# Asymptotics for positive singular solutions to subcritical sixth order
equations

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

We classify the local asymptotic behavior of positive singular solutions to a
class of subcritical sixth order equations on the punctured ball. Initially,
using a version of the integral moving spheres technique, we prove that
solutions are asymptotically radially symmetric solutions with respect to the
origin. We divide our approach into some cases concerning the growth of
nonlinearity. In general, we use an Emden–Fowler change of variables to
translate our problem to a cylinder. In the lower critical regime, this is not
enough, thus, we need to introduce a new notion of change of variables. The
difficulty is that the cylindrical PDE in this coordinate system is
nonautonomous. Nonetheless, we define an associated nonautonomous Pohozaev
functional, which can be proved to be asymptotically monotone. In addition, we
show a priori estimates for these two functionals, from which we extract
compactness properties. With this ingredients, we can perform an asymptotic
analysis technique to prove our main result.

###### Key words and phrases:

Tri-Laplacian, Lower critical exponent, Sixth order equation, Local asymptotic
behavior, Emden–Fowler solutions

###### 2020 Mathematics Subject Classification:

35J60, 35B09, 35J30, 35B40

This research is partially supported by São Paulo Research Foundation (FAPESP)
#2020/07566-3 and #2021/15139-0 and Natural Sciences and Engineering Research
Council of Canada (NSERC)

###### Contents

  1. 1 Introduction
  2. 2 Emden–Fowler coordinates
    1. 2.1 Autonomous case
    2. 2.2 Non-autonomous case
  3. 3 Pohozaev functionals
    1. 3.1 Autonomous case
    2. 3.2 Non-autonomous case
  4. 4 A priori upper bounds
  5. 5 Asymptotic radial symmetry
    1. 5.1 Kelvin transform
    2. 5.2 Integral representation
    3. 5.3 Sliding technique
  6. 6 The limiting Pohozaev levels
  7. 7 Local asymptotic behavior
    1. 7.1 Serrin–Lions case
    2. 7.2 Gidas–Spruck case
    3. 7.3 Aviles case
  8. A Tri-Laplacian coefficients
    1. A.1 Polar coordinates
    2. A.2 Autonomous case
    3. A.3 Nonautonomous case
    4. A.4 Upper critical case
    5. A.5 Lower critical case

##  1\. Introduction

We study (classical) positive singular solutions $u\in
C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$ with $n\geqslant 7$ (which will always
be assumed so forth) to the following family of subcritical sixth order PDEs

$\displaystyle(-\Delta)^{3}u=f_{p}(u)\quad{\rm in}\quad B^{*}_{R}.$
($\mathcal{P}_{6,p,R}$)

Here $B_{R}\setminus\\{0\\}\subset\mathbb{R}^{n}$ is the punctured ball of
radius $R>0$, $\Delta^{3}=\Delta\circ\Delta\circ\Delta$ is the tri-Laplacian,
and the nonlinearity $f\in C^{1}(B_{R})$ is given by

$f_{p}(u):=|u|^{p-1}u\quad{\rm with}\quad
p\in(1,2_{\\#}]\cup(2_{\\#},2^{\\#}-1),$

where $2_{\\#}:=\frac{n}{n-6}$ and $2^{\\#}:=\frac{2n}{n-6}$ are,
respectively, the lower and upper critical exponents in the sense of the
compact Sobolev embedding of $H^{3}(\mathbb{R}^{n})$.

We say that a positive solution $u\in C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$
has a removable singularity at the origin if $\lim_{x\rightarrow
0}u(x)<+\infty$, that is, it can be continuously extended to the origin;
otherwise, we say that it is a non-removable singularity. These are called
non-singular and singular solutions, respectively.

Let us mention that the S. Luo et al. [24, 25] studied homogeneous solutions
to ($\mathcal{P}_{6,p,R}$) in the blow-limit case $R=+\infty$, that is,

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

On this subject, they proved the following classification result

######  Theorem A.

Let $u\in C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$ be a positive singular
solution to ($\mathcal{P}_{6,p,\infty}$) with $p\in(1,2^{\\#}-1)$. Assume that
$u$ is homogeneous of degree $-\gamma_{p}$, where $\gamma_{p}:=\frac{6}{p-1}$.

  * (a)

If $p\in(1,2_{\\#}]$, then $u\equiv 0$.

  * (b)

If $p=(2_{\\#},2^{\\#}-1)$, then

$u(x)={K}_{0}(n,p)^{\frac{1}{p-1}}|x|^{-\gamma_{p}},$ (1)

where

${K}_{0}(n,p)=\gamma_{p}\left(\gamma_{p}+2\right)\left(\gamma_{p}+4\right)\left(n-2-\gamma_{p}\right)\left(n-4-\gamma_{p}\right)\left(n-6-\gamma_{p}\right).$

In the light of the last theorem, it makes sense to divide our analysis into
three cases, namely, the Serrin–Lions case $p\in(1,2_{\\#})$, Aviles case
$p=2_{\\#}$, and the Gidas–Spruck case $p\in(2_{\\#},2^{\\#}-1)$. Our main
result in this manuscript classifies the local behavior of positive solutions
to ($\mathcal{P}_{6,p,R}$) in these situations.

######  Theorem 1.

Let $u\in C^{6}(\mathbb{R}^{n}\\{0\\})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$). Assume that $-\Delta u\geqslant 0$ and
$\Delta^{2}u\geqslant 0$. Then, it follows

$u(x)=(1+\mathcal{O}(|x|))\overline{u}(x)\quad{\rm as}\quad x\rightarrow 0,$

where $\overline{u}(r)=\mathop{\ooalign{$\int$\cr$-$}}_{\partial
B_{R}}|u(r\theta)|\mathrm{d}\theta$ is the spherical average of $u$. Moreover,

  * (a)

if $p\in(1,2_{\\#})$, then

$u(x)\simeq|x|^{6-n}\quad{\rm as}\quad x\rightarrow 0;$

  * (b)

if $p=2_{\\#}$, then

$u(x)=(1+\mathrm{o}(1))\widehat{K}_{0}(n)^{\frac{n-6}{6}}|x|^{6-n}(\ln|x|)^{\frac{6-n}{6}}\quad{\rm
as}\quad x\rightarrow 0,$

where

$\widehat{K}_{0}(n)=\frac{4}{3}(n-2)(n-4)(n-6)^{2};$ (2)

  * (c)

if $p\in(2_{\\#},2^{\\#}-1)$, then

$u(x)=(1+\mathrm{o}(1))K_{0}(n,p)^{\frac{1}{p-1}}|x|^{-\frac{6}{p-1}}\quad{\rm
as}\quad x\rightarrow 0.$

When $p\in(1,2_{\\#})$, we show that solutions behave like the fundamental
solution, and so, the origin is a removable singularity. When
$p\in(2_{\\#},2^{\\#}-1)$, we prove that solutions to ($\mathcal{P}_{6,p,R}$)
behave near the isolated singularity like the homogeneous solutions to the
blow-up limit equation ($\mathcal{P}_{6,p,\infty}$), which are classified by
(1). In the lower critical case $p=2_{\\#}$, which is the so-called sixth
order Serrin exponent, we observe that since $K_{0}(n,2_{\\#})=0$, it follows
that ($\mathcal{P}_{6,p,\infty}$) does not have non-trivial homogeneous
solutions. This explains why this situation has different blow-up rate near
the singularity, which is given by a homogeneous term times a log-correction
factor.

Now let us compare our results to the ones in the fourth and second order
cases.

