. This operator is known to
satisfy the transformation law below

$P_{2m}(g)(\phi)=u^{-\frac{n+2m}{n-2m}}P_{2m}(g)(u\phi).$ (1.15)

It is straightforward to see that one recovers (1.14) by substituting
$\phi\equiv 1$ into (1.15). The transformation rules (1.14) and (1.15)
motivate us to define the so-called total curvature functional
$\mathcal{Q}_{2m}:[g]\rightarrow\mathbb{R}$ and Yamabe-type functional
$\mathcal{Y}^{+}_{2m}:[g]\rightarrow\mathbb{R}$ for $m=1,2,3$ as

$\mathcal{Q}_{2m}(\bar{g}):=\frac{\int_{M}Q_{2m}({\bar{g}})\mathrm{d}\mathrm{v}_{\bar{g}}}{{\rm
vol}_{\bar{g}}(M)^{\frac{n-2m}{n}}}=\frac{\frac{2}{n-2m}\int_{M}uP_{2m}(g)u\mathrm{d}\mathrm{v}_{g}}{\left(\int_{M}u^{\frac{2n}{n-2m}}\mathrm{d}\mathrm{v}_{g}\right)^{\frac{n-2m}{n}}}$

and

$\displaystyle\mathcal{Y}^{+}_{2m}(M,g):=\inf_{\bar{g}\in[g]}\mathcal{Q}_{2m}(\bar{g})=\inf_{\bar{g}\in[g]}\frac{\int_{M}Q_{2m}({\bar{g}})\mathrm{d}\mathrm{v}_{\bar{g}}}{{\rm
vol}_{\bar{g}}(M)^{\frac{n-2m}{n}}}=\inf_{u\in\mathcal{C}_{+}^{\infty}(M)}\frac{\frac{2}{n-2m}\int_{M}uP_{2m}(g)u\mathrm{d}\mathrm{v}_{g}}{\left(\int_{M}u^{\frac{2n}{n-2m}}\mathrm{d}\mathrm{v}_{g}\right)^{\frac{n-2m}{n}}},$

where $\mathrm{d}\mathrm{v}_{g}$ denotes the standard volume measure. This
quantity is a higher order analog of the famous Yamabe invariant. Following
this notation, for $m=1,2,3$ we set the differential invariant

$\mathbb{Y}_{2m}^{+}(M):=\sup_{[g]\in\mathfrak{c}}\mathcal{Y}_{2m}^{+}(M,g)=\sup_{[g]\in\mathfrak{c}}\inf_{\widetilde{g}\in[g]}\left\\{\frac{\int_{M}Q_{2m}({\bar{g}})\mathrm{d}\mathrm{v}_{\bar{g}}}{{\rm
vol}_{\bar{g}}(M)^{\frac{n-2m}{n}}}\right\\},$

where $\mathfrak{c}$ denotes the space of conformal classes on the underlying
manifold.

Notice that we require all test functions in the infimum for
$\mathcal{Y}_{2m}^{+}$must all be positive. Naturally, one may also define

$\mathcal{Y}_{2m}(M,g):=\inf_{\mathcal{C}^{\infty}(M)\setminus\\{0\\}}\frac{\frac{2}{n-2m}\int_{M}uP_{2m}(g)u\mathrm{d}\mathrm{v}_{g}}{\left(\int_{M}|u|^{\frac{2n}{n-2m}}\mathrm{d}\mathrm{v}_{g}\right)^{\frac{n-2m}{n}}},$

and clearly $\mathcal{Y}_{2m}(M,g)\leqslant\mathcal{Y}_{2m}^{+}(M,g)$.

We assume that $(M^{n},g)$ is closed (compact and without boundary). We seek
conformal metrics of the form $\bar{g}=\mathrm{u}^{{4}/{(n-6)}}g$. We
prescribe the resulting metric to have constant $Q_{6}$-curvature, which we
normalize to be

$Q_{n}=Q_{6}(g_{0})=\frac{n(n^{4}-20n^{2}+64)}{2^{5}},$ (1.16)

where $g_{0}=g_{\mathbb{S}^{n}}$ denotes the standard round metric on the
sphere. In addition, using the transformation law (1.15), the condition that
$\bar{g}=\mathrm{u}^{{4}/{(n-6)}}g$ satisfies $Q_{6}(g)=Q_{n}$ on $M$ is
equivalent to the finding $\mathrm{u}\in\mathcal{C}^{\infty}(M)$ solution to
the the sixth order GJMS equation

$P_{6}({g})\mathrm{u}=c_{n}\mathrm{u}^{\frac{n+6}{n-6}}\quad\mbox{on}\quad M,$
($\mathcal{Q}_{6,g}$)

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 in
(1). We remark that the nonlinearity on the right-hand side of
($\mathcal{Q}_{6,g}$) has critical growth with respect to the compact
embedding of the Beppo–Levi space
$\mathcal{D}^{3,2}(\mathbb{R}^{n})\rightarrow L^{2^{\\#}}(\mathbb{R}^{n})$,
where $2^{\\#}=\frac{2n}{n-6}$. It is well known that this embedding is not
compact, reflecting the conformal invariance of the PDE ($\mathcal{Q}_{6,g}$).

As usual, we need to analyze the conformal invariant to find solutions to the
last equation. Namely, we define the sixth order Yamabe invariant
$\mathcal{Y}_{6}:[g]\rightarrow\mathbb{R}$ as

$\mathcal{Y}_{6}(M,g):=\inf_{u\in
L^{\frac{2n}{n-6}}(M)\setminus\\{0\\}}\frac{\frac{2}{n-6}\int_{M}uP_{6}(g)u\mathrm{d}\mathrm{v}_{g}}{\left(\int_{M}|u|^{\frac{2n}{n-6}}\mathrm{d}\mathrm{v}_{g}\right)^{\frac{n-6}{n}}},$
(1.17)

where $\bar{g}=u^{{4}/{(n-6)}}g$ is a conformal metric with $u\in
L^{\frac{2n}{n-6}}(M)$ is a conformal factor. The main results in this paper
are based on the existence of a positive Green function for the sixth order
GJMS operator, which can be stated as

  1. ($\mathcal{G}_{+}$)

For each $p\in M$, there exists a positive Green function, denoted by
$G_{6}(g)_{p}:M\times M\rightarrow\mathbb{R}$, for the sixth order GJMS
operator $P_{6}(g):\mathcal{C}^{\infty}(M)\rightarrow\mathcal{C}^{\infty}(M)$
with a pole at $p$. In other terms, it satisfies

$P_{6}(g)G_{6}(g)_{p}=\delta_{p}\quad{\rm in}\quad M,$

where $\delta_{p}$ denotes the Dirac measure centered at $p$.

In the light of the last definition, we can use the positive Green function
for the sixth order GJMS operator to define the dual conformal invariant. This
quantity is a counterpart to the new invariant introduced by Hang and Yang
[30] for the fourth order setting. As a matter of fact, assuming that
$(M^{n},g)$ satisfies ($\mathcal{G}_{+}$), we define the dual sixth order
Yamabe invariant $\Theta_{6}:[g]\rightarrow\mathbb{R}$ as

$\Theta_{6}(M,g):=\sup_{f\in
L^{\frac{2n}{n+6}}(M)\setminus\\{0\\}}\frac{\int_{M}fG_{6}(g)f\mathrm{d}{\rm
v}_{g}}{{\left(\int_{M}f^{\frac{2n}{n+6}}\mathrm{d}\mathrm{v}_{g}\right)^{\frac{n-6}{n}}}}=\sup_{\bar{g}\in[g]}\frac{\int_{M}Q_{6}({\bar{g}})\mathrm{d}\mathrm{v}_{\bar{g}}}{\|Q_{6}({\bar{g}})\|^{2}_{L^{\frac{2n}{n+6}}(M)}},$
(1.18)

where $\bar{g}=f^{{4}/{(n+6)}}g$ is a conformal metric with $f\in
L^{\frac{2n}{n+6}}(M)$ is a dual conformal factor. In some sense,
$\Theta_{6}(M,g)$ is comparable with the reciprocal
$\mathcal{Y}_{6}^{+}(M,g)^{-1}$. The advantages of considering this quantity
are that if a maximizer for (1.18) exists, it is smooth and does not change
its sign, and the associated conformal metric has constant sixth order
$Q$-curvature.

One needs a priori bound on the sixth order Yamabe invariant to apply the
topological bifurcation strategy for coverings. This type of inequality is
related to the asymptotic of the Green function and the concept of mass for a
GJMS operator [35, 39]. Below we state the positive mass theorem of Qing and
Raske [44] (see also [39]). For this, we work in conformal normal coordinates
[14, 27] around $p$, i.e. $r:=\mathrm{d}_{g}(x,p)$.

######  Lemma A.

Let $(\widetilde{M}^{n},\tilde{g})$ be a closed locally conformally flat
manifold with $n\geqslant 7$ satisfying $R(\tilde{g})\geqslant 0$. Suppose
that ($\mathcal{G}_{+}$) holds. Then, for any $p\in M$, one has

$G_{6}(g)_{p}(r)=C_{n}r^{6-n}+m_{6}(g)_{p}+\mathcal{O}(r)\quad{\rm as}\quad
r\rightarrow 0,$

where $m_{6}(g)\in\mathbb{R}$ is a constant. Moreover, if the Poincaré
exponent of the Kleinian group $\Gamma\subset{\rm iso}(M)$ satisfies

$0<\delta(\Gamma)<\frac{n-6}{2},$ ($\mathcal{QR}$)

where
$\delta(\Gamma):=\inf\\{\delta>0:\sum_{\gamma\in\Gamma}\left|\gamma^{\prime}(x)\right|^{\delta}<\infty\
{\rm for\ all}\ x\in\mathbb{S}^{n}\\}$, then $m_{6}(g)\geqslant 0$, with
equality $m_{6}(g)=0$ implying
$(\widetilde{M}^{n},\tilde{g})\simeq(\mathbb{R}^{n},\delta)$.

Based on the positivity and topological properties above, we have the Aubin-
type result below

######  Proposition A.

Let $(\widetilde{M}^{n},\tilde{g})$ be a closed locally conformally flat
manifold with $n\geqslant 7$ satisfying $R(\tilde{g})\geqslant 0$. Suppose
that ($\mathcal{G}_{+}$) and ($\mathcal{QR}$) hold. Then, both the statements
below hold and are equivalent

  1. ($\mathcal{A}$)

The infimum in (1.17) is attained at some smooth function
$u\in\mathcal{C}^{\infty}(M)$, and the conformal metric
$\bar{g}=u^{{4}/{(n-6)}}g$ has constant sixth order $Q$-curvature. Moreover,
one has

$\mathcal{Y}_{6}(M,g)\leqslant\mathcal{Y}_{6}(\mathbb{S}^{n},g_{0}),$

with equality if and only if $(M^{n},g)$ is conformally equivalent to the
round sphere $(\mathbb{S}^{n},g_{0})$;

  2. ($\mathcal{A}^{\prime}$)

The supremum in (1.18) is attained at some smooth function
$f\in\mathcal{C}^{\infty}(M)$, and the conformal metric
$\bar{g}=f^{{4}/{(n+6)}}g$ has constant sixth order $Q$-curvature. Moreover,
one has

$\Theta_{6}(M,g)\geqslant\Theta_{6}(\mathbb{S}^{n},g_{0}),$

with equality if and only if $(M^{n},g)$ is conformally equivalent to the
round sphere $(\mathbb{S}^{n},g_{0})$.

We recall that on Einstein manifolds, the sixth order GJMS operator expresses
as (3.2). In this case, one can directly apply Bishop–Gromov volume comparison
to guarantee that the geometric inequalities above are satisfied. This follows
if one assumes the validity of the positive mass theorem (which can be
recovered by assuming ($\mathcal{QR}$)) in the low-dimensional cases $n=7,8,9$
(c.f. [44, 39]) or from the local structure around a non-vanishing point of
the Weyl tensor in remaining cases $n\geqslant 10$ (c.f. [18]). Recently,
there has been some progress in proving the validity of ($\mathcal{G}_{+}$).
Indeed, Jeffrey Case informed us of a sufficient condition for ($\mathcal{A}$)
to be satisfied for a special Einstein product introduced by Gover and Leitner
[24]. Namely, when the underlying manifold satisfies the property below

  1. ($\mathcal{GL}$)

$(M^{n},g)=(M^{n_{1}}\times M^{n_{2}},g_{1}\oplus g_{2})$ with
$(M^{n_{1}},g_{1})$ and $(M^{n_{2}},g_{2})$ being both Einstein and

$n_{1}(n_{1}-1)R(g_{1})+n_{2}(n_{2}-1)R(g_{2})=0.$

with $R(g_{2})>0$ and $n_{2}\gg 1$ sufficiently large, then
($\mathcal{G}_{+}$) is true. In addition, he also expects that
($\mathcal{A}^{\prime}$) holds in this case.

Our objective is to prove nonuniqueness results for constant sixth order
$Q$-curvature metrics. Although these results look, at first glance, seemingly
different, they are both based on bifurcation techniques and global geometric
inequalities above, namely ($\mathcal{A}$) and ($\mathcal{A}^{\prime}$).

The first main result of this paper is based on [46, Theorem 1] and uses a
bifurcation technique to obtain the existence of infinitely many on the
product of two spheres.

######  Theorem 1.1.

Let $(\mathbb{S}^{1}\times\mathbb{S}^{n-1},g_{\mathbb{S}^{1}}\oplus
g_{\mathbb{S}^{n-1}})$ with $n\geqslant 7$. Then, there exists infinitely many
nonhomothetic conformal metrics on ${\rm
Met}^{\infty}(\mathbb{S}^{1}\times\mathbb{S}^{n-1})$ with constant sixth order
$Q$-curvature. Moreover, one has

$\lim_{\ell\rightarrow+\infty}\mathcal{Y}_{6}^{+}(\mathbb{S}^{1}\times\mathbb{S}^{n-1},g_{\ell})=\mathcal{Y}_{6}^{+}(\mathbb{S}^{n},g_{0}).$

Consequently, it follows the equality

$\mathbb{Y}_{6}^{+}(\mathbb{S}^{1}\times\mathbb{S}^{n-1})=\mathbb{Y}_{6}^{+}(\mathbb{S}^{n})=\frac{n(n^{4}-20n^{2}+64)}{2^{5}}\omega_{n}^{6/n}$

cannot be realized by a smooth metric on
$\mathbb{S}^{n-1}\times\mathbb{S}^{1}$.