First, we consider positive solutions $u\in
C^{4}(\mathbb{R}^{n}\setminus\\{0\\})$ with $n\geqslant 5$ to the family of
fourth order equations

$\displaystyle(-\Delta)^{2}u=f_{p}(u)\quad{\rm in}\quad B_{R}^{*},$
($\mathcal{P}_{4,R,p}$)

where $R<+\infty$, $\Delta^{2}=\Delta\circ\Delta$ is the bi-Laplacian, and
$p\in(1,2_{**}]\cup(2_{**},2^{**}-1)$, where $2_{**}=\frac{n}{n-4}$ and
$2^{**}=\frac{2n}{n-4}$. Notice that ($\mathcal{P}_{4,R,p}$) is subcritical in
the sense of the compact Sobolev embedding $H^{2}(\mathbb{R}^{n})$. On this
subject, we should mention that R. Soranzo [29] for $p\in(1,2_{**})$, H. Yang
[34] and Z. Guo et al. [17] for $p\in(2_{**},2^{**}-1)$, and the first-named
author and J. M. do Ó [2] for $p=2_{**}$ study qualitative properties for
positive solutions to ($\mathcal{P}_{4,R,p}$). We have the result below

######  Theorem B.

Let $u\in C^{4}(\mathbb{R}^{n}\\{0\\})$ be a positive singular solution to
($\mathcal{P}_{4,R,p}$). Assume that $-\Delta u\geqslant 0$. Then, it follows

$u(x)=(1+\mathcal{O}(|x|))\overline{u}(x)\quad{\rm as}\quad x\rightarrow 0.$

Moreover,

  * (a)

if $p\in(1,2_{**})$, then

$u(x)\simeq|x|^{4-n}\quad{\rm as}\quad x\rightarrow 0;$

  * (b)

if $p=2_{**}$, then

$u(x)=(1+\mathrm{o}(1))\widehat{K}_{4,0}(n)^{\frac{n-4}{4}}|x|^{4-n}(\ln|x|)^{\frac{4-n}{4}}\quad{\rm
as}\quad x\rightarrow 0,$

where

$\widehat{K}_{4,0}(n)=\frac{(n-2)(n-4)^{2}}{2};$

  * (c)

if $p\in(2_{**},2^{**}-1)$, then

$u(x)=(1+\mathrm{o}(1))K_{4,0}(n,p)^{\frac{1}{p-1}}|x|^{-\frac{4}{p-1}}\quad{\rm
as}\quad x\rightarrow 0.$

where

$\displaystyle
K_{4,0}(n,p)=\frac{4}{p-1}\left(\frac{4}{p-1}+2\right)\left(n-2-\frac{4}{p-1}\right)\left(n-4-\frac{4}{p-1}\right).$

Second, we consider positive singular solutions $u\in
C^{2}(\mathbb{R}^{n}\setminus\\{0\\})$ with $n\geqslant 3$ the second order
equation below

$\displaystyle-\Delta u=f_{p}(u)\quad{\rm in}\quad B_{R}^{*},$
($\mathcal{P}_{2,R,p}$)

where $R<+\infty$, $\Delta$ is the Laplacian, and
$p\in(1,2_{*}]\cup(2_{*},2^{*}-1)$, where $2_{*}=\frac{n}{n-2}$ and
$2^{*}=\frac{2n}{n-2}$ Notice that ($\mathcal{P}_{2,R,p}$) is subcritical in
the sense of the compact Sobolev embedding of $H^{1}(\mathbb{R}^{n})$. All the
aforementioned classification results were inspired by the classical theorems
of J. Serrin [27] and P.-L. Lions [23], B. Gidas and J. Spruck [14], and P.
Aviles [4] on the study of positive singular solutions to the second order
semi-linear PDE ($\mathcal{P}_{2,R,p}$)

######  Theorem C.

Let $u\in C^{2}(\mathbb{R}^{n}\\{0\\})$ be a positive singular solution to
($\mathcal{P}_{2,R,p}$). Then, it follows

$u(x)=(1+\mathcal{O}(|x|))\overline{u}(x)\quad{\rm as}\quad x\rightarrow 0,$

Moreover,

  * (a)

if $p\in(1,2_{*})$, then

$u(x)\simeq|x|^{2-n}\quad{\rm as}\quad x\rightarrow 0;$

  * (b)

if $p=2_{*}$, then

$u(x)=(1+\mathrm{o}(1))\widehat{K}_{2,0}(n)^{\frac{n-2}{2}}|x|^{2-n}(\ln|x|)^{\frac{2-n}{2}}\quad{\rm
as}\quad x\rightarrow 0,$

where

$\widehat{K}_{2,0}(n)=\frac{(n-2)^{2}}{2};$

  * (c)

if $p\in(2_{*},2^{*}-1)$, then

$u(x)=(1+\mathrm{o}(1))K_{2,0}(n,p)^{\frac{1}{p-1}}|x|^{-\frac{2}{p-1}}\quad{\rm
as}\quad x\rightarrow 0,$

where

$\displaystyle K_{2,0}(n,p)=\frac{2}{p-1}\left(n-2-\frac{2}{p-1}\right).$

This type of asymptotic analysis extends to a rich class of strongly coupled
second order systems as studied in [13, 6].

The main difference between the asymptotic analysis for the critical and
subcritical regimes occurs because of the change in the monotonicity
properties of the Pohozaev functional, which in classifies the type of
stability for singular solutions to ($\mathcal{P}_{6,p,R}$) around a blow-up
(shrink-down) limit solution. This method is inspired by Fleming’s tangent
cone analysis for minimal hypersurfaces [11, 10]. In the critical case, it can
be showed that the Pohozaev functional becomes constant, that is, blow-up
limit solutions are stable. In contrast, in the subcritical case, they are
asymptotically stable. This discrepancy is caused by the sign-changing
behavior of the coefficients of the tri-Laplacian in Emden–Fowler coordinates
(or logarithm cylindrical coordinates), which are suitable for this problem
(see Remark 3.1).

The proof of Theorem 1 is divided into two parts. First, we prove the
asymptotic symmetry of singular solutions to ($\mathcal{P}_{6,p,R}$) in the
punctured ball. Second, we use some ODE analysis and the monotonicity
properties of the Pohozaev functional to study the asymptotic behavior for
solutions on the cylinder. The strategy strongly relies on the growth of the
subcritical nonlinearity, which, in the lower critical situation $p=2_{\\#}$,
turns out to be far different from the other regime. We should emphasize that
the superharmonicity conditions are not necessary in the range
$p\in(2_{\\#},2^{\\#}-1)$, which is not the case otherwise. Third, we define
two homological-type invariants that satisfy suitable monotonicity properties.
This can be used to study the local asymptotic behavior of positive singular
solutions near the isolated singularity. We need to subdivide our approach
with respect to the growth of the nonlinearity into three cases; namely,
Serrin–Lions case, Aviles case, and Gidas–Spruck case.

We remark that in a companion paper the same authors study the asymptotics for
positive singular solutions in the other limit situation $p=2^{\\#}-1$.
Indeed, together with [32] and Theorems 1 and A, this would provide a holistic
picture of the qualitative behavior of solutions to ($\mathcal{P}_{6,p,R}$) in
the broader range $p\in(1,2^{\\#}-1]$. These results are inspired by the
classical literature for semi-linear second order equations due to J. Serrin
[27], P.-L. Lions [23], P. Aviles [4], B. Gidas and J. Spruck [14], and L. A.
Caffarelli et al. [5] with an improvement given by N. Korevaar et al. [20].
For more results on asymptotic analysis, we refer the interested reader to
[19].

We should observe that ($\mathcal{P}_{6,p,R}$), ($\mathcal{P}_{4,R,p}$), and
($\mathcal{P}_{2,R,p}$) are particular cases of a more general class of
equations, which we describe as follows. More precisely, considering
(classical) positive singular solutions $u\in
C^{2m}(\mathbb{R}^{n}\setminus\\{0\\})$ with $n\geqslant 2m:=N$ to the
following family of subcritical even order poly-harmonic PDEs

$\displaystyle(-\Delta)^{m}u=f_{p}(u)\quad{\rm in}\quad B^{*}_{R}.$
($\mathcal{P}_{N,p,R}$)

Here $B_{R}\setminus\\{0\\}\subset\mathbb{R}^{n}$ is the punctured ball of
radius $R>0$, $\Delta^{m}=\Delta\circ\dots\circ\Delta$ is the poly-Laplacian,
and the nonlinearity $f_{p}\in C^{1}(B_{R})$ is given by

$f_{p}(u):=|u|^{p-1}u\quad{\rm with}\quad
p\in\left[\frac{n}{n-2m},\frac{n+2m}{n-2m}\right):=[2_{m,*},2_{m}^{*}),$

where $2_{m,*}:=\frac{n}{n-2m}$ and $2^{m}_{*}:=\frac{2n}{n-2m}$ are,
respectively, the lower and upper critical exponents with respect to the
compact Sobolev embedding of $H^{m}(\mathbb{R}^{n})$. In this regard, it is
natural to expect a classification result for the local asymptotic behavior of
singular solutions to ($\mathcal{P}_{N,p,R}$) near isolated singularities in
the sense of Theorem 1.

Let us state as the following conjecture

######  Conjecture 2.

Let $u$ be a positive singular solution to ($\mathcal{P}_{N,p,R}$) with
$p\in(1,2_{m}^{*}-1)$. Assume that $(-\Delta u)^{j}\geqslant 0$ for any
$j=1,\dots,m-1$. Then, it follows

$u(x)=(1+\mathcal{O}(|x|))\overline{u}(x)\quad{\rm as}\quad x\rightarrow 0,$

Moreover,

  * (a)

if $p\in(1,2^{*}_{m})$, then

$u(x)\simeq|x|^{N-n}\quad{\rm as}\quad x\rightarrow 0;$

  * (b)

if $p=2_{m,*}$, then

$u(x)=(1+\mathrm{o}(1))\widehat{K}_{0}(n)^{\frac{n-N}{N}}|x|^{N-n}(\ln|x|)^{\frac{N-n}{N}}\quad{\rm
as}\quad x\rightarrow 0,$

where

$\widehat{K}_{N,0}(n)=\frac{2^{m-2}(m-1)!}{m}\prod_{j=0}^{m-1}\left(n-2j\right)(n-N)^{2}.$

  * (c)

if $p\in(2_{m,*},2_{m}^{*}-1)$, then

$u(x)=(1+\mathrm{o}(1))K_{0}(n,p)^{\frac{1}{p-1}}|x|^{-\frac{N}{p-1}}\quad{\rm
as}\quad x\rightarrow 0.$

######  Remark 3.

In a recent paper, it was independently proved by X. Huang, Y. Li, and H. Yang
[18] that the super poly-harmonic condition can be removed. We also refer to
Q. Ngô and D. Ye [26] for more details on the blow-up limit case $R=+\infty$.
We keep this condition in our manuscript because of its natural relations with
curvature sign conditions for the upper critical situation $p=2^{\\#}-1$. They
also provided some upper bound estimates that will be important in our
methods.

Here is our plan for the rest of the paper. In Section 2, we introduce both
the autonomous and nonautonomous Emden–Fowler coordinates. In Section 3, we
define the associated Pohozaev functionals, and we prove their (asymptotic)
monotonicity properties. In Section 4, we prove some a priori upper bound
estimates. In Section 5, we perform a variant of the integral moving spheres
method and prove that solutions are asymptotically radially symmetric. In
Section 6, we study the limit values of the Pohozaev functional under blow-up
and shrink-down sequences. In Section 7, we use the monotonicity formulas and
some asymptotic analysis to prove the classification of the local asymptotic
behavior in Theorem 1.

##  2\. Emden–Fowler coordinates

In this section, we define the Emden–Fowler change of variables, which is used
to transform the singular PDE ($\mathcal{P}_{6,p,R}$) problem into an ODE
problem.

###  2.1. Autonomous case

We define a classical change of variables, which transforms
($\mathcal{P}_{6,p,R}$) into an ODE with constant coefficients.

######  Definition 2.1.

Let us define the sixth order autonomous Emden–Fowler change of variables
$($or cylindrical logarithm coordinates$)$ given by

$v(t,\theta)=r^{\gamma_{p}}u(r,\sigma),\quad\mbox{where}\quad t=\ln
r,\quad\sigma=\theta=x|x|^{-1}.$

Using this coordinate system and performing a lengthy computation, we arrive
at the following sixth order nonlinear PDE on the cylinder
${\mathcal{C}}_{T}:=(-\infty,T)\times\mathbb{S}^{n-1}$ with $T=\ln R<+\infty$,

$-P^{3}_{\rm cyl}v=f_{p}(v)\quad{\rm on}\quad{\mathcal{C}}_{T}.$
($\mathcal{C}_{p,T}$)

Here $P^{3}_{\rm cyl}$ is the tri-Laplacian written in cylindrical coordinates
given by

$\displaystyle P^{3}_{\rm cyl}$
$\displaystyle=\partial_{t}^{(6)}+K_{5}(n,p)\partial_{t}^{(5)}+K_{4}(n,p)\partial_{t}^{(4)}+K_{3}(n,p)\partial_{t}^{(3)}+K_{2}(n,p)\partial_{t}^{(2)}+K_{1}(n,p)\partial_{t}+K_{0}(n,p)$
$\displaystyle+2\partial^{(4)}_{t}\Delta_{\theta}+J_{3}(n,p)\partial^{(3)}_{t}\Delta_{\theta}+J_{2}(n,p)\partial^{(2)}_{t}\Delta_{\theta}+J_{1}(n,p)\partial_{t}\Delta_{\theta}+J_{0}(n,p)\Delta_{\theta}$
$\displaystyle+3\partial^{(2)}_{t}\Delta^{2}_{\theta}+L_{1}(n,p)\partial_{t}\Delta^{2}_{\theta}+L_{0}(n,p)\Delta^{2}_{\theta}+\Delta^{3}_{\theta},$

where $K_{j}(n,p),J_{j}(n,p),L_{j}(n,p)$ for $j=0,1,2,3,4,5$ are dimensional
constants given by (A.2) and (A.2).

######  Definition 2.2.

Let us consider the autonomous Emden–Fowler transformation as follows

$\mathfrak{F}:C_{c}^{\infty}(B_{R}^{*})\rightarrow
C_{c}^{\infty}(\mathcal{C}_{T})\quad\mbox{given
by}\quad\mathfrak{F}(u)=e^{\gamma_{p}t}u(e^{t},\theta):=v.$ (2.1)

###  2.2. Non-autonomous case

In the lower critical case, because of the vanishing of the coefficient
$K_{0}(n,2_{\\#})$, we already know that the situation changes dramatically.
In this fashion, we define the so-called nonautonomous Emden–Fowler change of
variables [4, 13, 31, 28].