Notice that $\mathbb{Y}_{6}^{+}(\mathbb{S}^{n})$ coincides with the best
Sobolev constant of the compact embedding of the Beppo–Levi space
$\mathcal{D}^{3,2}(\mathbb{R}^{n})\rightarrow L^{2^{\\#}}(\mathbb{R}^{n})$. As
a by-product of Lemma A and Theorem 1.1, we prove that the sixth order Yamabe
invariant on $\mathbb{S}^{1}\times\mathbb{S}^{n-1}$ coincides with the one of
the round sphere. To the best of our knowledge, this provides the first class
of examples for which the sixth order Yamabe invariant is directly computed;
such a fact is remarkable and of independent interest.

######  Corollary 1.2.

Let $(\mathbb{S}^{1}\times\mathbb{S}^{n-1},g_{\mathbb{S}^{1}}\oplus
g_{\mathbb{S}^{n-1}})$ with $n\geqslant 7$. Suppose that ($\mathcal{G}_{+}$)
holds. Then, it follows

$\mathcal{Y}_{6}(M,g)=\mathcal{Y}^{*}_{6}(M,g)\leqslant\mathcal{Y}_{6}(\mathbb{S}^{n},g_{0}),$

where

$\mathcal{Y}_{6}^{*}(M,g)=\inf_{\bar{g}\in[g],Q_{2}(\bar{g})>0,Q_{4}(\bar{g})>0}\mathcal{Q}_{6}(\bar{g})$

with the equality implying $(M^{n},g)=(\mathbb{S}^{n},g_{0})$. Moreover, one
has

$\mathbb{Y}_{6}(\mathbb{S}^{n-1}\times\mathbb{S}^{1})=\mathbb{Y}_{6}^{+}(\mathbb{S}^{n-1}\times\mathbb{S}^{1})=\mathcal{Y}_{6}^{*}(\mathbb{S}^{n},g_{0}).$

The constant $Q$-curvature problem can also be posed on noncompact manifolds
(possibly with a simple structure at infinity). The natural boundary condition
is the completeness of the metric. For instance, this is trivially satisfied
by metrics that descend to a compact quotient $M\setminus\Gamma$; we call
these periodic solutions with period $\Gamma\subset{\rm iso}(M)$; this is
inspired by the classical work of Schoen and Yau [49]. More precisely, we seek
for complete metrics on $M\setminus\Lambda$ of the form
$\bar{g}=\mathrm{u}^{{4}/{(n-6)}}g$, where $\Lambda^{k}\subset M^{n}$ is a
$k$-dimensional submanifold. Namely, when $(M^{n},g)$ is a compact locally
conformally flat manifold with positive scalar curvature $R_{g}\geqslant 0$,
one has that $\Lambda:=\mathbb{S}^{n}\setminus\mathscr{D}(\widetilde{M})$ can
be seen as the complement of image of the developing map
$\mathscr{D}:\widetilde{M}\hookrightarrow\mathbb{S}^{n}$, where
$\widetilde{M}$ is the universal cover of $M$.

To $g$ to be complete on $M^{n}\setminus\Lambda^{k}$, one has to impose
$\liminf_{\mathrm{d}_{g}(p,\Lambda)}\mathrm{u}(p)=+\infty$. As before, writing
$\bar{g}=\mathrm{u}^{{4}/{(n-6)}}g$, one has that $Q_{6}(g)=Q_{n}$ on
$M\setminus\Lambda$ is equivalent to finding
$\mathrm{u}\in\mathcal{C}^{\infty}(M\setminus\Lambda)$ solving

$P_{6}({g})\mathrm{u}=c_{n}\mathrm{u}^{\frac{n+6}{n-6}}\quad\mbox{on}\quad
M\setminus\Lambda.$ ($\mathcal{Q}_{6,g,\Lambda}$)

We denote by $\Omega:=M\setminus\Lambda$ and ${\rm
Met}^{\infty}(\Omega)=\cup_{j=1}^{\infty}{\rm Met}^{j}(\Omega)$ the set of
smooth metrics over $\Omega$, where ${\rm Met}^{j}(\Omega)$ denotes the set of
$\mathcal{C}^{j}$ metrics over $\Omega$ with $j\in\mathbb{N}$ is equipped with
the Gromov-Hausdorff topology. In this direction, it is convenient to
introduce some terminology. For any $g\in{\rm Met}^{\infty}(\Omega)$, let us
define the moduli space of complete metrics below

$\displaystyle\mathcal{M}_{6,\Lambda}({g})=\left\\{\bar{g}\in[g]:\
\text{$\bar{g}$ is complete on $\Omega$}\ {\rm and}\ Q_{6}(g)\equiv
Q_{n}\right\\}\subset{\rm Met}^{\infty}(\Omega).$

This set is called the moduli space of constant sixth order $Q$-curvature
metrics singular at $\Lambda$. When $\Lambda=\varnothing$, we shall denote
$\mathcal{M}_{6,\Lambda}({g})=\mathcal{M}_{6}({g})$. Also when $g=g_{0}$, we
simply denote $\mathcal{M}_{6,\Lambda}({g}_{0})=\mathcal{M}_{6,\Lambda}$.

Our second main result is inspired by [11, Theorem B] and exploits the
abundance of discrete cocompact groups on symmetric spaces to find infinitely
many periodic solutions with different periods on noncompact manifolds,
reflecting a genuinely noncompact phenomenon.

######  Theorem 1.3.

Let $(M^{n},g)$ be a closed Riemannian manifold with $n\geqslant 7$. Assume
that ($\mathcal{G}_{+}$) and ($\mathcal{A}^{\prime}$) hold. Suppose that
$(M^{n}_{\infty},g_{\infty})\rightarrow(M^{n},g)$ is a Riemannian covering
whose group of deck transformations has infinite profinite completion. Then,
there exists infinitely many nonhomothetic conformal periodic metrics on
$\mathcal{M}_{6}(g_{C}\oplus g_{\mathcal{S}})$ with constant positive sixth
order $Q$-curvature.

As a by-product of the proof, we have the following result

######  Corollary 1.4.

Let $(C^{n_{1}}\times\mathcal{S}^{n_{2}},g_{C}\oplus g_{\mathcal{S}})$ be a
complete noncompact Riemannian manifold with $n_{1}+n_{2}=n\geqslant 7$.
Assume that ($\mathcal{G}_{+}$) and ($\mathcal{A}^{\prime}$) hold. Assume that
$(C^{n_{1}},g_{C})$ is a closed manifold with constant scalar curvature, and
$(\mathcal{S}^{n_{2}},g_{\mathcal{S}})$ is a simply-connected symmetric space
of noncompact or Euclidean type. Then, there exists infinitely many
nonhomothetic conformal periodic metrics on $\mathcal{M}_{6}(g_{C}\oplus
g_{\mathcal{S}})$ with constant positive sixth order $Q$-curvature.

An immediate consequence of Corollary 1.4 is that one can find infinitely many
complete metrics with constant $Q$-curvature on the round sphere blowing up
along an equatorial subsphere.

######  Corollary 1.5.

Let $(\mathbb{S}^{n}\setminus\mathbb{S}^{k},g_{0})$ be the standard round
sphere singular along a minimal equatorial $k$-subsphere with $n\geqslant 7$
and $0\leqslant k<\frac{n-6}{2}$. Suppose that ($\mathcal{G}_{+}$) holds.
Then, there exist infinitely many complete conformal metrics on
$\mathcal{M}_{6,\mathbb{S}^{k}}$ with constant positive sixth order
$Q$-curvature.

If one assumes that the results by Case and Malchiodi [16] holds, then
assumption ($\mathcal{G}_{+}$) in Corollary 1.2 could be changed by the
dimension condition $n\geqslant 10$. Furthermore, it is not hard to check that
the manifolds satisfying ($\mathcal{GL}$) would provide a large class of
manifolds to apply this general bifurcation technique on Riemannian coverings.

The restriction on the dimension of the singular set $0\leqslant k<{(n-6)/2}$
is sharp in the sense that the Poincaré exponent of the holonomy
representation $\Gamma\subset{\rm iso}({M})\subset\mathbb{H}^{k+1}$ is not
well-defined otherwise [43] for its fourth order counterpart. We also remark
that the positivity condition ($\mathcal{G}_{+}$) is usually hard to verify,
and it is taken as a hypothesis in some recent papers (c.f. [36, 31]).

In some exceptional cases, this property can be proved. For instance, in the
locally conformally flat case, this assumption is intimately related with the
size of the Poincaré exponent, which by the injectivity of the developing map,
coincides with the Hausdorff dimension of the singular set, that is,
$0<\delta(\Gamma)<k$. Furthermore, observe that for $k=0$, the singular set is
a finite set of points. In this case, in [2] it is proved compactness for this
moduli space of metrics under certain conditions on the asymptotic necksize of
the isolated singularities.

The proof of our main theorems is mainly based on variational bifurcation
theory and topological methods [50, 34, 40]. The primary sources of
difficulties rely on the need for maximum principle enjoyed by this type of
operator. We are restricted to manifolds of dimension $n\geqslant 7$, as the
cases $3\leqslant n\leqslant 5$ and especially $n=6$ require a separate
discussion. Recently, for the case of constant fractional $Q$-curvature
metrics $\sigma\in(0,1)$, Bettiol, González, and Maalaoui [7] also used
bifurcation techniques to prove similar results on round spheres. In a
companion paper [3], we intend to study constant sixth order $Q$-curvature
metrics on closed manifolds.

Let us compare our results with the existing literature. In his pioneering
work, Yamabe [52] was the first to introduce these invariants and study the
problem of finding a constant scalar curvature metric in a given conformal
class. After that, Aubin [5] proved that
$\mathcal{Y}_{2}(M,g)\leqslant\mathcal{Y}_{2}(\mathbb{S}^{n},g_{0})$. He also
showed that whenever
$\mathcal{Y}_{2}(M,g)<\mathcal{Y}_{2}(\mathbb{S}^{n},g_{0})$ is satisfied, one
can find a smooth metric with constant scalar curvature metric $\bar{g}\in[g]$
such that $\mathcal{Q}_{2}(\bar{g})=\mathcal{Y}_{2}(M,g)$. At last, Schoen
[47] proved that $\mathcal{Y}_{2}(M,g)<\mathcal{Y}_{2}(\mathbb{S}^{n},g_{0})$
for each conformal class $[g]\neq[g_{0}]$. In particular,
$\mathcal{Y}_{2}(M,g)<\mathcal{Y}_{2}(\mathbb{S}^{n},g_{0})$ whenever $M$ is
not equivalent to the round sphere.

Nevertheless, the equality $\mathbb{Y}_{2}(M)=\mathbb{Y}_{2}(\mathbb{S}^{n})$
may occur even when $M$ is not the sphere. In particular, he found in [48] an
explicit sequence of metrics $\\{g_{\ell}\\}_{\ell\in\mathbb{N}}\subset{\rm
Met}^{\infty}(\mathbb{S}^{1}\times\mathbb{S}^{n-1})$ such that

$\mathcal{Y}_{2}(\mathbb{S}^{1}\times\mathbb{S}^{n-1},g_{\ell})\rightarrow\mathcal{Y}_{2}(\mathbb{S}^{n},g_{0})\quad{\rm
as}\quad\ell\rightarrow\infty,$

and so
$\mathbb{Y}_{2}(\mathbb{S}^{1}\times\mathbb{S}^{n-1})=\mathbb{Y}_{2}(\mathbb{S}^{n})$.
As the underlying manifolds are not diffeomorphic, the equality above cannot
be realized by a smooth metric on $\mathbb{S}^{1}\times\mathbb{S}^{n-1}$. As a
consequence of the maximum principle, the Yamabe invariant
$\mathcal{Y}_{2}(M,g)$ is always nonnegative. Therefore, the minimizers of the
functional $\mathcal{Q}_{2}$ over all nontrivial functions in $H^{1}(M)$ will
be automatically positive.

Nonetheless, minimizers of $\mathcal{Y}_{2m}(M,g)$ with $m=2,3$, if exist,
might change sign. On the fourth order setting, Esposito and Robert [20]
showed that $\mathcal{Y}_{4}^{+}(M,g)<+\infty$ for each conformal class $[g]$.
Gursky, Hang and Lin [28] proved that if $n\geqslant 6$ and if
$\mathcal{Y}_{2}(M,g)>0$ and $\mathcal{Y}_{4}^{+}(M,g)>0$ then

$\mathcal{Y}_{4}(M,g)=\mathcal{Y}_{4}^{+}(M,g)=\mathcal{Y}_{4}^{*}(M,g):=\inf_{\bar{g}\in[g],Q_{2}(\bar{g})>0}\mathcal{Q}_{4}(\bar{g}).$

Thereafter, Hang and Yang [30] proved that if $\mathcal{Y}_{2}(M,g)>0$ and
$Q_{4}(g)\geqslant 0$ with $Q_{4}(g)\not\equiv 0$, then

$\mathcal{Y}_{4}(M,g)=\mathcal{Y}_{4}^{*}(M,g)\leqslant\mathcal{Y}_{4}(\mathbb{S}^{n},g_{0})$
(1.19)

and that equality implies $(M^{n},g)\simeq(\mathbb{S}^{n},g_{0})$. Also, under
these hypotheses, there exists a smooth, constant $Q$-curvature metric
$\bar{g}\in[g]$ such that $\mathcal{Q}_{4}(\bar{g})=\mathcal{Y}_{4}^{*}(M,g)$.
The result referenced above implies the equality
$\mathbb{Y}_{4}^{+}(\mathbb{S}^{n-1}\times\mathbb{S}^{1})=\mathcal{Y}_{4}^{*}(\mathbb{S}^{n})$.
More precisely, they have introduced a new dual conformal invariant for any
$(M^{n},g)$ be a closed Riemannian manifold with $n\geqslant 5$. Furthermore,
they obtained

$\mathcal{Y}_{4}(M,g)=\mathcal{Y}_{4}^{*}(M,g)\leqslant\mathcal{Y}_{4}(\mathbb{S}^{n},g_{0}).$
(1.20)

Based on (1.19), by using classical bifurcation techniques on the universal
cover of Schoen [48], Ratzkin [46] proved nonuniqueness results for fourth
order $Q$-curvature metrics on the product
$\mathbb{S}^{1}\times\mathbb{S}^{n-1}$. Almost concomitantly, Bettiol et al.
[11] applied a topological method to extend this construction for much broader
situations; this depends on the reversed Aubin inequality in (1.20). These
result are inspired by the classical works on the scalar curvature equation
[52, 5, 48]. We also refer the reader to [10, 9, 8, 47] for more result using
a related bifurcation technique.