######  Definition 2.3.

Let us define the sixth order nonautonomous Emden–Fowler change of variables
$($or cylindrical logarithm coordinates$)$ given by

$w(t,\theta)=r^{6-n}(\ln r)^{\frac{6-n}{6}}u(r,\sigma),\quad{\rm where}\quad
t=\ln r\quad{\rm and}\quad\sigma=\theta=x|x|^{-1}.$ (2.2)

Using this coordinate system and performing a lengthy computation, we arrive
at

$-\widetilde{P}^{3}_{\rm cyl}w=t^{-1}|w|^{2_{\\#}-1}w\quad{\rm
on}\quad{\mathcal{C}}_{T}.$ ($\widetilde{\mathcal{C}}_{T}$)

Here $\widetilde{P}^{3}_{\rm cyl}$ is the tri-Laplacian written in
nonautonomous Emden–Fowler coordinates given by

$\displaystyle\widetilde{P}^{3}_{\rm cyl}$
$\displaystyle=\partial_{t}^{(6)}+\widetilde{K}_{5}(n,t)\partial_{t}^{(5)}+\widetilde{K}_{4}(n,t)\partial_{t}^{(4)}+\widetilde{K}_{3}(n,t)\partial_{t}^{(3)}+\widetilde{K}_{2}(n,t)\partial_{t}^{(2)}+\widetilde{K}_{1}(n,p)\partial_{t}+\widetilde{K}_{0}(n,p)$
$\displaystyle+\widetilde{J}_{4}(n,t)\partial^{(4)}_{t}\Delta_{\theta}+\widetilde{J}_{3}(n,t)\partial^{(3)}_{t}\Delta_{\theta}+\widetilde{J}_{2}(n,t)\partial^{(2)}_{t}\Delta_{\theta}+\widetilde{J}_{1}(n,t)\partial_{t}\Delta_{\theta}+\widetilde{J}_{0}(n,t)\Delta_{\theta}$
$\displaystyle+\widetilde{L}_{2}(n,t)\partial^{(2)}_{t}\Delta^{2}_{\theta}+\widetilde{L}_{1}(n,t)\partial_{t}\Delta^{2}_{\theta}+\widetilde{L}_{0}(n,t)\Delta^{2}_{\theta}+\Delta^{3}_{\theta},$

where $\widetilde{K}_{j}(n,t),\widetilde{J}_{j}(n,t),\widetilde{L}_{j}(n,t)$
for $j=0,1,2,3,4,5$ are real-valued functions given by (A.3) and (A.3).

Analogously to the standard autonomous case, we also consider a transformation
that sends a singular solution to ($\mathcal{P}_{6,p,R}$) with $p=2_{\\#}$
into solutions to a nonautonomous ODE on the cylinder.

######  Definition 2.4.

Let us consider the non-autonomous Emden–Fowler transformation as follows

$\tilde{\mathfrak{F}}:C_{c}^{\infty}(B_{R}^{*})\rightarrow
C_{c}^{\infty}(\mathcal{C}_{T})\quad\mbox{given
by}\quad\tilde{\mathfrak{F}}(u)=e^{(6-n)t}t^{\frac{6-n}{6}}u(e^{t},\theta):=w.$
(2.3)

##  3\. Pohozaev functionals

Next, we define two central characters in our studies, the so-called
autonomous and nonautonomous Pohozaev functionals. The heuristics are that
these Pohozaev functionals classify whether or not the blow-up limit solutions
in cylindrical coordinates are (asymptotically) stable equilibrium points of
the associated sixth order ODE. More precisely, from the dynamical systems
point of view, it works as a Lyapunov functional. Let us remark that to prove
the Pohozaev invariant to be well-defined, one needs to use some upper
estimates and asymptotic symmetry that will be proved independently in Section
5 (see Lemmas 4.5 and 4.6).

###  3.1. Autonomous case

Initially, let us define the Pohozaev functional associated to the PDE
equation on the cylinder given by ($\mathcal{C}_{p,T}$), which will play a
central role in our analysis (for more details on this class of invariants, we
refer to [12, 16, 34, 35]).

######  Remark 3.1.

By direct computations, notice that when $p\in(2_{\\#},2^{\\#}-1)$, one has
the sign relations

$K_{5}(n,p),K_{1}(n,p),L_{1}(n,p)\geqslant 0,\quad{\rm and}\quad
K_{3}(n,p),J_{3}(n,p),J_{1}(n,p)\leqslant 0.$

In addition, we can explicitly compute these coefficients in the limiting
situations $p=2_{\\#}$ and $p=2^{\\#}-1$ (A.5) and (A.4). This change in sign
behavior explains why one needs to use distinct methods when the power
parameter $p\in(1,+\infty)$ changes.

With respect to Conjecture 2, we speculate that a relationship like this
should hold for the coefficients of the poly-Laplacian written in Emden–Fowler
coordinates.

######  Definition 3.2.

For any $v\in C^{6}(\mathbb{R})$ positive solution to ($\mathcal{C}_{p,T}$)
with $p\in(2_{\\#},2^{\\#}-1)$, let us define its cylindrical Pohozaev
functional by

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

Here $\mathbb{S}_{t}^{n-1}=\\{t\\}\times\mathbb{S}^{n-1}$ is the cylindrical
ball with volume element given by $\mathrm{d}\theta=e^{-2t}\mathrm{d}\sigma$,
where $\mathrm{d}\sigma_{r}$ is the volume element of the ball of radius $r>0$
in $\mathbb{R}^{n}$. In addition, we set

$\displaystyle\mathcal{H}(t,\theta,v):=\mathcal{H}_{{\rm
rad}}(t,\theta,v)+\mathcal{H}_{\rm ang}(t,\theta,v),$

where

$\displaystyle\mathcal{H}_{{\rm rad}}(t,\theta,v)$
$\displaystyle:=\left(v^{(5)}v^{(1)}-v^{(4)}v^{(2)}+\frac{1}{2}{v^{(3)}}^{2}\right)+K_{5}\left(v^{(4)}v^{(1)}-v^{(3)}v^{(2)}\right)$
$\displaystyle+{K_{4}}\left(v^{(3)}v^{(1)}-\frac{1}{2}{v^{(2)}}^{2}\right)+K_{3}v^{(2)}v^{(1)}+\frac{K_{2}}{2}{v^{(1)}}^{2}+\frac{K_{0}}{2}v^{2}+\frac{|v|^{p+1}}{p+1}$

and

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

######  Remark 3.3.

Using the inverse of the Emden–Fowler transformation, one can also construct
the spherical Pohozaev functional given by $\mathcal{P}_{\rm
sph}(r,u):=\left(\mathcal{P}_{\rm
cyl}\circ\mathfrak{F}^{-1}\right)\left(t,v\right)$. From [19, Propositon A.1],
it follows that $\mathcal{P}_{\rm sph}$ also satisfies a monotonicity
property.

We deduce a monotonicity formula for the logarithmic cylindrical Pohozaev
functional $\mathcal{P}_{\rm cyl}(t,v)$, which will be essential to show that
the limit Pohozaev invariant is well-defined as $t\rightarrow-\infty$.

######  Proposition 3.4.

Let $v\in C^{6}(\mathbb{R})$ be a positive solution to ($\mathcal{C}_{p,T}$)
with $p\in(2_{\\#},2^{\\#}-1)$. If $-\infty<t_{1}\leqslant t_{2}<T$, then
$\mathcal{P}_{\rm cyl}(t_{2},v)-\mathcal{P}_{\rm cyl}(t_{1},v)\leqslant 0$.
More precisely, we have the monotonicity formula

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{P}_{\rm
cyl}(t,v)=\int_{\mathbb{S}_{t}^{n-1}}\left(-K_{5}|\partial_{t}^{(3)}v|^{2}+K_{3}|\partial_{t}^{(2)}v|^{2}-K_{1}|\partial_{t}v|^{2}-J_{3}|\partial^{(2)}_{t}\nabla_{\theta}v|^{2}-L_{1}|\partial_{t}\Delta_{\theta}v|^{2}\right)\mathrm{d}\theta.$
(3.1)

In particular, ${\mathcal{P}}_{\rm cyl}(t,v)$ is nonincreasing, and so
$\mathcal{P}_{\rm
cyl}(-\infty,v):=\lim_{t\rightarrow-\infty}{\mathcal{P}}_{\rm cyl}(t,v)$
exists.