In the light of [29], we speculate that both geometric inequalities above hold
under milder conditions of the underlying manifold.

######  Conjecture 1.

Let $(M^{n},g)$ be a closed Riemannian manifold with $n\geqslant 7$. Assume
that

$\mathcal{Y}_{2}(M,g)\geqslant 0,\quad\mathcal{Y}_{4}(M,g)\geqslant
0,\quad{\rm and}\quad Q_{6}({g})\geqslant 0\quad{\rm but}\quad
Q_{6}({g})\not\equiv 0.$ ($\mathcal{HY}$)

Then, ($\mathcal{G}_{+}$) is satisfied. In particular, both ($\mathcal{A}$)
and ($\mathcal{A}^{\prime}$) also hold.

It is not hard to see that
$(\mathbb{S}^{1}\times\mathbb{S}^{n-1},g_{\mathbb{S}^{1}}\oplus
g_{\mathbb{S}^{n-1}})$ satisfies ($\mathcal{HY}$). This fact somehow supports
the conjecture above. Consequently, all the methods to prove the existence of
infinitely many conformal metrics with constant sixth order $Q$-curvature can
be adapted promptly

######  Conjecture 2.

Let $(M^{n},g)=(C^{n_{1}}\times\mathcal{S}^{n_{2}},g_{C}\oplus
g_{\mathcal{S}})$ be a complete noncompact Riemannian manifold with
$n_{1}+n_{2}=n\geqslant 7$. Suppose that ($\mathcal{HY}$) holds. Assume that
$(C^{n_{1}},g_{C})$ is a closed manifold with constant scalar curvature, and
$(\mathcal{S}^{n_{2}},g_{\mathcal{S}})$ is a simply-connected symmetric space
of noncompact or Euclidean type. Then, there exists infinitely many
nonhomothetic conformal periodic metrics on $\mathcal{M}_{6}(g_{C}\oplus
g_{\mathcal{S}})$ with constant positive sixth order $Q$-curvature.

Let us now describe the plan for the rest of the manuscript. In Section 2, we
establish some notations that will be used throughout the paper. In Section 3,
we use a bifurcation technique based on the jump of the Morse index on the
universal cover to prove Theorem 1.1. In Section 4, we generalize the latter
approach to the case of an infinite tower of Riemannian covers to give proof
for Corollary 1.4. In conclusion, we also explain how these main results are
interconnected.

##  2\. Notation

Let us establish some standard terminology and definitions. In what follows,
we will always be using Einstein’s summation convection.

  * •

$\delta=g_{\mathbb{R}^{n}}$ denotes the standard Euclidean metric;

  * •

$g_{0}=g_{\mathbb{S}^{n}}$ denotes the standard round metric;

  * •

$(e_{i})_{i=1}^{n}$ denotes a local coordinate frame;

  * •

${\rm Rm}_{g}\in\mathfrak{T}^{3}_{1}(M)$ denotes the Riemannian curvature
tensor,

  * •

$\accentset{\circ}{\rm Rm}_{g}\in\mathfrak{T}^{4}_{0}(M)$ denotes the
covariant Riemann curvature tensor,

  * •

${\rm Ric}_{g}={\rm tr}_{g}\accentset{\circ}{\rm
Rm}_{g}\in\mathfrak{T}^{2}_{0}(M)$ denotes the Ricci curvature tensor , which
can be expressed as ${\rm Ric}_{jk}=\accentset{\circ}{\rm
Rm}_{ijk}^{i}=g^{i\ell}\accentset{\circ}{\rm Rm}_{ijk\ell}$;

  * •

$R_{g}={\rm tr}_{g}{\rm Ric}_{g}\in\mathfrak{T}^{0}_{0}(M)$ denotes the scalar
curvature given by $R_{g}=g^{ij}{\rm Ric}_{ij}$;

  * •

$\mathfrak{T}^{r}_{s}(M)$ denotes the set of $(r,s)$-type tensor over $M$ with
$\mathfrak{T}^{0}_{0}(M)=\mathcal{C}^{\infty}(M)$;

  * •

$\Delta_{g}:=g^{ij}\nabla_{i}\nabla_{j}$ denotes the Laplace–Beltrami
operator;

  * •

$\nabla_{g}$ denotes the Levi–Civita connection;

  * •

${\rm tr}_{g}:\mathfrak{T}^{r}_{s}(M)\rightarrow\mathfrak{T}^{r-2}_{s}(M)$
denotes the trace operator.

  * •

$a_{1}\lesssim a_{2}$ if $a_{1}\leqslant Ca_{2}$, $a_{1}\gtrsim a_{2}$ if
$a_{1}\geqslant Ca_{2}$, and $a_{1}\simeq a_{2}$ if $a_{1}\lesssim a_{2}$ and
$a_{1}\gtrsim a_{2}$.

  * •

$u=\mathcal{O}(f)$ as $x\rightarrow x_{0}$ for
$x_{0}\in\mathbb{R}\cup\\{\pm\infty\\}$, if $\limsup_{x\rightarrow
x_{0}}(u/f)(x)<\infty$ is the Big-O notation;

  * •

$u=\mathrm{o}(f)$ as $x\rightarrow x_{0}$ for
$x_{0}\in\mathbb{R}\cup\\{\pm\infty\\}$, if $\lim_{x\rightarrow
x_{0}}(u/f)(x)=0$ is the little-o notation;

  * •

$u\simeq\widetilde{u}$, if $u=\mathcal{O}(\widetilde{u})$ and
$\widetilde{u}=\mathcal{O}(u)$ as $x\rightarrow x_{0}$ for
$x_{0}\in\mathbb{R}\cup\\{\pm\infty\\}$;

  * •

$\mathcal{C}^{j,\alpha}(M)$, where $j\in\mathbb{N}$ and $\alpha\in(0,1)$, is
the classical Hölder space; we simply denote $\mathcal{C}^{j}(M)$ when
$\alpha=0$.

  * •

$W^{j,q}(M)$ is the classical Sobolev space, where $j\in\mathbb{N}$ and
$q\in[1,+\infty]$; when $j=0$ we simply denote $L^{q}(M)$ when $q=2$, we
simply denote $H^{j}(M)$;

  * •

$\gamma_{n}=\frac{n-6}{2}$ is the Fowler rescaling exponent;

  * •

$2^{\\#}=\frac{2n}{n-6}$ is the critical Sobolev exponent.

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

######  Definition 2.1.

Let $(M^{n},g)$ be a closed Riemannian manifold. We define the following
operations with tensors

  * (a)

cross product $\times:{\rm Sym}_{2}(M)\times{\rm Sym}_{2}(M)\rightarrow{\rm
Sym}_{2}(M)$ given by

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

  * (b)

dot product $\cdot:{\rm Sym}_{2}(M)\times{\rm
Sym}_{2}(M)\rightarrow\mathbb{R}$ given by

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

  * (c)

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$\cr}}{}}:{\rm
Sym}_{2}(M)\times{\rm Sym}_{2}(M)\rightarrow\mathfrak{T}^{4}_{0}(M)$ given by

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

  * (d)

dot operator $\cdot:{\rm Sym}_{2}(M)\rightarrow{\rm Sym}_{2}(M)$ given by

$(\accentset{\circ}{\rm Rm}\cdot h)_{jk}:=R_{ijk\ell}h^{i\ell}.$

  * (e)

$L^{2}$-formal adjoint of the Lie derivative $\delta_{g}:{\rm
Sym}_{2}(M)\rightarrow\mathbb{R}$ given by

$\left(\delta_{g}h\right)_{i}:=-\left(\operatorname{div}_{g}h\right)_{i}=-\nabla_{g}^{j}h_{ij}.$

##  3\. Toy model

This section is devoted to proving Theorem 1.1. Instead, we will prove a
slightly stronger proposition based on the validity of Aubin’s lemma in
($\mathcal{A}$). Our proof follows the strategy in [48] and [46] and relies
strongly on the explicit construction of the Delaunay metrics on the cylinder
from Proposition B.

###  3.1. Variational structure

We begin with the well-known variational characterization of constant
$Q$-curvature metrics. let us recall the continuous classical Sobolev
embedding

$W^{3,2}(M)\hookrightarrow L^{2^{\\#}}(M),$ (3.1)

where $2^{\\#}={2n}/{(n-6)}$ with $n\geqslant 7$ is the critical Sobolev
exponent.

######  Remark 3.1.

We emphasize that assuming that $(M^{n},g)$ is Einstein with $n\geqslant 7$,
the operator on the left-hand of ($\mathcal{Q}_{6,g}$) has a much simpler form
than (1) $($c.f. [22, 23]$)$. Namely, let us denote by
$P_{6}(g):W^{3,2}(M)\rightarrow\mathbb{R}$ given by

$\displaystyle P_{6}(g)$
$\displaystyle:=(-\Delta_{g}+a_{n}R_{g})(-\Delta_{g}+b_{n}R_{g})(-\Delta_{g}+c_{n}R_{g}).$
(3.2)

where $a_{n},b_{n},c_{n}>0$ given by

$\displaystyle a_{n}:=\frac{(n-6)(n+4)}{4n(n-1)},\quad
b_{n}:=\frac{(n-4)(n+2)}{4n(n-1)},\quad{\rm and}\quad
c_{n}:=\frac{n-2}{4(n-1)}$

are also dimensional constants.

######  Definition 3.2.

Let $(M^{n},g)$ be a closed Riemannian Einstein manifold with $n\geqslant 7$.
We define the bilinear form $\mathcal{E}_{6}(g):W^{3,2}(M)\times
W^{3,2}(M)\rightarrow\mathbb{R}$ given by

$\mathcal{E}_{6}(g)(u_{1},u_{2})=\displaystyle\int_{M}u_{2}P_{6}(g)u_{1}\mathrm{d}\mathrm{v}_{g},$

where $\mathrm{d}\mathrm{v}_{g}$ denotes the standard volume measure. We set
$\mathcal{E}_{6}(g)(u,u)=\mathfrak{Q}_{6}(g)(u)$ to be its associated
quadratic form. More explicitly, the functional
$\mathfrak{Q}_{6}(g):W^{3,2}(M)\rightarrow\mathbb{R}$ is defined as

$\displaystyle\mathfrak{Q}_{6}(g)(u)$
$\displaystyle=-\int_{M}|\nabla_{g}\Delta_{g}u|^{2}\mathrm{d}\mathrm{v}_{g}+A_{n}\int_{M}|\Delta_{g}u|^{2}\mathrm{d}\mathrm{v}_{g}-B_{n}\int_{M}|\nabla_{g}u|^{2}\mathrm{d}\mathrm{v}_{g}+C_{n}\int_{M}|u|^{2}\mathrm{d}\mathrm{v}_{g}.$
(3.3)

with

$\displaystyle A_{n}$ $\displaystyle:=\frac{3n^{2}-6n-32}{4n(n-1)}$
$\displaystyle B_{n}$
$\displaystyle:=\frac{3n^{4}-12n^{3}-52n^{2}+128n+192}{16n^{2}(n-1)^{2}}$
(3.4) $\displaystyle C_{n}$
$\displaystyle:=\frac{(n^{4}-20n^{2}+64)(n-6)}{64n^{2}(n-1)^{3}}$

being dimensional constants.

######  Remark 3.3.

Notice that for any $u_{1},u_{2}\in W^{3,2}(M)$, it is not hard to see that
from (3.2) and the divergence theorem, it holds

$\displaystyle\mathcal{E}_{6}(g)(u_{1},u_{2})=\int_{M}u_{2}P_{6}(g)u_{1}\mathrm{d}\mathrm{v}_{g}=\int_{M}u_{1}P_{6}(g)u_{2}\mathrm{d}\mathrm{v}_{g}=\mathcal{E}_{6}(g)(u_{2},u_{1}),$

which shows that $\mathcal{E}_{6}(g):M\times M\rightarrow\mathbb{R}$ is a
symmetric bilinear form. It is well-known the symmetry property together with
the fact that the sixth order GJMS operator is self-adjoint holds in general
for any metric, which dates back to the work [26].

######  Definition 3.4.

Let $(M^{n},g)$ be a closed Riemannian manifold with $n\geqslant 7$. The total
sixth order curvature functional $\mathcal{Q}_{6}:[g]\rightarrow\mathbb{R}$ is
given by

$\mathcal{Q}_{6}(\bar{g})={{\rm
vol}_{\bar{g}}(M)^{\frac{6-n}{n}}}{\int_{M}Q_{6}({\bar{g}})\mathrm{d}\mathrm{v}_{\bar{g}}}.$
(3.5)

In terms of conformal factors, we have
$\mathcal{Q}_{6}:W^{3,2}_{+}(M)\rightarrow\mathbb{R}$ is given by

$\mathcal{Q}_{6}(u)={\frac{2}{n-6}\|u\|^{2}_{L^{2^{\\#}}(M)}}{\int_{M}uP_{6}(g)u\mathrm{d}\mathrm{v}_{g}}={\frac{2}{n-6}\|u\|^{2}_{L^{2^{\\#}}(M)}}\mathfrak{Q}_{6}(g)(u).$
(3.6)

is the sixth order Yamabe functional, where $W_{+}^{3,2}(M)$ denotes the
subspace of $($almost everywhere$)$ positive functions in $W^{3,2}(M)$.

######  Lemma 3.5.

Let $(M^{n},g)$ be a closed Riemannian manifold with $n\geqslant 7$. The sixth
order Yamabe-type functional
$\mathcal{Q}_{6}:W^{3,2}_{+}(M)\rightarrow\mathbb{R}$ given by (3.6) is
$\mathcal{C}^{1}$ with $D\mathcal{Q}_{6}:W^{3,2}_{+}(M)\rightarrow\mathbb{R}$
given by

$D\mathcal{Q}_{6}(u)\phi=\frac{4}{n-6}\|u\|_{L^{2^{\\#}}(M)}\int_{M}\phi\left(P_{6}(g)u-\|u\|_{L^{2^{\\#}}(M)}^{-2^{\\#}}\mathfrak{Q}_{6}(g)(u)u^{\frac{n+6}{n-6}}\right)\mathrm{d}\mathrm{v}_{g}.$
(3.7)

###### Proof.

Let $u\in W_{+}^{3,2}(M)\cap\mathcal{C}^{0}(M)$. Using the compactness of
$(M^{n},g)$, we can choose a test function $\phi\in W^{3,2}(M)$ such that
$0<\sup|\phi|<2^{-1}\inf_{M}u$. In particular, $u+t\phi\in W_{+}^{3,2}(M)$ for
$0<t<3/2$. On the one hand, by Taylor expanding (3.3) until second order, we
obtain

$\displaystyle\mathfrak{Q}_{6}(g)(u+t\phi)$
$\displaystyle=\mathfrak{Q}_{6}(g)(u)+2t\mathcal{E}_{6}(g)(\phi,u)+t^{2}\mathfrak{Q}_{6}(g)(\phi)$
(3.8) $\displaystyle=\int_{M}uP_{6}(g)u\mathrm{d}\mathrm{v}_{g}+2t\int_{M}\phi
P_{6}(g)u\mathrm{d}\mathrm{v}_{g}+t^{2}\int_{M}\phi
P_{6}(g)\phi\mathrm{d}\mathrm{v}_{g}.$

On the other hand, we have

$\displaystyle\|u+t\phi\|_{L^{2^{\\#}}(M)}^{2}$
$\displaystyle=\left(\int_{M}(u+t\phi)^{\frac{2n}{n-6}}\mathrm{d}\mathrm{v}_{g}\right)^{\frac{6-n}{n}}$
(3.9) $\displaystyle=\|u\|_{L^{2^{\\#}}(M)}^{2}\left(\int_{M}\phi
u^{\frac{n+6}{n-6}}\mathrm{d}\mathrm{v}_{g}\right)+\mathcal{O}\left(t^{2}\|\phi\|_{W^{3,2}(M)}^{2}\right)$
$\displaystyle=\|u\|_{L^{2^{\\#}}(M)}^{-2}\left(1-2t\|u\|_{L^{2^{\\#}}(M)}^{-2^{\\#}}\int_{M}\phi
u^{\frac{n+6}{n-6}}\mathrm{d}\mathrm{v}_{g}+\mathcal{O}\left(t^{2}\|\phi\|_{W^{3,2}(M)}\right)\right).$

Therefore, by combining (3.8) and (3.9), we find

$\displaystyle\mathcal{Q}_{6}(u+t\phi)$
$\displaystyle=\frac{2}{(n-6)}\|u\|_{L^{2^{\\#}}(M)}^{-2}\left[\mathfrak{Q}_{6}(g)(u)+2t\int_{M}\phi\left(P_{6}(g)u-\mathfrak{Q}_{6}(g)(u)\|u\|_{L^{2^{\\#}}(M)}^{-2^{\\#}}u^{\frac{n+6}{n-6}}\right)\mathrm{d}\mathrm{v}_{g}+\mathcal{O}(t^{2}\|\phi\|_{W^{3,2}(M)}^{2})\right],$

which completes the proof. ∎

######  Lemma 3.6.

Let $(M^{n},g)$ be a closed Riemannian manifold with $n\geqslant 7$. The total
sixth order curvature $\mathcal{Q}_{6}:[g]\rightarrow\mathbb{R}$ given by
(3.5) is $\mathcal{C}^{1}$ with first variation
$\mathrm{d}\mathcal{Q}_{6}:T_{\tilde{g}}[g]\simeq\mathcal{C}^{\infty}(M)\rightarrow\mathbb{R}$
given by

$\mathrm{d}\mathcal{Q}_{6}(\bar{g})\tilde{g}=\frac{n-6}{2}\int_{M}\phi\left(Q_{6}({\bar{g}})-\overline{Q}_{6}({\bar{g}})\right)\mathrm{d}\mathrm{v}_{g},$

where $\overline{Q}_{6}({\bar{g}})$ denotes the spherical average of
$Q_{6}({\bar{g}})$ In particular, metric $\bar{g}=u^{{4}/{(n-6)}}g$ is a
critical point of the functional $\mathcal{Q}_{6}:[g]\rightarrow\mathbb{R}$ if
and only if its sixth order $Q$-curvature $Q_{6}({\bar{g}})$ is constant.

###### Proof.

Initially, assuming that $\bar{g}=u^{4/(n-6)}\in[g]$ is a critical point of
$\mathcal{Q}_{6}$, one has

$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\mathcal{Q}_{6}\left((u+t\phi)^{\frac{4}{n-6}}g\right)=0\quad{\rm
for\ each}\quad\phi\in W^{3,2}(M)$

such that the norm $\|\phi\|_{L^{2^{\\#}}(M)}\ll 1$ is small enough. The last
identity combined with (1.15) yields

$P_{6}(g)u={\mathfrak{Q}_{6}(g)(u)u^{\frac{n+6}{n-6}}}{\|u\|_{L^{2^{\\#}}(M)}^{-2^{\\#}}}.$

Hence, substituting this value into (3.7) gives us

$Q_{6}({\bar{g}})=\frac{2}{(n-6)}{\mathfrak{Q}_{6}(g)(u)}{\|u\|_{L^{2^{\\#}}(M)}^{-2^{\\#}}},$

which is indeed a constant.

Conversely, suppose $Q_{6}({\bar{g}})=\bar{Q}$ is a constant, in which case
(1.15) gives us

$P_{6}(g)u=\frac{n-6}{2}\bar{Q}u^{\frac{n+6}{n-6}}$

and so

$\mathfrak{Q}_{6}(g)(u)=\frac{n-6}{2}\bar{Q}\|u\|_{L^{2^{\\#}}(M)}^{2^{\\#}}.$

Substituting this value into (3.7) yields

$\displaystyle\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\mathcal{Q}_{6}\left((u+t\phi)^{\frac{4}{n-6}}g\right)$
$\displaystyle=\frac{4}{(n-6)\|u\|_{L^{2^{\\#}}(M)}^{2}}\int_{M}\phi\left(P_{6}(g)u-\frac{n-6}{2}\bar{Q}^{\frac{n+6}{n-6}}\right)\mathrm{d}\mathrm{v}_{g}$
$\displaystyle=\frac{4}{(n-6)\|u\|_{L^{2^{\\#}}(M)}^{2}}\int_{M}\phi\left(\frac{n-6}{2}\bar{Q}u^{\frac{n+6}{n-6}}-\frac{n-6}{2}\bar{Q}u^{\frac{n+6}{n-6}}\right)\mathrm{d}\mathrm{v}_{g}$
$\displaystyle=0.$

From which, we conclude that $\bar{g}=u^{{4}/{(n-6)}}g$ is a critical point of
$\mathcal{Q}_{6}$.

The proof of the lemma is concluded. ∎

######  Remark 3.7.

By scale invariance, constant sixth order $Q$-curvature metrics can also be
characterized as follows

$\|u\|_{L^{2^{\\#}}(M)}=1.$

It is easy to recover the constant $Q$-curvature equation
($\mathcal{Q}_{6,g}$) as the Euler-Lagrange equation of this constrained
variational problem and compute the value of the $Q$-curvature of
$g=u^{{4}/{(n-6)}}g$ in terms of the Lagrange multiplier associated to the
critical point $u$.

###  3.2. Cylindrical coordinates

This section is devoted to constructing a change of variables that transforms
the local singular PDE ($\mathcal{P}_{6,R}$) problem into an ODE problem with
constant coefficients.

Let $\mathbb{S}^{n}\hookrightarrow\mathbb{R}^{n+1}$ denote the $n$-dimensional
sphere, which we equip with the standard round metric
$g_{0}=g_{\mathbb{S}^{n}}$ given by the pullback of the standard Euclidean
metric $\delta=g_{g_{\mathbb{R}^{n}}}$ under the stereographic projection
$\Pi$. For any $k\in\mathbb{R}$ with $0\leqslant k\leqslant n$, we seek
complete metrics on $\mathbb{S}^{n}\setminus\Lambda^{k}$ of the form
$g=\mathrm{u}^{{4}/{(n-6)}}g_{0}$, where either
$\Lambda^{k}\subset\mathbb{S}^{n}$ is a smooth $k$-dimensional submanifold
(singular case) or $\Lambda=\varnothing$ (non-singular case). In order to $g$
to be complete on $\mathbb{S}^{n}\setminus\Lambda^{k}$, one has to impose
$\liminf_{\mathrm{d}(p,\Lambda)}\mathrm{u}(p)=+\infty$. Furthermore, we
prescribe the resulting metric to have constant sixth order curvature, which
we normalize as in (1.16) to be $Q_{6}(g)=Q_{6}(g_{0})\equiv Q_{n}$. We focus
on the case $k=0$, and so $\Lambda^{0}:=\\{p_{1},\dots,p_{N}\\}$ for some
$N\in\mathbb{N}$. From now on, we assume $\Lambda^{0}:=\\{p,-p\\}$ with
$p=\mathbf{e}_{1}$ the north pole and denote $\Lambda^{0}=\Lambda$.

In more analytical terms, the condition that $g=\mathrm{u}^{{4}/{(n-6)}}g_{0}$
satisfies $Q_{6}(g)=Q_{n}$ on $\mathbb{S}^{n}\setminus\Lambda$ is equivalent
to the PDE

$P_{6}({g_{0}})\mathrm{u}=c_{n}\mathrm{u}^{\frac{n+6}{n-6}}\quad\mbox{on}\quad\mathbb{S}^{n}\setminus\Lambda,$
($\mathcal{Q}_{6,g_{0}}$)

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

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

In general, the GJMS operators are conformally covariant, and in particular if
$g=\mathrm{u}^{4/(n-6)}g_{0}$ is conformal to $g_{0}$ then

$P_{6}({g})\phi=\mathrm{u}^{-\frac{n+6}{n-6}}P_{6}({g_{0}})(\mathrm{u}\phi)\quad\mbox{for
all}\quad\phi\in\mathcal{C}^{\infty}(\mathbb{S}^{n}\setminus\Lambda).$ (3.10)

Again, we remark that the nonlinearity on the right-hand side of
($\mathcal{Q}_{6,g_{0}}$) has critical growth with respect to the Sobolev
embedding (3.1). It is well known that this embedding is not compact,
reflecting the conformal invariance of the PDE ($\mathcal{Q}_{6,g_{0}}$).

It will be convenient to transfer the PDE ($\mathcal{Q}_{6,g_{0}}$) to
Euclidean space, which we can do using the standard stereographic projection
(with the north pole in $\mathbb{S}^{n}\setminus\Lambda$, and in particular, a
non-singular point of any of the metrics we consider). After stereographic
projection, we can write

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

where $\delta$ is the Euclidean metric.

In these coordinates, we have
$g=\mathrm{u}^{{4}/{(n-6)}}g_{0}=(\mathrm{u}\cdot u_{\rm
sph})^{{4}/{(n-6)}}\delta$. Thus,
$u\in\mathcal{C}^{\infty}(\mathbb{R}^{n}\setminus\\{0\\})$ given by
$u=\mathrm{u}\cdot u_{\rm sph}$ is a positive singular solution to the
transformed equation

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

where $\Delta$ is the usual flat Laplacian and
$\Pi(\Lambda):=\\{0\\}\subset\mathbb{R}^{n}$ is the image of the singular set
$\Lambda\subset\mathbb{S}^{n}$ under the stereographic projection. As a
notational shorthand, we adopt the convention that $\mathrm{u}$ refers to a
conformal factor relating the metric $g$ to the round metric, i.e.
$g=\mathrm{u}^{{4}/{(n-6)}}g_{0}$, 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 $u=\mathrm{u}\cdot u_{\rm
sph}$. In this Euclidean setting, the transformation law (3.10) in particular
implies the scaling law for ($\mathcal{Q}_{6,\delta}$), namely if $u$ solves
($\mathcal{Q}_{6,\delta}$) then so does the rescaling

$u_{\lambda}(x):=\lambda^{\frac{n-6}{2}}u(\lambda x)$

for any $\lambda>0$.

######  Definition 3.8.

We define the sixth order, a logarithmic cylindrical change of variables as
$\mathfrak{F}:\mathcal{C}^{\infty}(\mathbb{B}_{R}^{*})\rightarrow\mathcal{C}^{\infty}(\mathcal{C}_{T})$
given by

$\mathfrak{F}(u)(t,\theta)=e^{\frac{6-n}{2}t}u(e^{-t}\theta)=v(t,\theta)\quad{\rm
with}\quad\theta=x/|x|.$ (3.11)

where $R>0$ and $T=-\ln R$ and
$\mathcal{C}_{T}=(T,+\infty)\times\mathbb{S}^{n-1}$. We also set its the
inverse transform
$\mathfrak{F}^{-1}:\mathcal{C}^{\infty}(\mathcal{C}_{T})\rightarrow\mathcal{C}^{\infty}(\mathbb{B}_{R}^{*})$
is given by

$\mathfrak{F}^{-1}(v)(x)=|x|^{\frac{6-n}{2}}v(-\ln|x|,\theta)=u(x).$

The exponent $\gamma_{n}:=\frac{n-6}{2}$ is sometimes called Fowler rescaling
exponent.

Using this change of variables, we arrive at the following sixth order
nonlinear PDE on the cylinder

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

Here $P_{\rm cyl}$ is the sixth order GJMS operator associated to the
cylindrical metric $g_{\rm cyl}=\mathrm{d}t^{2}+\mathrm{d}\theta^{2}$ on
$\mathcal{C}_{\infty}:=\mathbb{R}\times\mathbb{S}^{n-1}$, and it is given by

$\displaystyle P_{\rm cyl}:=P_{\rm rad}+P_{\rm ang},$ (3.12)

where

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

and

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

with

$\displaystyle K_{0}=2^{-8}(n-6)^{2}(n-2)^{2}(n+2)^{2}$ $\displaystyle
K_{2}=2^{-4}(3n^{4}-24n^{3}+72n^{2}-96n+304)$ $\displaystyle
K_{4}=2^{-2}(3n^{2}-12n+44)$ $\displaystyle
J_{0}=2^{-3}(3n^{4}-18n^{3}-192n^{2}+1864n-3952)$ (3.13) $\displaystyle
J_{1}=2^{-1}(3n^{3}+3n^{2}-244n+620)$ $\displaystyle J_{2}=2n^{2}+13n-68$
$\displaystyle J_{3}=2(n+1)$ $\displaystyle L_{0}=2^{-2}(3n^{2}-12n-20)$

being dimensional constants.