###### Proof.

Initially, observe that (3.1) follows by a direct computation, which consists
multiplying ($\mathcal{C}_{p,T}$) by $v^{(1)}$, and integrating by parts.
Hence, using Remark 3.1, we find $\partial_{t}\mathcal{P}_{\rm
cyl}(t,v)\leqslant 0$, which proves that the cylindrical Pohozaev functional
is nonincreasing. Finally, since by Lemma 4.2, the Pohozaev functional is
bounded above, the full existence of limit follows. ∎

######  Remark 3.5.

Using the inverse of cylindrical transform, that is, $\mathcal{P}_{\rm
sph}=\mathcal{P}_{\rm cyl}\circ\mathfrak{F}^{-1}$, it follows that
$\mathcal{P}_{\rm sph}(0,u)=\lim_{r\rightarrow 0}\mathcal{P}_{\rm
sph}(r,u)=\lim_{t\rightarrow-\infty}{\mathcal{P}}_{\rm cyl}(t,v).$ The last
equality implies that the Pohozaev invariant is well-defined in the punctured
ball when $R\rightarrow 0$.

###  3.2. Non-autonomous case

In the lower critical case $p=2_{\\#}$, since $K_{0,\\#}:=K_{0}(n,2_{\\#})=0$
and $\gamma_{2_{\\#}}=n-6$, a new cylindrical transformation was defined.
Concerning this nonautonomous cylindrical transformation, we compute its
associated Pohozaev functional, which in this situation has some time-
dependent terms.

######  Definition 3.6.

For any $w\in C^{6}(\mathbb{R})$ be a positive solution to
($\widetilde{\mathcal{C}}_{T}$), let us define its cylindrical nonautonomous
Pohozaev functional by

$\displaystyle\widetilde{\mathcal{P}}_{\rm cyl}(t,w)$
$\displaystyle=\displaystyle\int_{\mathbb{S}_{t}^{n-1}}\widetilde{\mathcal{H}}(t,\theta,w)\mathrm{d}\theta.$

Here

$\displaystyle\widetilde{\mathcal{H}}(t,\theta,w):=\widetilde{\mathcal{H}}_{{\rm
rad}}(t,\theta,w)+\widetilde{\mathcal{H}}_{\rm ang}(t,\theta,w),$

where

$\displaystyle\widetilde{\mathcal{H}}_{{\rm rad}}(t,\theta,w)$
$\displaystyle:=t\left(w^{(5)}w^{(1)}-w^{(4)}w^{(2)}+\frac{1}{2}{w^{(3)}}^{2}\right)+t\widetilde{K}_{5}\left(w^{(4)}w^{(1)}-w^{(3)}w^{(2)}\right)$
$\displaystyle+t\widetilde{K}_{4}\left(w^{(3)}w^{(1)}-\frac{1}{2}{w^{(2)}}^{2}\right)+t\widetilde{K}_{3}w^{(2)}w^{(1)}+\frac{t\widetilde{K}_{1}}{2}{w^{(1)}}^{2}+\frac{t\widetilde{K}_{0}}{2}w^{2}+\frac{n-6}{2(n-3)}|w|^{2_{\\#}+1}$

and

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

######  Remark 3.7.

Initially, notice that the nonautonomous Pohozaev invariant is well-defined
for any $t\in(-\infty,T)$ since $u\in C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$
is smooth away from the origin. In addition, due to Proposition 5.5, for
$R<\infty$, we get that $|w(t,\theta)-\overline{w}(t)|=\mathcal{O}(e^{\beta
t})$ as $t\rightarrow-\infty$, for some $\beta>0$, where $\overline{w}(t)$ is
the average of $w(t,\theta)$ over $\theta\in\mathbb{S}_{t}^{n-1}$ with
$w=\widetilde{\mathfrak{F}}(u)$ given by (2.3). In particular, one can find
some large $T_{0}\gg 1$ and $C>0$ $($independent of $t$$)$ satisfying

$\left|\nabla_{\theta}w(t,\theta)\right|+\left|\Delta_{\theta}w(t,\theta)\right|+\left|\nabla_{\theta}\Delta_{\theta}w(t,\theta)\right|+\left|\Delta^{2}_{\theta}w(t,\theta)\right|\leqslant
Ce^{\beta t}\quad{\rm in}\quad\mathcal{C}_{T_{0}}.$ (3.2)

Moreover, from the sharp estimate in Proposition 3.4 and the gradient estimate
(4.4), it follows

$\sum_{j=0}^{5}|\partial_{t}^{(j)}w(t,\theta)|\leqslant C\quad{\rm
in}\quad\mathcal{C}_{T_{0}}.$ (3.3)

Before we prove the monotonicity formula, we need to show an auxiliary result

######  Lemma 3.8.

Let $w\in C^{6}(\mathbb{R})$ be a positive solution to
($\widetilde{\mathcal{C}}_{T}$). Then,
$\lim_{t\rightarrow-\infty}w(t,\theta)\in\\{0,\widehat{K}_{0}(n)^{\frac{n-6}{6}}\\}$,
where $\widehat{K}_{j}(n)=\lim_{t\rightarrow-\infty}t\widetilde{K}_{j}(n,t)$
for $j=0,1,2,3,4,5$. Moreover,

$\lim_{t\rightarrow-\infty}{w}^{(j)}(t,\theta)=\lim_{t\rightarrow-\infty}\overline{w}^{(j)}(t)=0\quad{\rm
for\ all}\quad j\geqslant 1.$

###### Proof.

Indeed, using the asymptotic symmetry in Proposition 5.5 and the estimates in
Remark 3.7, we have that the cylindrical transformation of spherical average
$\overline{w}=\widetilde{\mathfrak{F}}(\overline{u})$ satisfies,

$\overline{P}_{\rm
cyl}\overline{w}=-|\overline{w}|^{2_{\\#}-1}\overline{w}+\mathcal{O}(\overline{w}(t)e^{t})\quad{\rm
as}\quad t\rightarrow-\infty,$

where

$\overline{P}_{\rm
cyl}=t\partial_{t}^{(6)}+\widehat{K}_{5}(n)\partial_{t}^{(5)}+\widehat{K}_{4}(n)\partial_{t}^{(4)}+\widehat{K}_{3}(n)\partial_{t}^{(3)}+\widehat{K}_{2}(n)\partial_{t}^{(2)}+\widehat{K}_{1}(n)\partial_{t}+\widehat{K}_{0}(n).$

Thus, decomposing the left-hand side of the last equation into a product of
three second order operators, we can adapt the arguments in [16, Section 2]
(see also [1, Section 4]) for the last sixth order asymptotic identity. In
this fashion, we split the proof of the claim into two steps:

Step 1: Either $w(t,\theta)=\mathrm{o}(1)$ or
$w(t,\theta)=\widehat{K}_{0}(n)^{\frac{n-6}{6}}+\mathrm{o}(1)$ as
$t\rightarrow-\infty$.

As a matter of fact, the conclusion follows directly from the same argument in
[3, Lemma 4.10] to the limit ODE ($\widetilde{\mathcal{C}}_{T}$) with
$T=\infty$.

Step 2: $\partial_{t}^{(j)}w(t,\theta)=\mathrm{o}(1)$ as $t\rightarrow-\infty$
for all $j\geqslant 1$.