######  Remark 3.9.

The following decomposition holds

$P_{\rm rad}=L_{\lambda_{1}}\circ L_{\lambda_{2}}\circ L_{\lambda_{3}},$

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

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

For the proof, we refer the reader to [4, Proposition 2.7].

###  3.3. Delaunay metrics

This section presents some particular model metrics on the moduli space. Let
$\Lambda=\\{p_{1},p_{2}\\}\subset\mathbb{S}^{n}$, 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
$\mathrm{u}:\mathbb{S}^{n}\backslash\\{p_{1},p_{2}\\}\rightarrow(0,+\infty)$
determines a metric $g\in\mathcal{M}_{6,\Lambda}$ and after composing with a
stereographic projection it corresponds to a singular solution to
($\mathcal{P}_{6,\infty}$).

Let us start with 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 [51]. Later on, in [32], it is
proved that blow-up limit solutions exist. Finally, based on a topological
shooting method, the first and last authors recently classified all possible
solutions to this limit equation [4].

One can merge these classification results into the statement below

######  Proposition B.

Let $u\in\mathcal{C}^{6}(\mathbb{R}^{n}\setminus\\{0\\})$ 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

$u_{x_{0},\varepsilon}(x)=\left(\frac{2\varepsilon}{1+\varepsilon^{2}|x-x_{0}|^{2}}\right)^{\frac{n-6}{2}}.$
(3.14)

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
$\varepsilon_{0}\in(0,\varepsilon_{*}]$ and $T\in(0,T_{\varepsilon_{0}}]$ such
that

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

Here $\varepsilon_{*}=K_{0}^{(n-6)/6}$, $T_{\varepsilon}\in\mathbb{R}$ 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 $K_{4},K_{2},K_{0},\varepsilon_{*}>0$ are dimensional constants
$\varepsilon_{0}\in(0,\varepsilon_{*}]$ $($See (3.2) and (3.17)$)$. These are
called $($sixth order$)$ Emden–Fowler solutions.

Applying the cylindrical transform (3.11) to this PDE, in turn, yields

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

where we recall $\mathcal{C}_{\infty}:=\mathbb{R}\times\mathbb{S}^{n}$ is the
cylinder. Next, using that all solutions to ($\mathcal{Q}_{6,g_{0}}$) are
radially symmetric with respect to the origin, ($\mathcal{C}_{\infty}$)
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, one can quickly find two equilibrium solutions.
First, the cylindrical solution

$v_{\rm cyl}(t)\equiv\varepsilon_{*}>0,$ (3.15)

which is the only constant solution, where $\varepsilon_{*}>0$ is given by
(3.17). Second, the spherical solution

$v_{\rm sph}(t,\theta)=\cosh(t)^{\frac{6-n}{2}}.$ (3.16)

Undoing the cylindrical change of variables, we have

$u_{\rm sph}(x)=\left(\frac{1+|x|^{2}}{2}\right)^{\frac{6-n}{2}}$

which is non-singular at the origin and is a particular case of (3.14) with
$\varepsilon=1$ and $x_{0}=0$. Also, for the cylindrical solution, it follows

$u_{\rm cyl}(x)=\varepsilon_{*}|x|^{\frac{6-n}{2}},$

which is singular at the origin.

In this setting, Proposition B classifies all positive solutions
$v_{\varepsilon_{0}}\in\mathcal{C}^{6}(\mathbb{R})$ to
($\mathcal{O}_{6,\infty}$) in terms of the necksize
$\varepsilon_{0}\in(0,\varepsilon_{*}]$, where
$\varepsilon_{0}=\min_{\mathbb{R}}v\in(0,\varepsilon_{*}]$. Henceforth, we
keep $\varepsilon_{0}=\varepsilon$ for simplicity. Varying the parameter
$\varepsilon$ from its maximal value of $\varepsilon_{*}$ to $0$, we see that
the Delaunay solutions in Proposition B (b) interpolate between the
cylindrical solution $v_{\rm cyl}$ and the spherical solution $v_{\rm sph}$.
Therefore, we denote the minimal period of $v_{\varepsilon}$ by
$T_{\varepsilon}$. In other words, we have that entire solutions to
($\mathcal{O}_{6,\infty}$) are either: the constant cylindrical solution
$v_{\rm cyl}$, translates of the spherical solution $v_{\rm sph}$, or
translates of a Delaunay solution $v_{\varepsilon}$ for some
$\varepsilon\in(\varepsilon_{*},1)$.

######  Lemma B.

Let $v\in\mathcal{C}^{6}(\mathbb{R})$ be a solution to
($\mathcal{O}_{6,\infty}$). One of the following three alternatives holds:

  * (a)

$v\equiv\pm\varepsilon_{*}$ or $v\equiv 0$;

  * (b)

$v(t)=\pm c_{n}(2\cosh(t-T))^{\frac{6-n}{2}}$ for some $T\in\mathbb{R}$;

  * (c)

$v(t)=v_{\varepsilon}(t-T)$ for some $\varepsilon>0\in(0,\varepsilon_{*})$ on
is a periodic solution.

###### Proof.

See [4, Lemma 5.1]. ∎

######  Definition 3.10.

The conformal metrics

$g_{\rm sph}=u_{\rm sph}^{\frac{4}{n-6}}\delta\quad{\rm and}\quad g_{\rm
cyl}=u_{\rm cyl}^{\frac{4}{n-6}}\delta$

will be called spherical and cylindrical metrics, respectively. For each
$\varepsilon\in(0,\varepsilon_{*})$, the conformal metric given by

$g_{\varepsilon}=u_{\varepsilon}^{\frac{4}{n-6}}\delta$

will be called a Delaunay metric. The family of Delaunay metrics, denoted by
$\\{g_{\varepsilon}\\}_{\varepsilon\in(0,\varepsilon_{*})}$ interpolates
between the spherical metric $g_{0}=g_{\rm sph}$ and the cylindrical metric
$g_{\varepsilon_{*}}=g_{\rm cyl}$.

###  3.4. Hamiltonian energy

We now turn to a discussion of the existence and specific form of a family of
homological integral invariants of solutions of equation
($\mathcal{Q}_{6,g_{0}}$). Let us start with the classical definition of
Hamiltonian energy for a solution $v\in\mathcal{C}^{6}(\mathcal{C}_{\infty})$
to ($\mathcal{C}_{\infty}$).

######  Definition 3.11.

For any real function $v\in\mathcal{C}^{6}(\mathcal{C}_{\infty})$, let us
define its Hamiltonian energy $($with respect to ($\mathcal{C}_{\infty}$)$)$
$\mathcal{H}:\mathcal{C}_{\infty}\times\mathcal{C}^{6}(\mathcal{C}_{\infty})\rightarrow\mathbb{R}$
by

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

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.

With this definition in hands, it is direct to prove that this energy is
conserved along with solutions to ($\mathcal{O}_{6,\infty}$), that is, the
following identity holds

$\dfrac{\mathrm{d}}{\mathrm{d}t}\mathcal{H}(t,v)\equiv 0.$

In other terms, there exists $\mathcal{H}_{v}\in\mathbb{R}$ such that

$\mathcal{H}(v)(t)=\mathcal{H}(t,v)\equiv\mathcal{H}(v):=\mathcal{H}_{v}.$

######  Remark 3.12.

Notice that setting

$G(v):=F(v)-\frac{K_{0}}{2}v^{2}$

the real roots of $G$ are precisely the only equilibrium solutions to
($\mathcal{O}_{6,\infty}$), namely

$v\equiv 0\quad{\rm and}\quad v\equiv\pm\varepsilon_{*},$

where

$\varepsilon_{*}:=K_{0}^{\frac{n-6}{12}}=\left(\frac{(n-6)(n-2)(n+2)}{8}\right)^{\frac{n-6}{6}}.$
(3.17)

is a dimensional constant.

######  Remark 3.13.

By computing this energy on the cylindrical and spherical solutions, we find

$\mathcal{H}_{\rm sph}:=\mathcal{H}\left(v_{\rm sph}\right)=0\quad{\rm
and}\quad\mathcal{H}_{\rm cyl}:=\mathcal{H}\left(v_{\rm cyl}\right)<0.$

Furthermore, we see that the level set

$\\{\mathcal{H}\equiv
0\\}\cap\\{v^{(1)}=0\\}\cap\\{v^{(3)}=0\\}\cap\\{v^{(5)}=0\\}$

consists entirely of the (homoclinic) solution curve of the spherical solution
and the origin. Whereas, for each $H\in(0,-\mathcal{H}_{\rm cyl})$ the level
set

$\\{\mathcal{H}\equiv
H\\}\cap\\{v^{(1)}=0\\}\cap\\{v^{(3)}=0\\}\cap\\{v^{(5)}=0\\}$

is a closed curve associated to the Delaunay solution $v_{\varepsilon}$ for
some $\varepsilon\in(\varepsilon_{*},1)$.

Our last lemma will be referred to as the strong comparison result to reflect
that for two bounded solutions to be equal, they only need to coincide until
the first order at the origin. Now we prove that the Hamiltonian energy is a
parameter that orders bounded solutions in the $(v,v^{(1)})$-phase plane.

######  Lemma C.

Let $v_{1},v_{2}\in\mathcal{C}^{6}(\mathbb{R})$ be bounded solutions to
($\mathcal{O}_{6,\infty}$). Suppose that $v_{1}(0)=v_{2}(0)$. The following
holds:

  * (i)

If $v_{1}^{(1)}(0)>v_{2}^{(1)}(0)\geqslant 0$ or
$v_{1}^{(1)}(0)<v_{2}^{(1)}(0)\leqslant 0$, then
$\mathcal{H}(v_{1})>\mathcal{H}(v_{2})$.

  * (ii)

If $v_{1}^{(1)}(0)<v_{2}^{(1)}(0)\leqslant 0$ or
$v_{2}^{(1)}(0)>v_{1}^{(1)}(0)\geqslant 0$, then
$\mathcal{H}(v_{1})<\mathcal{H}(v_{2})$.

  * (iii)

If $v_{1}(0)=v_{2}(0)$, then $\mathcal{H}(v_{1})=\mathcal{H}(v_{2})$.

In particular, we see

$\lim_{\varepsilon\nearrow 1}\mathcal{H}(\varepsilon)=0,$

where $\mathcal{H}(\varepsilon):=\mathcal{H}(v_{\varepsilon})$ is the
Hamiltonian energy of the associated Emden–Fowler solution.

###### Proof.

See [4, Lemmas 4.20 and 4.21]. ∎

###  3.5. Linearized operator

Now we study the linearized operator around Delaunay solutions. The heuristics
are that when this operator is Fredholm, its indicial roots determine the rate
at which singular solutions to the nonlinear problem ($\mathcal{C}_{T}$)
converge to this limit solution near the isolated singularity. Here, we borrow
some ideas from [6, 37].

First, let us consider the following nonlinear operator $\mathcal{N}_{\rm
cyl}:H^{6}(\mathcal{C}_{T})\rightarrow L^{2}(\mathcal{C}_{T})$ given by

$\mathcal{N}_{\rm cyl}(v):=-P_{\rm cyl}v+c_{n}v^{\frac{n+6}{n-6}}.$

Then, we compute its linearization around Delaunay solutions classified above,
as follows

######  Lemma 3.14.

The linearization of $\mathcal{N}_{\rm cyl}:H^{6}(\mathcal{C}_{T})\rightarrow
L^{2}(\mathcal{C}_{T})$ around an Emden–Fowler solution $v_{\varepsilon,T}$ is
given by

$\mathscr{L}_{\rm cyl}[v_{\varepsilon}](\psi):=-P_{\rm
cyl}v+\hat{c}_{n}v_{\varepsilon}^{\frac{12}{n-6}}\psi,$ (3.18)

where $\hat{c}_{n}:=\frac{n+6}{n-6}c_{n}$. From now on, we simply denote
$\mathscr{L}_{\rm cyl}[v_{\varepsilon}](\psi):=\mathscr{L}_{\rm
cyl}^{\varepsilon}(\psi)$. Notice that $\mathscr{L}_{\rm
cyl}^{\varepsilon}(\psi)=\mathrm{d}\mathcal{N}_{\rm
cyl}[v_{\varepsilon}](\psi)$ is the Fréchet derivative of the nonlinear
functional $\mathcal{N}_{\rm cyl}$.

###### Proof.

By definition, we have

$\displaystyle\mathscr{L}_{\rm
cyl}[v_{\varepsilon}](\psi)=\frac{\mathrm{d}}{\mathrm{d}t}\Big{|}_{t=0}\mathcal{N}_{\rm
cyl}(v_{\varepsilon}+t\psi).$ (3.19)

Next, by routine computations, it follows

$\displaystyle\mathcal{N}_{\rm cyl}(v_{\varepsilon}+t\psi)-\mathcal{N}_{\rm
cyl}(v_{\varepsilon})$ $\displaystyle=-P_{\rm cyl}v_{\varepsilon}+tP_{\rm
cyl}\psi-(v_{\varepsilon}+t\psi)^{\frac{n+6}{n-6}}+P_{\rm
cyl}v_{\varepsilon}+v_{\varepsilon}^{\frac{n+6}{n-6}}$ $\displaystyle=tP_{\rm
cyl}\psi-(v_{\varepsilon}+t\psi)^{\frac{n+6}{n-6}}+v_{\varepsilon}^{\frac{n+6}{n-6}},$

which yields

$\displaystyle\mathcal{N}_{\rm cyl}(v_{\varepsilon}+t\psi)-\mathcal{N}_{\rm
cyl}(v_{\varepsilon})=tP_{\rm cyl}\psi-tc_{n}\left(-P_{\rm
cyl}v+\frac{n+6}{n-6}c_{n}v_{\varepsilon}^{\frac{12}{n-6}}\psi\right)+\mathcal{O}(t^{2})\quad{\rm
as}\quad t\rightarrow 0$

which by taking $t\rightarrow 0$ implies (3.19). ∎

Unfortunately, the linearized operator is not generally Fredholm since it has
no closed range [38, Theorem 5.40]. This issue is caused by nontrivial
elements on its kernel; these are called the Jacobi fields [6]. Therefore, we
must introduce suitable weighted Sobolev and Hölder spaces on which the
linearized operator has a well-defined right-inverse, up to a discrete set on
the complex plane. For more details, see [41, Section 2].