Indeed, we can adapt the proof in [3, Lemma 4.8] to the ODE
($\widetilde{\mathcal{C}}_{T}$) with $T=\infty$.

Steps 1 and 2 together give the proof for the first claim. ∎

To prove the full existence of $\widetilde{\mathcal{P}}_{\rm cyl}(-\infty,w)$,
we shall verify the estimates in the next lemma, which is a sixth order
version of [4, Lemma 3.2]. Due to the appearance of higher order derivative
terms, we give different proof to the ones in the lemma quoted above. Namely,
our proof is based on Lemma 3.8 combined with a simple L’Hôspital rule.
Initially, by differentiating $\widetilde{\mathcal{H}}(t,\theta,w)$ with
respect to $t$, integrating by parts over $\mathbb{S}_{t}^{n-1}$ (using
differentiation under the integral sign one can even omit the dependence on
$t$), and using ($\widetilde{\mathcal{C}}_{T}$), we find

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}\widetilde{\mathcal{P}}_{\rm
cyl}(t,w)=\Xi_{\rm rad}(t,w)+\Xi_{\rm ang}(t,w).$

Here

$\displaystyle\quad\Xi_{\rm
rad}(t,w):=\int_{\mathbb{S}^{n-1}}\partial_{t}\widetilde{\mathcal{H}}_{{\rm
rad}}(t,\theta,w)\mathrm{d}\theta\quad{\rm and}\quad\Xi_{\rm
ang}(t,w):=-\int_{\mathbb{S}^{n-1}}\partial_{t}\widetilde{\mathcal{H}}_{{\rm
ang}}(t,\theta,w)\mathrm{d}\theta,$ (3.4)

where

$\displaystyle\partial_{t}\widetilde{\mathcal{H}}_{{\rm rad}}(t,\theta,w)$
$\displaystyle=w^{(5)}w^{(1)}-w^{(4)}w^{(3)}+{w^{(3)}}^{2}+\mathfrak{p}_{5}w^{(4)}w^{(1)}-\mathfrak{p}_{5}w^{(3)}w^{(2)}$
$\displaystyle+\mathfrak{p}_{4}w^{(3)}w^{(1)}-\mathfrak{p}_{4}{w^{(2)}}^{2}+\mathfrak{p}_{3}w^{(2)}w^{(1)}+\mathfrak{p}_{2}{w^{(2)}}^{2}+\mathfrak{p}_{1}{w^{(1)}}^{2}+\mathfrak{p}_{0}w^{2}.$

The real-valued functions $\mathfrak{p}_{j}(n,\cdot):(-\infty,\ln
R)\rightarrow\mathbb{R}$ are given by (A.3). More explicitly, we have

$\displaystyle\mathfrak{p}_{0}(n,t)$
$\displaystyle:=\frac{1}{36}(3n^{3}-48n^{2}+228n-320)t^{-2}+\mathcal{O}(t^{-4})\quad{\rm
as}\quad t\rightarrow-\infty,$ (3.5) $\displaystyle\mathfrak{p}_{1}(n,t)$
$\displaystyle:=\frac{1}{864}(n-6)(3456n^{2}-20736n+27648)t$ (3.6)
$\displaystyle-\frac{1}{864}(n-6)(864n^{3}-10368n^{2}+44928n-64512)+\mathcal{O}(t^{-1})\quad{\rm
as}\quad t\rightarrow-\infty,$ $\displaystyle\mathfrak{p}_{2}(n,t)$
$\displaystyle:=-(3n^{3}-48n^{2}+228n-320)+\mathcal{O}(t^{-2})\quad{\rm
as}\quad t\rightarrow-\infty,$ (3.7) $\displaystyle\mathfrak{p}_{3}(n,t)$
$\displaystyle:=-\frac{1}{2}(n^{3}-30n^{2}+212n-408)+\mathcal{O}(t^{-2})\quad{\rm
as}\quad t\rightarrow-\infty,$ (3.8) $\displaystyle\mathfrak{p}_{4}(n,t)$
$\displaystyle:=\frac{1}{2}(3n^{2}-42n+124)+\mathcal{O}(t^{-2})\quad{\rm
as}\quad t\rightarrow-\infty,$ (3.9) $\displaystyle\mathfrak{p}_{5}(n,t)$
$\displaystyle:=-\frac{1}{2}{(n-6)}.$ (3.10)

Now we use the asymptotic estimate proved in Lemma 3.8 to prove the result
below

######  Lemma 3.9.

Let $w\in C^{6}(\mathbb{R})$ be a positive solution to
($\widetilde{\mathcal{C}}_{T}$). Then, $\lim_{t\rightarrow-\infty}\Xi_{\rm
rad}(t,w)=0.$

###### Proof.

Notice that

$\displaystyle\Xi_{\rm rad}(t,w)$
$\displaystyle:=I_{51}+I_{42}+I_{33}+I_{41}+I_{32}+I_{31}+I_{22}+I_{21}+I_{22}+I_{11}+I_{00}.$

Next, we estimate each term of the last identity separately by steps.

Step 1:
$I_{00}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{0}(n,t)w(t,\theta)^{2}\mathrm{d}\theta=0$.

Using (3.5), we find that $\lim_{t\rightarrow-\infty}\mathfrak{p}_{0}(n,t)=0$,
which yields

$\displaystyle\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{0}(n,t)w^{(1)}(t,\theta)^{2}\mathrm{d}\theta=\left(\lim_{t\rightarrow-\infty}\mathfrak{p}_{0}(n,t)\right)\left(\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(1)}(t,\theta)^{2}\mathrm{d}\theta\right)=0.$

This gives us the desired conclusion.

Step 2:
$I_{11}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{1}(n,t)w^{(1)}(t,\theta)^{2}\mathrm{d}\theta=0$.

The difference in this case is
$\lim_{t\rightarrow-\infty}\mathfrak{p}_{1}(n,t)=-\infty$. However, using
(3.6), we can decompose
$\mathfrak{p}_{1}(n,t)=\widetilde{\mathfrak{p}}_{1}(n,t)+\widehat{\mathfrak{p}}_{1}(n,t)$,
where $\lim_{t\rightarrow-\infty}\widetilde{\mathfrak{p}}_{1}(n,t)<+\infty$
and
$\widehat{\mathfrak{p}}_{1}(n,t)=\frac{1}{864}(n-6)(3456n^{2}-20736n+27648)t$.
With this notation, we set

$\displaystyle\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{1}(n,t)w^{(1)}(t,\theta)^{2}\mathrm{d}\theta:=\widetilde{I}_{11}+\widehat{I}_{11}.$

Notice that since
$\lim_{t\rightarrow-\infty}\widetilde{\mathfrak{p}}_{1}(n,t)=0$, we trivially
see that $\widetilde{I}_{11}=0$. Moreover, in order to estimate
$\widehat{I}_{11}$, we use the L’Hôspital rule as follows
$\widehat{I}_{11}=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\widehat{\mathfrak{p}}_{1}(n,t)w(t,\theta)^{2}\mathrm{d}\theta=0,$
which proves this step.

Step 3:
$I_{22}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{2}(n,t)w^{(2)}(t,\theta)^{2}\mathrm{d}\theta=0.$

As before, by (3.7), it holds that
$\lim_{t\rightarrow-\infty}\mathfrak{p}_{2}(n,t)<\infty$ and using

$\displaystyle\lim_{t\rightarrow\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{2}(n,t)w^{(2)}(t,\theta)^{2}\mathrm{d}\theta=\left(\lim_{t\rightarrow-\infty}\mathfrak{p}_{2}(n,t)\right)\left(\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(2)}(t,\theta)^{2}\mathrm{d}\theta\right)=0,$

the proof of this step follows promptly.

Step 4:
$I_{21}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{3}(n,t)w^{(2)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta=0$.

In fact, using (3.8), one has that
$\lim_{t\rightarrow-\infty}\mathfrak{p}_{3}(n,t)<+\infty$. Thus,

$\displaystyle\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{3}(n,t)w^{(2)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta=\left(\lim_{t\rightarrow-\infty}\mathfrak{p}_{3}(n,t)\right)\left(\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(2)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta\right)=0,$

which by Lemma 3.8 leads to the conclusion.

Step 5:
$I_{22}:=-\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{4}(n,t)w^{(2)}(t,\theta)^{2}\mathrm{d}\theta=0$.

Indeed, by (3.9), we obtain that
$\lim_{t\rightarrow-\infty}\mathfrak{p}_{4}(n,t)<+\infty$. Thus,

$\displaystyle-\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{4}(n,t)w^{(2)}(t,\theta)^{2}\mathrm{d}\theta=-\left(\lim_{t\rightarrow-\infty}\mathfrak{p}_{4}(n,t)\right)\left(\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(2)}(t,\theta)^{2}\mathrm{d}\theta\right)=0,$

which by Lemma 3.8 concludes the argument.

Step 6:
$I_{31}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{4}(n,t)w^{(3)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta=0$.

Using the same argument as before, we find

$\displaystyle\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{4}(n,t)w^{(3)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta=\left(\lim_{t\rightarrow-\infty}\mathfrak{p}_{4}(n,t)\right)\left(\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(3)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta\right)=0,$

which finishes this step.

Step 7:
$I_{32}:=-\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{5}(n,t)w^{(3)}(t,\theta)w^{(2)}(t,\theta)\mathrm{d}\theta=0$.

Indeed, by (3.10), we obtain that
$\lim_{t\rightarrow-\infty}\mathfrak{p}_{4}(n,t)<+\infty$. Thus,

$\displaystyle-\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{5}(n,t)w^{(3)}(t,\theta)w^{(2)}(t,\theta)\mathrm{d}\theta=-\left(\lim_{t\rightarrow-\infty}\mathfrak{p}_{5}(n,t)\right)\left(\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(3)}(t,\theta)w^{(2)}(t,\theta)\mathrm{d}\theta\right)=0,$

which by Lemma 3.8 concludes the argument.

Step 8:
$I_{41}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{5}(n,t)w^{(4)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta=0$.

Using the same argument as before, we find

$\displaystyle\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}\mathfrak{p}_{5}(n,t)w^{(4)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta=\left(\lim_{t\rightarrow-\infty}\mathfrak{p}_{5}(n,t)\right)\left(\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(4)}(t,\theta)w^{(1)}(t,\theta)\mathrm{d}\theta\right)=0,$

which finishes this step.

Step 9:
$I_{33}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(3)}(t,\theta)^{2}\mathrm{d}\theta=0$.

It follows directly by Lemma 3.8.

Step 10:
$I_{42}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(4)}(t,\theta)w^{(2)}(t,\theta)\mathrm{d}\theta=0$.

It follows directly by Lemma 3.8.

Step 11:
$I_{51}:=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}^{n-1}}w^{(5)}(t,\theta),w^{(1)}(t,\theta)\mathrm{d}\theta=0$.

It follows directly by Lemma 3.8.

Finally, putting together all the steps the proof is concluded. ∎

The next proposition is the monotonicity of this new Pohozaev functional.

######  Proposition 3.10.

Let $w\in C^{6}(\mathbb{R})$ be a positive solution to
($\widetilde{\mathcal{C}}_{T}$). Then, one has

  * (i)

if $n=7,8$, then there exists $T_{*}\gg 1$ such that
$\widetilde{\mathcal{P}}_{\rm cyl}(t,w)$ is non-decreasing for $t>T_{*}$.

  * (ii)

if $n\geqslant 9$, then there exists $T^{*}\gg 1$ such that
$\widetilde{\mathcal{P}}_{\rm cyl}(t,w)$ is nonincreasing for $t>T^{*}$.
Moreover, $\widetilde{\mathcal{P}}_{\rm
cyl}(-\infty,w):=\lim_{t\rightarrow-\infty}\widetilde{\mathcal{P}}_{\rm
cyl}(t,w)$ exists.

###### Proof.

Initially, using Remark 3.7 and Lemma 3.8 one can find $t_{0}\gg 1$ such that
$\operatorname{sign}(\partial_{t}\widetilde{\mathcal{H}}_{{\rm
rad}}(t,w))=\operatorname{sign}(\mathfrak{p}_{0}(n,t)|w|^{2})$ for
$|t|>t_{0}$. Besides, we directly verify for $n\geqslant 9$ that there exists
$t_{1}\gg 1$ sufficiently large such that $\mathfrak{p}_{0}(n,t)<0$ for
$|t|>t_{1}$, which, by taking $T^{*}:=\max\\{t_{1},t_{0}\\}$, implies that
$\widetilde{\mathcal{P}}_{\rm cyl}(t,w)$ is nonincreasing for $|t|>T^{*}$.
However, for $n=7,8$, there exists $t_{1}\gg 1$ sufficiently large such that
$\mathfrak{p}_{0}(n,t)<0$ for $|t|>t_{2}$; thus, setting
$T_{*}:=\max\\{t_{0},t_{2}\\}$, we get that $\widetilde{\mathcal{P}}_{\rm
cyl}(t,w)$ is nondecreasing for $|t|>T_{*}$. Hence, the existence of
$\widetilde{\mathcal{P}}_{\rm cyl}(-\infty,w)$ follows since
$\widetilde{\mathcal{P}}_{\rm cyl}(t,w)$ is uniformly bounded both from above
and below as $t\rightarrow-\infty$, and, by (3.2), it gives us
$\lim_{t\rightarrow-\infty}\Xi_{\rm ang}(t,w)=0$, where $\Xi_{\rm ang}$ is
defined by (3.4). This combined with (3.3) yields that
$\liminf_{t\rightarrow-\infty}\widetilde{\mathcal{P}}_{\rm cyl}(t,w)<+\infty$.
The proof is concluded. ∎

##  4\. A priori upper bounds

This subsection is devoted to provide a priori upper bounds for positive
singular solutions to ($\mathcal{P}_{6,p,R}$) with $p\in(1,2^{\\#}-1)$. Our
strategy relies on the classification of the limit solutions to
($\mathcal{P}_{6,p,R}$) combined with a blow-up argument. When
$p\in(2_{\\#},2^{\\#}-1)$, most of our arguments in this section are similar
in spirit to the ones in [18], so we omit them here. Nevertheless, for
$p=2_{\\#}$, our proof is completely new and based on [29]. Notice that by
scaling invariance, we may assume $R=1$ without loss of generality. Also, all
constants in this section depend only on $n$ and $p$.