######  Definition 3.15.

Given $j,q\geqslant 1$ and $\beta\in\mathbb{R}$, for any $v\in
L_{\mathrm{loc}}^{q}(\mathcal{C}_{T})$ define the following weighted Lebesgue
norm

$\|v\|_{L_{\beta}^{q}(\mathcal{C}_{T})}^{q}=\int_{0}^{\infty}\int_{\mathbb{S}^{n-1}}e^{-2\beta
t}|v(t,\theta)|^{q}\mathrm{d}\theta\mathrm{d}t.$

Also, let us define the weighted Lebesgue space by

$L_{\beta}^{q}(\mathcal{C}_{T})=\left\\{v\in
L_{\mathrm{loc}}^{q}(\mathcal{C}_{T}):\|v\|_{L_{\beta}^{q}(\mathcal{C}_{T})}<\infty\right\\}.$

Similarly, consider the Sobolev spaces $W_{\beta}^{k,q}(\mathcal{C}_{T})$ of
functions with $k$ weak derivatives in $L^{q}$ having finite weighted norms.
Here we also denote the Hilbert spaces

$W^{j,2}_{\beta}(\mathcal{C}_{T})=H^{j}_{\beta}(\mathcal{C}_{T})\quad{\rm
and}\quad W^{j,q}(\mathcal{C}_{T})=W^{j,q}(\mathcal{C}_{T}).$

Notice that when $\beta=0$, we recover the classical Sobolev spaces.

######  Definition 3.16.

Given $j\geqslant 1$, $\beta\in\mathbb{R}$ and $\zeta\in(0,1)$, for any
$u\in\mathcal{C}_{\mathrm{loc}}^{0,\beta}(\mathcal{C}_{T})$ define the
following norm

$\|v\|_{\mathcal{C}_{\beta}^{0,\zeta}(\mathcal{C}_{T})}=\sup_{T>1}\sup\left\\{\frac{e^{-\beta
t_{1}}|v(t_{1},\theta_{1})|-e^{-\beta
t_{2}}|v(t_{2},\theta_{2})|}{\mathrm{d}_{\rm
cyl}((t_{1},\theta_{1}),(t_{2},\theta_{2}))^{\zeta}}:(t_{1},\theta_{1}),(t_{2},\theta_{2})\in\mathcal{C}_{T-1,T+1}\right\\}.$

For $m=0$, we weighted Holdër space by

$\mathcal{C}_{\beta}^{0,\zeta}(\mathcal{C}_{T})=\left\\{v\in\mathcal{C}_{\mathrm{loc}}^{0,\beta}(\mathcal{C}_{T}):\|v\|_{\mathcal{C}_{\mathrm{loc}}^{0,\beta}(\mathcal{C}_{T})}<\infty\right\\}.$

One can similarly define higher order weighted Holdër spaces
$\mathcal{C}_{\beta}^{m,\zeta}(\mathcal{C}_{T})$ for $m\geqslant 1$.

######  Definition 3.17.

The Jacobi fields in the kernel of $\mathscr{L}_{\rm
cyl}^{\varepsilon}:H^{6}_{\beta}(\mathcal{C}_{T})\rightarrow
L^{2}_{\beta}(\mathcal{C}_{T})$, are the solutions $\psi\in
H^{6}_{\beta}(\mathcal{C}_{T})$ of the following sixth order linear equation,

$\mathscr{L}_{\rm cyl}^{\varepsilon}(\psi)=0\quad{\rm
on}\quad\mathcal{C}^{n}_{T}.$

###  3.6. Fourier eigenmodes

We study the kernel of a linearized operator around a Delaunay solution by
decomposing into its Fourier eigenmodes, a separation of variables technique.

First, let us consider $\\{\lambda_{j},\chi_{j}(\theta)\\}_{j\in\mathbb{N}}$
the eigendecomposition of the Laplace–Beltrami operator on $\mathbb{S}^{n-1}$
with the normalized eigenfunctions,

$\Delta_{\theta}\chi_{j}(\theta)=-\lambda_{j}\chi_{j}(\theta).$ (3.20)

Here the eigenfunctions $\\{\chi_{j}(\theta)\\}_{j\in\mathbb{N}}$ are called
spherical harmonics with associated sequence of eigenvalues
$\\{\lambda_{j}\\}_{j\in\mathbb{N}}$ given by $\lambda_{j}=j(j+n-2)$ counted
with multiplicity $\mathfrak{m}_{j}$, which are defined by

$\mathfrak{m}_{0}=1\quad\mbox{and}\quad\mathfrak{m}_{j}=\frac{(2j+n-2)(j+n-3)!}{(n-2)!j!}.$

In particular, we have $\lambda_{0}=0,\quad\lambda_{1}=\dots=\lambda_{n}=n-1$,
$\lambda_{j}\geqslant 2n$, if $j>n$ and $\lambda_{j}\leqslant\lambda_{j+1}$.
Moreover, these eigenfunctions are the restrictions to $\mathbb{S}^{n-1}$ of
homogeneous harmonic polynomials in $\mathbb{R}^{n}$. Here we denote by
$V_{j}$ the eigenspace spanned by ${\chi_{j}(\theta)}$. Next, using (3.20), it
is easy to observe

$\Delta^{2}_{\theta}\chi_{j}(\theta)=\lambda_{j}^{2}\chi_{j}(\theta)\quad{\rm
and}\quad\Delta^{3}_{\theta}\chi_{j}(\theta)=-\lambda_{j}^{3}\chi_{j}(\theta).$
(3.21)

is the spectral data of $\Delta^{2}_{\theta}$ and $-\Delta^{3}_{\theta}$,
respectively.

Now, using the expression (3.12) in the formula (3.18), we get

$\displaystyle\mathscr{L}_{\rm cyl}^{\varepsilon}(\psi)$
$\displaystyle:=\psi^{(6)}-K_{4}\psi^{(4)}+K_{2}\psi^{(2)}-K_{0}\psi-\hat{c}_{n}v_{\varepsilon}^{\frac{12}{n-6}}\psi$
$\displaystyle+2\Delta_{\theta}\psi^{(4)}-J_{3}\Delta_{\theta}\psi^{(3)}+J_{2}\Delta_{\theta}\psi^{(2)}-J_{1}\Delta_{\theta}\psi^{(1)}+J_{0}\Delta_{\theta}\psi+3\Delta^{2}_{\theta}\psi^{(2)}-L_{0}\Delta^{2}_{\theta}\psi+\Delta^{3}_{\theta}\psi.$

It is convenient to rearrange the terms as follows

$\displaystyle\mathscr{L}_{\rm cyl}^{\varepsilon}(\psi)$
$\displaystyle:=\psi^{(6)}-K_{4}\psi^{(4)}+2\Delta_{\theta}\psi^{(4)}+K_{2}\psi^{(2)}+J_{2}\Delta_{\theta}\psi^{(2)}-J_{3}\Delta_{\theta}\psi^{(3)}+3\Delta^{2}_{\theta}\psi^{(2)}-J_{1}\Delta_{\theta}\psi^{(1)}$
$\displaystyle-K_{0}\psi+J_{0}\Delta_{\theta}\psi-
L_{0}\Delta^{2}_{\theta}\psi+\Delta^{3}_{\theta}\psi-\hat{c}_{n}v_{\varepsilon}^{\frac{12}{n-6}}\psi.$

Next, using (3.20) and (3.21), we can project this operator on the
eigenspaces, which gives us

$\displaystyle\mathscr{L}_{\rm cyl}^{\varepsilon,j}(\psi)$
$\displaystyle=\psi^{(6)}+(-K_{4}+2\lambda_{j})\psi^{(4)}+\lambda_{j}J_{3}\psi^{(3)}+(K_{2}+2J_{2}\lambda_{j})\psi^{(2)}+\lambda_{j}J_{1}\psi^{(1)}$
(3.22)
$\displaystyle+\left(-K_{0}+J_{0}\lambda_{j}-L_{0}\lambda_{j}^{2}+\lambda_{j}^{3}-\hat{c}_{n}v_{\varepsilon}^{\frac{12}{n-6}}\right)\psi\quad{\rm
for}\quad j\in\mathbb{N}.$

Moreover, for any $\phi\in L^{2}(\mathbb{S}^{n-1})$, we write

$\phi(t,\theta)=\sum_{j\in\mathbb{N}}\phi_{j}(t)\chi_{j}(\theta),\quad\mbox{where}\quad\phi_{j}(t)=\int_{\mathbb{S}^{n-1}}\phi(t,\theta)\chi_{j}(\theta)\mathrm{d}\theta.$

In other terms, $\phi_{j}$ is the projection of $\phi$ on the eigenspace
$V_{j}$. Thus, to understand the kernel of $\mathscr{L}_{\rm
cyl}^{\varepsilon}$, we consider the induced family of ODEs

$\mathscr{L}_{\rm cyl}^{\varepsilon,j}(\psi_{j})=0\quad{\rm for}\quad
j\in\mathbb{N}.$

###  3.7. Jacobi fields

Now we investigate the growth/decay rate in which the Jacobi fields on the
kernel of the linearized operator grow/decay. For this, we transform the sixth
order equation (3.22) into a first order equation on $\mathbb{R}^{6}$. More
precisely, defining
$\Psi=(\psi,\psi^{(1)},\psi^{(2)},\psi^{(3)},\psi^{(4)},\psi^{(5)})$, we
conclude that (3.22) is equivalent to the first order system

$\Psi^{\prime}_{j}(t)=N_{\varepsilon,j}(t)\Psi_{j}(t)\quad{\rm
in}\quad\mathbb{R}^{6}.$

Here

$N_{\varepsilon,j}(t)=\left[{\begin{array}[]{cccccc}0&1&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ 0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
0&B_{j}&C_{j}&D_{j}&E_{j}&F_{\varepsilon,j}(t)\\\ \end{array}}\right],$

where

$\displaystyle B_{j}$ $\displaystyle=-K_{4}+2\lambda_{j}$ $\displaystyle
C_{j}$ $\displaystyle=\lambda_{j}J_{3}$ $\displaystyle D_{j}$
$\displaystyle=K_{2}+2J_{2}\lambda_{j}$ $\displaystyle E_{j}$
$\displaystyle=\lambda_{j}J_{1}$ $\displaystyle F_{\varepsilon,j}(t)$
$\displaystyle=-K_{0}+J_{0}\lambda_{j}-L_{0}\lambda_{j}^{2}+\lambda_{j}^{3}-\hat{c}_{n}v_{\varepsilon}^{\frac{12}{n-6}}(t)$

are the coefficients of the projected Jacobi equation in (3.22).

Notice that as a consequence of Proposition B, one has that
$N_{\varepsilon,j}(t)$ is a $T_{\varepsilon}$-periodic matrix. Hence, the
monodromy matrix associated with this ODE system with periodic coefficients is
given by
$M_{\varepsilon,j}(t)=\exp{\int_{0}^{t}N_{\varepsilon,j}(\tau)\mathrm{d}\tau}$.
Finally, we define the Floquet exponents, denoted by
$\widetilde{\mathfrak{I}}_{j}^{\varepsilon}$, as the complex frequencies
associated to the eigenvectors of $M_{\varepsilon,j}(t)$, which forms a six-
dimensional basis for the kernel of $\mathscr{L}_{\rm cyl}^{\varepsilon,j}$.
Using Abel’s identity, we get that $N_{\varepsilon,j}(t)$ is constant, which
yields

$\det(M_{\varepsilon,j}(t))=\exp{\int_{0}^{T}\operatorname{tr}{N}_{\varepsilon,j}(\tau)\mathrm{d}\tau}=\exp\left(-{\int_{0}^{T}C_{\varepsilon,j}(\tau)\mathrm{d}\tau}\right)=1.$

Since $N_{\varepsilon,j}(t)$ has real coefficients, all its eigenvalues are
pairs of complex conjugates, and so

$\widetilde{\mathfrak{I}}_{j}^{\varepsilon}=\\{\pm\rho_{1,\varepsilon,j},\pm{\rho}_{2,\varepsilon,j},\pm{\rho}_{3,\varepsilon,j}\\}\subset\mathbb{C}\quad{\rm
for}\quad j\in\mathbb{N},$

where

$\rho_{\ell,\varepsilon,j}=\alpha_{\ell,\varepsilon,j}+i\beta_{\ell,\varepsilon,j}\quad{\rm
for}\quad\ell=1,2,3.$

Then, the set of indicial roots of $\mathscr{L}_{\rm cyl}^{\varepsilon,j}$ are
given by

$\mathfrak{I}_{j}^{\varepsilon}=\\{\pm\beta_{1,\varepsilon,j},\pm{\beta}_{2,\varepsilon,j},\pm{\beta}_{3,\varepsilon,j}\\}\subset\mathbb{R}\quad{\rm
for}\quad j\in\mathbb{N}.$

Moreover, for any $\psi\in\ker(\mathscr{L}_{\rm cyl}^{\varepsilon,j})$, we
have

$\psi=b^{+}_{1}\psi_{\varepsilon,j}^{1,+}+b_{1}^{-}\psi_{\varepsilon,j}^{-}+b_{2}^{+}\psi_{\varepsilon,j}^{2,+}+b_{2}^{-}\psi_{\varepsilon,j}^{2,-}+b_{3}^{+}\psi_{\varepsilon,j}^{3,+}+b_{3}^{-}\psi_{\varepsilon,j}^{3,-}\quad{\rm
for}\quad j\in\mathbb{N},$

where

$\psi^{1,\pm}_{\varepsilon,j}(t)=e^{\pm\rho_{1,\varepsilon,j}t},\quad{\psi}_{\varepsilon,j}^{2,\pm}(t)=e^{\pm{\rho}_{2,\varepsilon,j}t}\quad{\rm
and}\quad{\psi}_{\varepsilon,j}^{3,\pm}(t)=e^{\pm{\rho}_{3,\varepsilon,j}t}\quad{\rm
for}\quad j\in\mathbb{N}.$

Hence, the exponential decay/growth rate of the Jacobi fields is controlled by
quantity below

$\beta^{*}_{\varepsilon,j}:=\min\\{|\beta_{1,\varepsilon,j}|,|\beta_{2,\varepsilon,j}|,|\beta_{3,\varepsilon,j}|\\}.$

Therefore, the asymptotic properties of $\mathscr{L}_{\rm
cyl}^{\varepsilon,j}$ are obtained by the study of
$\mathfrak{I}^{\varepsilon}$.

Let us define indicial root function $\beta:(0,1)\rightarrow(0,+\infty)$ given
by $\beta(\varepsilon)=\beta^{*}_{\varepsilon,0}$ is the lowest indicial root
of

$\displaystyle\mathscr{L}_{\rm cyl}^{\varepsilon,0}(\psi)$
$\displaystyle=\psi^{(6)}-K_{4}\psi^{(4)}+K_{2}\psi^{(2)}+\left(-K_{0}+\hat{c}_{n}v_{\varepsilon}^{\frac{12}{n-6}}\right)\psi.$

Then, the (fundamental) period $T:(0,1)\rightarrow(0,+\infty)$ is given by

$T(\varepsilon)=\frac{2\pi}{\beta(\varepsilon)}.$

Let us keep the notation $\beta(\varepsilon)=\beta_{\varepsilon}$ and
$T(\varepsilon)=T_{\varepsilon}$.

The next lemma relates the fundamental period of the Delaunay solution with
the indicial roots of the linearized operator around this solution. From this,
we show that the period function blows up. We emphasize that this will be
enough for later performing our bifurcation technique.

######  Lemma 3.18.

The period function $T:(0,1)\rightarrow(0,+\infty)$ is an increasing and
satisfies

$\displaystyle\lim_{\varepsilon\nearrow 1}T(\varepsilon)=\infty\quad{\rm
and}\quad\lim_{\varepsilon\searrow 1}T(\varepsilon)=T_{\rm cyl}.$ (3.23)

Here $T_{\rm cyl}:=T(\varepsilon_{*})$ is the fundamental period of the
cylindrical solution given by

$T_{\rm cyl}=\frac{2\pi}{\beta_{\rm cyl}}$ (3.24)

where $\beta_{\rm cyl}\in\mathbb{R}$ is the largest indicial root associated
with the cylindrical solution. Moreover, one has

$\sup_{\varepsilon\in(0,1)}\|v_{\varepsilon}\|_{L^{\infty}(\mathbb{R})}\leqslant
1.$

###### Proof.

First, the monotonicity in (3.23) is a direct consequence of Proposition B and
Lemma C. In addition, $\beta_{\rm cyl}=\Im(\rho)\in\mathbb{R}$ is the greatest
imaginary part of complex solutions to the algebraic equation

$z^{6}-K_{4}z^{4}+K_{2}z^{2}+\left(-K_{0}+\hat{c}_{n}\varepsilon_{*}^{\frac{12}{n-6}}\right)=0\quad{\rm
in}\quad\mathbb{C}.$

Using the change of variables $\rho=z^{2}$, we arrive at

$\mathfrak{p}(\rho)=0\quad{\rm in}\quad\mathbb{C},$

where $\mathfrak{p}:\mathbb{C}\rightarrow\mathbb{C}$ is the indicial
polynomial given by

$\mathfrak{p}(\rho)=\rho^{3}-K_{4}\rho^{2}+K_{2}\rho+\left(-K_{0}+\hat{c}_{n}\varepsilon_{*}^{\frac{12}{n-6}}\right)=0\quad{\rm
in}\quad\mathbb{C}.$

Therefore, it is not hard to check that the last equation has one real
solution and two complex solutions, which are conjugate. Thus, we find

$\beta_{\rm
cyl}=\beta^{*}_{\varepsilon_{*},0}=\max_{\rho\in\mathfrak{p}^{-1}(\\{0\\})}\Im(\rho)<+\infty.$
(3.25)

The proof is then concluded. ∎

###  3.8. Bifurcation via Morse index

In this section, we prove Theorem 1.1. Notice that since
$(\mathbb{S}^{1}\times\mathbb{S}^{n-1},g_{\mathbb{S}^{1}}\oplus
g_{\mathbb{S}^{n-1}})$ is not Einstein, one cannot directly guarantee that
($\mathcal{G}_{+}$) is in force. However, this can be overcome using the lcf
property combined with ($\mathcal{QR}$).

###### Proof of Theorem 1.1.

Initially, since
$(\mathbb{S}^{1}\times\mathbb{S}^{n-1},g_{\mathbb{S}^{1}}\oplus
g_{\mathbb{S}^{n-1}})$ is locally conformally flat and
$R(g_{\mathbb{S}^{1}}\oplus g_{\mathbb{S}^{n-1}})\geqslant 0$, it follows from
Proposition A that ($\mathcal{A}$) holds as well.

Next, for each $T>0$, we consider its associated $T$-periodic Emden–Fowler
solution denoted by
$v_{T}:=v_{\varepsilon}\in\mathcal{C}^{6}(\mathcal{C}_{T})$ solving
($\mathcal{C}_{T}$) such that $\varepsilon:=v_{T}(0)\in(0,1)$ is the Delaunay
(or necksize) parameter. Thus, let us define a conformal metric on
$\overline{\mathcal{I}}_{T}\times\mathbb{S}^{n-1}$ with
$\overline{\mathcal{I}}\subset\mathbb{R}$ denoting the closed interval
$\overline{\mathcal{I}}_{T}:=[-T/2,T/2]$ of the form
$g_{v_{\varepsilon}}=v_{\varepsilon}^{{4}/{(n-6)}}g_{\rm cyl}$, where we used
the identification

$\mathbb{S}_{T}\times\mathbb{S}^{n-1}\simeq\overline{\mathcal{I}}_{T}\times\mathbb{S}^{n-1}:=\mathcal{C}^{n}_{T}.$
(3.26)

Now notice that since $T\rightarrow+\infty$ as $\varepsilon\rightarrow 0$, one
has $v_{T}\rightarrow v_{\rm cyl}$, where $v_{\rm cyl}$ is the cylindrical
solution given by (3.15). Hence, one has that $g_{T}\rightarrow g_{\infty}$ as
$T\rightarrow+\infty$, where $g_{\infty}=g_{\rm cyl}$ is the cylindrical
metric, which is such that

$Q_{2}({g_{\rm cyl}})\equiv(n-1)(n-2),\quad Q_{4}({g_{\rm
cyl}})\equiv\frac{(n-1)(n^{2}-2n-2)}{8},\quad{\rm and}\quad Q_{6}({g_{\rm
cyl}})\equiv Q_{n},$

where we recall that $Q_{n}>0$ is given by (1.16). Hence, applying
($\mathcal{A}$), we get

$\mathcal{Y}_{6}^{+}(\mathbb{S}_{T}^{1}\times\mathbb{S}^{n-1},g_{\rm
cyl})=\mathcal{Y}_{6}(\mathbb{S}_{T}^{1}\times\mathbb{S}^{n-1},g_{\rm
cyl})<\mathcal{Y}_{6}^{+}(\mathbb{S}^{n},g_{0}).$

Furthermore, we recall that from Lemma 3.6, we know that each critical point
of $\mathcal{Q}_{6}$ in the conformal class $[g_{\rm cyl}]$ must be a constant
$Q$-curvature metric on $\mathbb{S}_{T}^{1}\times\mathbb{S}^{n-1}$. We pull
this constant $Q$-curvature metric on
$\mathbb{S}_{T}^{1}\times\mathbb{S}^{n-1}$ back to its universal cover
$\mathbb{R}\times\mathbb{S}^{n-1}:=\mathcal{C}^{n}_{\infty}$, obtaining a
smooth, positive $T$-periodic function
$v_{T}:\mathcal{C}_{\infty}\rightarrow(0,\infty)$ satisfying
($\mathcal{O}_{6,\infty}$). From Lemma B, it holds that one the three
alternatives hold: either the constant cylindrical solution $v_{\rm cyl}$,
translates of the spherical solution $v_{\rm sph}$, or translates of a
Delaunay solution $v_{\varepsilon}$ for some
$\varepsilon\in(\varepsilon_{*},1)$.

For any $T>0$, let us denote by $\mu(T)\in\mathbb{N}$, the number of
$Q$-curvature metrics in the conformal class $[g_{\rm cyl}]$ on
$\mathbb{S}_{T}^{1}\times\mathbb{S}^{n-1}$. Next, we express this number in
terms of the period $T\in(0,1)$.

Claim 1: It holds that $\mu(T):=\lfloor\frac{T}{T_{\rm cyl}}\rfloor:=\kappa$.

We proceed by induction in the following way. As before, we have the
normalization $Q_{6}(g)=Q_{n}$. The cylindrical solution $v_{\rm cyl}$ is the
only solution when $0<T\leqslant T_{\rm cyl}$, where $T_{\rm cyl}$ is given in
(3.24). For $T_{\rm cyl}<T\leqslant 2T_{\rm cyl}$ we have two constant
$Q$-curvature metrics, namely the cylinder and the Delaunay metric with
Delaunay parameter $\varepsilon\in(0,1)$ such that $T=T_{\varepsilon}$. When
$2T_{\rm cyl}<T\leqslant 3T_{\rm cyl}$ we obtain 3 constant $Q$-curvature
metrics, namely the cylindrical solution $v_{\rm cyl}$, the Delaunay solution
$v_{\varepsilon}$ such that $T_{\varepsilon}=T$, and the Delaunay solution
$v_{\alpha}$ such that $T_{\alpha}=T/2$. Continuing inductively, when
$(\kappa-1)T_{\rm cyl}<T\leqslant\kappa T_{\rm cyl}$, one can find
$\kappa\in\mathbb{N}$ distinct constant sixth order $Q$-curvature metrics.
Namely the cylindrical solution $v_{\rm cyl}$ together with the Delaunay
solution $v_{\varepsilon_{\ell}}$ with $T_{\varepsilon_{\ell}}=T/\ell$ for
each $\ell\in\\{1,2,\ldots,\kappa-1\\}$, which proves the first claim.

Next, we classify the blow-up behavior of the family above.

Claim 2: It follows

$\lim_{\varepsilon\nearrow+\infty}\|v_{\varepsilon}-v_{\rm
sph}\|_{\mathcal{C}^{\infty}_{\rm loc}(\mathbb{R})}=0$ (3.27)

uniformly on compact subsets, where $v_{\rm sph}$ is the spherical solution
given by (3.16).

In fact, for each $T>T_{\rm cyl}$ the Delaunay solution $v_{\varepsilon}$ such
that $T_{\varepsilon}=T$ are such that $v_{\varepsilon}(0)=\varepsilon$ and
${v}^{(1)}_{\varepsilon}(0)=0$. Thus, by Lemma C, these two initial conditions
uniquely determine a solution to ($\mathcal{O}_{6,\infty}$). Hence, from Lemma
3.18, we know that $\lim_{\varepsilon\nearrow 1}T_{\varepsilon}=+\infty$,
which implies the convergence in (3.27). Finally, since the estimate
$\|v_{\varepsilon}\|_{\mathcal{C}^{\infty}(\mathbb{R})}<1$ holds for all
$\varepsilon>0$, this convergence is uniform on compact subsets by the
Arzela–Ascoli theorem. This proves the second claim.

In conclusion, we prove the stability statement below.

Claim 3: $v_{\varepsilon_{1}}$ is the only stable critical point of
$\mathcal{Q}_{6}$ among $\\{v_{\rm
cyl},v_{\varepsilon_{1}},v_{\varepsilon_{2}},\ldots,v_{\varepsilon_{\ell-1}}\\}$.

Indeed, the function
$w_{\varepsilon_{\ell}}=\partial_{t}{v}_{\varepsilon_{\ell}}$ satisfies
$\mathscr{L}_{\varepsilon_{\ell}}(w_{\varepsilon_{\ell}})=0$ in $\mathbb{R}$,
where $\mathscr{L}_{\varepsilon_{\ell}}$ is the linearization of
($\mathcal{O}_{6,\infty}$) about $v_{\varepsilon_{\ell}}$. Furthermore,
observe that

$\left\\{t\in\overline{\mathcal{I}}_{T}:w_{\varepsilon_{\ell}}>0\right\\}=\bigcup_{j=-\lfloor\ell/2\rfloor}^{\lfloor\ell/2\rfloor}\left(jT_{\varepsilon_{\ell}},\left(\frac{2j+1}{2}\right)T_{\varepsilon_{\ell}}\right),$

where $\lfloor\ell/2\rfloor$ denotes the greatest non-negative integer less
than or equal to $\ell/2$. When $\ell\geqslant 2$ the number of nodal domains
combined with Strum-Liouville theory implies
$-\mathscr{L}_{\varepsilon_{\ell}}:\mathcal{C}^{6}(\mathbb{R})\rightarrow\mathcal{C}^{0}(\mathbb{R})$
has at least $\ell$ negative eigenvalues, and so
$v_{{\varepsilon_{\ell}}}\in\mathcal{C}^{6}(\mathbb{R})$ cannot be a stable
critical point of $\mathcal{Q}_{6}$. Furthermore the function
$w_{0}\in\mathcal{C}^{6}(\mathbb{R})$ defined as $w_{0}(t)=\cos(\mu_{\rm
cyl}t)$, where $\mu_{\rm cyl}>0$ is given by (3.25), satisfies
$\mathscr{L}_{\rm cyl}(w_{0})=0$, where $\mathscr{L}_{\rm
cyl}:\mathcal{C}^{6}(\mathbb{R})\rightarrow\mathcal{C}^{0}(\mathbb{R})$ is the
linearization of ($\mathcal{O}_{6,\infty}$) around $v_{\rm cyl}$. When
$T>2T_{\rm cyl}$ the function $w_{0}$ has at least 2 disjoint regions on which
it is positive, so $v_{\rm cyl}$ cannot be a stable critical point of
$\mathcal{Q}_{6}$ for $T\gg 1$ large. This proves Claim 2.

From Claims 2 and 3, we conclude that
$v_{\varepsilon}\in\mathcal{C}^{\infty}(\mathbb{S}_{T_{\varepsilon}}^{1}\times\mathbb{S}^{n-1})$
minimizes $\mathcal{Q}_{6}$ over the conformal class $[g_{\rm cyl}]$, which
yields

$\displaystyle\mathcal{Y}_{6}^{+}(\mathbb{S}_{T_{\varepsilon}}^{1}\times\mathbb{S}^{n-1},g_{\rm
cyl})$
$\displaystyle=\mathcal{Q}_{6}(g_{v_{\varepsilon}})=\frac{\frac{2}{n-6}\int_{\mathbb{S}^{1}\times\mathbb{S}^{n-1}}Q_{6}({g_{v_{\varepsilon}}})\mathrm{d}\mathrm{v}_{g_{v_{\varepsilon}}}}{\operatorname{vol}_{g_{v_{\varepsilon}}}(\mathbb{S}^{1}\times\mathbb{S}^{n-1})^{\frac{n-6}{n}}}$
$\displaystyle=\frac{2Q_{n}}{n-6}\frac{\operatorname{vol}_{g_{v_{\varepsilon}}}(\mathbb{S}^{1}\times\mathbb{S}^{n-1})}{\operatorname{vol}_{g_{v_{\varepsilon}}}\left(\mathbb{S}^{1}\times\mathbb{S}^{n-1}\right)^{\frac{n-6}{n}}}$
$\displaystyle=\frac{2Q_{n}}{n-6}\frac{\int_{\mathcal{I}_{T_{\varepsilon}}}\int_{\mathbb{S}^{n-1}}v^{\frac{2n}{n-6}}\mathrm{d}\theta\mathrm{d}t}{\left(\int_{\mathcal{I}_{T_{\varepsilon}}}\int_{\mathbb{S}^{n-1}}v^{\frac{2n}{n-6}}\mathrm{d}\theta\mathrm{d}t\right)^{\frac{n-6}{n}}}$
$\displaystyle=\frac{2Q_{n}\omega_{n-1}^{n/6}}{n-6}\left(\int_{\mathcal{I}_{T_{\varepsilon}}}v^{\frac{2n}{n-6}}\mathrm{d}t\right)^{n/6},$

where $\omega_{n-1}=|\mathbb{S}^{n-1}|$ denotes the $n$-dimensional Lebesgue
measure of the standard Euclidean sphere. Finally, we let $\varepsilon\nearrow
1$ in the last identity to see

$\displaystyle\mathbb{Y}_{6}^{+}\left(\mathbb{S}^{1}\times\mathbb{S}^{n-1}\right)$
$\displaystyle\geqslant\lim_{\varepsilon\nearrow
1}\mathcal{Y}_{6}^{+}(\mathbb{S}_{T_{\varepsilon}}^{1}\times\mathbb{S}^{n-1},g_{\rm
cyl})$
$\displaystyle=\frac{2Q_{n}\omega_{n-1}^{n/6}}{n-6}\left(\int_{-\infty}^{+\infty}v_{\rm
sph}(t)^{\frac{2n}{n-6}}\mathrm{d}t\right)^{n/6}$
$\displaystyle=\frac{2Q_{n}\omega_{n-1}^{n/6}}{n-6}\left(\int_{-\infty}^{+\infty}\cosh(t)^{-n}\mathrm{d}t\right)^{n/6}$
$\displaystyle=\frac{2Q_{n}\omega_{n-1}^{n/6}}{n-6}\left(\int_{0}^{+\infty}\left(\frac{1+r^{2}}{2}\right)^{-n}r^{n-1}\mathrm{d}r\right)^{n/6}$
$\displaystyle=\mathcal{Q}_{6}(g_{0})$
$\displaystyle=\mathcal{Y}_{6}^{+}(\mathbb{S}^{n},g_{0})$
$\displaystyle=\mathbb{Y}_{6}^{+}(\mathbb{S}^{n}),$

where $r=e^{-t}$. This, in turn, completes our first main result. ∎

##  4\. The noncompact case

This section is based on a topological argument from [10] to prove Theorem 1.3
as its corollaries.

###  4.1. Riemannian coverings

Notice that in this case the variational formulations as in Lemma 3.6 are no
longer available. To circumvent this issue and establish nonuniqueness results
in this context, we combine infinite towers of coverings with the existence of
a positive Green function ($\mathcal{G}_{+}$) and the reversed Aubin
inequality ($\mathcal{A}^{\prime}$).

###### Proof of Theorem 1.3.

From the assumption on the group of deck transformations, we can find an
infinite tower of Riemannian coverings satisfying

$(\widetilde{M}^{n}_{\infty},\tilde{g}_{\infty})\longrightarrow\cdots\longrightarrow(\widetilde{M}^{n}_{\ell},\tilde{g}_{\ell})\longrightarrow\cdots\longrightarrow(\widetilde{M}^{n}_{2},\tilde{g}_{2})\longrightarrow(\widetilde{M}^{n}_{1},\tilde{g}_{1})\longrightarrow(\widetilde{M}^{n}_{0},\tilde{g}_{0}),$

where $\widetilde{M}^{n}_{\ell}\rightarrow\widetilde{M}^{n}_{0}$ is an
$d_{\ell}$-sheeted covering and
$(\widetilde{M}^{n}_{0},\tilde{g}_{0})=(M^{n},g)$, with $d_{\ell}\geqslant 2$
and $d_{\ell}\nearrow+\infty$ as $\ell\rightarrow+\infty$. Since
$(\widetilde{M}^{n}_{\ell},g_{\ell})$ are locally isometric to
$(\widetilde{M}^{n}_{0},g_{0})$. Using the same argument as in [1, Lemma 3.6],
it is not hard to check that $(\widetilde{M}^{n}_{\ell},\tilde{g}_{\ell})$
also satisfies ($\mathcal{G}_{+}$) and ($\mathcal{A}^{\prime}$), which gives
us

${\Theta}_{6}(\widetilde{M}_{\ell},\tilde{g}_{\ell})=\frac{\int_{\widetilde{M}_{\ell}}Q_{6}(\tilde{g}_{\ell})\mathrm{d}\mathrm{v}_{g}}{\|Q_{6}(\tilde{g}_{\ell})\|_{L^{\frac{2n}{n+6}}(\widetilde{M}_{\ell})}}$

attains its maximum in each conformal class $[\tilde{g}_{\ell}]$ at some
metric $\bar{g}_{\ell}\in[\tilde{g}_{\ell}]$ for all $\ell\in\mathbb{N}$.
Denote by $\tilde{h}_{\ell}$ the pullback to $\widetilde{M}_{\ell}$ of the
metric $\bar{g}_{1}$, and note that $\tilde{h}_{\ell}\in[\tilde{g}_{\ell}]$.

Using that $Q_{1}=Q_{6}({\tilde{g}_{1}})>0$ is a positive constant and
$\tilde{h}_{\ell}$ is locally isometric to $\tilde{g}_{1}$, we get that
$Q_{6}({\tilde{g}_{\ell}})=Q_{1}$ for all $\ell\geqslant 1$, which yields

$\displaystyle{\Theta}_{6}(\widetilde{M}_{\ell},\tilde{g}_{\ell})=\frac{\int_{\widetilde{M}_{\ell}}Q_{6}({\tilde{h}_{\ell}})\mathrm{d}\mathrm{v}_{\tilde{h}_{\ell}}}{\|Q_{6}({\tilde{h}_{\ell}})\|^{2}_{L^{\frac{2n}{n+6}}(M)}}=Q_{1}^{-1}{\rm
vol}_{\tilde{g}_{1}}(\widetilde{M}_{1})^{-6/n}\left(\frac{d_{\ell}}{d_{1}}\right)^{-6/n}\searrow
0\quad{\rm as}\quad\ell\rightarrow+\infty.$

From this, one can also see

$\frac{2}{n-4}{\Theta}_{6}(\widetilde{M}_{\ell},\tilde{h}_{\ell})<\Theta_{6}(\mathbb{S}^{n},g_{0}),$

if $\ell\geqslant\ell_{0}$ for some $\ell_{0}\gg 1$ sufficiently large. Hence,
$h_{\ell_{0}}$ is not a maximizer of $\Theta_{6}$ in $[\tilde{g}_{\ell_{0}}]$.
By ($\mathcal{A}^{\prime}$), there exists a maximizer of $\Theta_{6}$ in
$[\tilde{h}_{\ell_{0}}]$, which is not homothetic to $[\tilde{h}_{\ell_{0}}]$.
The pullbacks to $\widetilde{M}^{n}_{\infty}$ of this maximizer and
$\tilde{h}_{\ell_{0}}$ are two nonhomothetic complete metrics with constant
positive $Q$-curvature in $[\tilde{g}_{\infty}]$. Now, we can do the same
procedure replacing $(\widetilde{M}^{n}_{1},\tilde{g}_{1})$ with
$(\widetilde{M}^{n}_{\ell_{0}},\tilde{g}_{\ell_{0}})$. Therefore, by
interactively repeating this process, the theorem is proved. ∎

######  Remark 4.1.

For example, as a consequence of Selberg–Malcev Lemma [45, Section 7.5], any
hyperbolic manifold has an infinite profinite completion due to the hence it
admits finite-sheeted Riemannian coverings by hyperbolic manifolds of
arbitrarily large volume. This fact also holds for symmetric spaces of
hyperbolic or Euclidean type in general. In this regard, the second summand
$(\mathcal{S}^{n_{2}},g_{\mathcal{S}})$ represents either the standard
Euclidean space $(\mathbb{R}^{n},g_{\mathbb{R}^{n}})$ or Hyperbolic space
$(\mathbb{H}^{n},g_{\mathbb{H}^{n}})$.

###  4.2. Bifurcation via infinite tower of Riemannian coverings

Here we apply the last proposition to prove the second main result of the
manuscript:

###### Proof of Corollary 1.4.

Initially, using a classical result from [12], it follows that every symmetric
space $\mathcal{S}^{n_{2}}$ of Euclidean type admits irreducible compact
quotients $\Sigma=\mathcal{S}^{n_{2}}/\Gamma$, and the same is valid for the
Euclidean space. Fixing, such a compact quotient $(\Sigma^{n_{2}},g_{\Sigma})$
with the induced locally symmetric metric. Since $\pi_{1}(\Sigma)$ is infinite
and residually finite, it has infinite profinite completion. Hence, it follows
that
$(M^{n}_{\infty},g_{\infty})=(C^{n_{1}}\times\mathcal{S}^{n_{2}},g_{C}\oplus
g_{\mathcal{S}})$, and
$(M^{n}_{0},g_{0})=(C^{n_{1}}\times\Sigma^{n_{2}},g_{C}\oplus g_{\Sigma}$).
Under the assumptions ($\mathcal{G}_{+}$) and ($\mathcal{A}^{\prime}$), the
proof is finished as a direct application of Theorem 1.3. ∎

###### Proof of Corollary 1.5.

First, we recall the conformal equivalence below

$\mathbb{S}^{n}\setminus\mathbb{S}^{k}\simeq\mathbb{S}^{n-k-1}\times\mathbb{H}^{k+1}\simeq\mathbb{S}^{n_{1}}\times\mathbb{H}^{n_{2}}=\widetilde{(\mathbb{S}^{n_{1}}\times\mathbb{M}^{n_{2}}(T))}:=\widetilde{M}^{n}_{T}$

where $n_{1}=n-k+1$, $n_{2}=k+1$ and
$\mathbb{M}^{n_{2}}(T)=\mathbb{H}^{n_{2}}\backslash\Gamma(T)$ being the
quotient manifold under the periodic Kleinian group
$\Gamma(T)\subset\mathbb{H}^{n_{2}+1}$ arising as holonomy representation of
the fundamental group $\pi_{1}(M)$ with
$(M^{n},g)=(\mathbb{S}^{n_{1}}\times\mathbb{H}^{n_{2}},g_{1}\oplus g_{2})$.
Second, using the locally conformally flat condition together with
$R(g)\geqslant 0$ and ($\mathcal{QR}$), one can apply Proposition A to
conclude that ($\mathcal{A}$) is also true, which combined with
($\mathcal{G}_{+}$) is enough to prove ($\mathcal{A}^{\prime}$). Finally, we
finish the proof by direct applying Corollary 1.4. In this case, the family
$\\{\widetilde{M}^{n}_{T}\\}_{T>0}$ is a tower of finite-sheeted regular
coverings with arbitrarily large volume. ∎

######  Remark 4.2.

Finally, let us come back to the case $k=0$, for which one has the following
conformal equivalence
$\mathbb{S}^{n}\setminus\mathbb{S}^{0}\simeq\mathbb{R}^{n}\setminus\\{0\\}\simeq\mathbb{S}^{n-1}\times\mathbb{R}$
with the identification $\mathbb{S}^{0}:=\\{p,-p\\}$ with
$p=\mathbf{e}_{1}\in\mathbb{S}^{n}$ the north pole. Notice that the proof
above can be adapted mutatis mutandis, using the compact quotients
$M^{n}_{T}=\mathbb{S}^{1}(T)\times\mathbb{S}^{n-1}$ as in (3.26), where
$\mathbb{S}^{1}(T)=\mathbb{R}/T\mathbb{Z}$, which as above form a similar
tower of finite-sheeted regular coverings with arbitrarily large volume.
Therefore, our technique produces minimizing solutions at each level that lift
to pairwise nonhomothetic periodic solutions in the universal covering
$\widetilde{M}^{n}_{T}=\mathbb{S}^{n-1}\times\mathbb{R}=\mathcal{C}^{n}_{\infty}$.
This, in turn, shows that Corollary 1.4 is indeed an extension of Theorem 1.1
to a much broader situation.

######  Remark 4.3.

Using this covering approach and motivated by the seminal work [49], it would
be interesting to understand under which conditions on the manifold
$(M^{n},g)$ the equality holds $\mathcal{M}_{6}(g_{C}\oplus
g_{\mathcal{S}})=\mathcal{M}_{6,\Lambda}(g)$, where $\Lambda^{k}\subset M^{n}$
is $k$-dimensional submanifold and we recall that
$g_{k}=g_{\mathbb{S}^{n}\setminus\mathbb{S}^{k}}$ is the standard round metric
singular along an equatorial subsphere. For instance, when $M$ descends to a
quotient $M/\Gamma$ for some finite subgroup $\Gamma\subset{\rm iso}(M)$. The
most simple example is the locally conformally flat manifold
$(M^{n},g)\simeq(\mathbb{S}^{n},g_{0})$ and
$\Lambda:=\mathbb{S}^{n}\setminus\mathscr{D}(\widetilde{M})$ is a Kleinian
group arising as the image of the developing map
$\mathscr{D}:\widetilde{M}\hookrightarrow\mathbb{S}^{n}$, which by definition
is conformal and injective. This class of manifolds recovers the example given
in Remark 4.2.

Acknowledgments. This paper was finished when the first-named author held a
Post-doctoral position at the University of British Columbia, whose
hospitality he would like to acknowledge. This project started when the first
and second named authors visited the College University of New York, which
they are thankful to R. G. Bettiol for the invitation. We would like to thank
J. Ratzkin for reading the final manuscript and his compelling contributions.
The authors warmly thank J. S. Case for his several remarks after the first
version of this preprint was on Arxiv.

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