######  Lemma 4.1.

Let $u\in C^{6}(B_{1})\cap$ $C(\bar{B}_{1})$ be a positive non-singular
solution to ($\mathcal{P}_{6,p,R}$) with $p\in(1,2^{\\#}-1)$ and $R=1$. Then,
there exists $C>0$ such that

$u(x)\leqslant C(1-|x|)^{-\gamma_{p}}\quad{\rm in}\quad B_{1}.$ (4.1)

###### Proof.

See [18, Lemma 3.1]. ∎

Using the last lemma, we can prove the auxiliary result below

######  Lemma 4.2.

Let $u\in C^{6}(B_{1}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(1,2^{\\#}-1)$ and $R=1$. Then, there exists
$C>0$ such that

$|u(x)|\leqslant C|x|^{-\gamma_{p}}\quad{\rm in}\quad B^{*}_{1/2}.$

###### Proof.

Fixing $x_{0}\in B^{*}_{1/2}$, let us define $r=\frac{1}{2}|x_{0}|$. Then,
since $\bar{B}_{r}\left(x_{0}\right)\subset B^{*}_{1}$, it is well-defined the
rescaled function given by

$\widetilde{u}_{r}(x)=r^{\gamma_{p}}u\left(rx+x_{0}\right)\quad\mbox{in}\quad\bar{B}_{1}.$

Since $u$ is a positive singular solution to ($\mathcal{P}_{6,p,R}$), we
obtain that $\widetilde{u}_{r}$ is a positive non-singular solution to

$(-\Delta)^{3}\widetilde{u}_{r}=f_{p}(\widetilde{u}_{r})\quad\mbox{in}\quad
B_{1},$

which is continuous up to the boundary. Therefore, we can apply Lemma 4.1 to
$\widetilde{u}_{r}$, which, by taking $x=0$ in (4.1), provides
$|\widetilde{u}_{r}(0)|\leqslant C$. Hence, by rewriting in terms of $u$, we
get $|{u}(x_{0})|\leqslant Cr^{-\gamma_{p}}$. At last, since $x_{0}\in
B^{*}_{1/2}$ is arbitrary and $r=\frac{1}{2}|x_{0}|$, the proof is finished. ∎

As a consequence, we have the following Harnack-type estimate

######  Corollary 4.3.

Let $u\in C^{6}(B_{1}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(1,2^{\\#}-1)$ and $R=1$. Then, there exists
$C>0$ (depending only on $n$ and $p$) such that

$\sum_{j=0}^{5}|x|^{\gamma_{p}+j}\left|D^{(j)}u(x)\right|\leqslant C\quad{\rm
in}\quad B^{*}_{1/2}.$

###### Proof.

See [18, Theorem 1.1]. ∎

The last lemma is a Harnack-type inequality

######  Lemma 4.4.

Let $u\in C^{6}(B_{1}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(1,2^{\\#}-1)$ and $R=1$. Assume that
$-\Delta u\geqslant 0$ and $\Delta^{2}u\geqslant 0$. Then, there exist
$c_{1},c_{2}>0$ and $r_{0}\in(0,1/16)$ such that

$\sup_{B_{r/2}\setminus\bar{B}_{3r/2}}u\leqslant
c_{1}\inf_{B_{r/2}\setminus\bar{B}_{3r/2}}u\quad{\rm in}\quad B^{*}_{1/16}.$

Moreover, the constant $c_{1}>0$ depends only on $n$, and $p$.

###### Proof.

See [18, Proposition 4.2]. ∎

We also establish some auxiliary results about upper bounds near the isolated
singularity.

######  Lemma 4.5.

Let $u\in C^{6}(B_{1}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(2_{\\#},2^{\\#}-1)$ and $R=1$, and
$v=\mathfrak{F}^{-1}(u)$ be its autonomous Emden–Fowler transformation given
by (2.1). Then, there exists $C>0$ such that

$|v|+|v^{(1)}|+|v^{(2)}|+|v^{(3)}|+|v^{(4)}|+|v^{(5)}|+|\nabla_{\theta}v|+|\Delta_{\theta}v|+|\nabla_{\theta}\Delta_{\theta}v|+|\Delta^{2}_{\theta}v|\leqslant
C\quad{\rm in}\quad\mathcal{C}_{\ln 2}.$

###### Proof.

First, by Lemma 4.2, we know that $v\in C^{6}(\mathcal{C}_{T})$ is uniformly
bounded. Moreover, using Corollary 4.3, one can find $C>0$ such that

$\displaystyle|v^{(1)}|+|\nabla_{\theta}v|$ $\displaystyle\leqslant
C\sum_{j=0}^{1}|x|^{\gamma_{p}}|D^{(j)}u(x)|\leqslant C,$
$\displaystyle|v^{(2)}|+|\Delta_{\theta}v|$ $\displaystyle\leqslant
C\sum_{j=0}^{2}|x|^{\gamma_{p}+j}|D^{(j)}u(x)|\leqslant C,$
$\displaystyle|v^{(3)}|+|\nabla_{\theta}\Delta_{\theta}v|$
$\displaystyle\leqslant
C\sum_{j=0}^{4}|x|^{\gamma_{p}+j}|D^{(j)}u(x)|\leqslant C,$
$\displaystyle|v^{(4)}|+|\Delta^{2}_{\theta}v|$ $\displaystyle\leqslant
C\sum_{j=0}^{4}|x|^{\gamma_{p}+j}|D^{(j)}u(x)|\leqslant C,$
$\displaystyle|v^{(5)}|$ $\displaystyle\leqslant
C\sum_{j=0}^{5}|x|^{\gamma_{p}+j}|D^{(j)}u(x)|\leqslant C,$

for $0<|x|<1/2$, which by a direct rescaling argument proves the lemma. ∎

In the lower critical case, we have the asymptotic upper bound below. Our
approach to proving this result follows the same lines of [29, Theorem 5].
Nonetheless, we write a short proof for the convenience of the reader.

######  Lemma 4.6.

Let $u\in C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$ be a positive singular
solution to ($\mathcal{P}_{6,p,R}$) with $p=2_{\\#}$. Assume that $-\Delta
u\geqslant 0$ and $\Delta^{2}u\geqslant 0$. Then, there exist $C_{0}(n)>0$ and
$0<r_{0}<R$ such that

$|\overline{u}(x)|\leqslant C_{0}(n)|x|^{6-n}(\ln|x|)^{\frac{6-n}{6}}\quad{\rm
for}\quad 0<|x|<r_{0}.$

###### Proof.

Note that for each $0<r<R$ with $r=|x|$, we get that the spherical average
$\overline{u}\in C^{6}(\mathbb{R})$ satisfies the following nonautonomous ODE

$r^{-6}\partial_{r}^{(6)}+M_{5}(n,r)\partial_{r}^{(5)}+M_{4}(n,r)\partial_{r}^{(4)}+M_{3}(n,r)\partial_{r}^{(3)}+M_{2}(n,r)\partial_{r}^{(2)}+M_{1}(n,r)\partial_{r}-f_{\\#}(u)\geqslant
0,$ (4.2)

where the coefficients $M_{j}(n,\cdot):(0,R)\rightarrow\mathbb{R}$ are given
by (A.1) and (A.1). Next, setting

$\psi_{0}=u,\quad\psi_{1}=-\Delta u,\quad{\rm and}\quad\psi_{2}=\Delta^{2}u.$

we can reformulate (4.2) as a system in radial coordinates and the divergent
form, we have

$\begin{cases}-\left(r^{n-1}\psi_{0}^{(1)}(r)\right)^{(1)}=r^{n-1}\psi_{0}(r)&\\\
-\left(r^{n-1}\psi_{1}^{(1)}(r)\right)^{(1)}=r^{n-1}\psi_{1}(r)&\text{ for
}r\in(0,R)\\\
-\left(r^{n-1}\psi_{2}^{(1)}(r)\right)^{(1)}=r^{n-1}f_{\\#}(u(r)).&\end{cases}$
(4.3)

In what follows, the proof will be divided into some steps

Step 1: Either $u\in C(B_{1})\cap H^{3}(B_{1})$ is a continuous weak solution
to ($\mathcal{P}_{6,p,R}$) with $p=2_{\\#}$, or $\lim_{r\rightarrow
0}u(r)=\lim_{r\rightarrow 0}-\Delta u(r)\lim_{r\rightarrow
0}\Delta^{2}u(r)=+\infty$. In particular,

$u^{(1)}(r)\leqslant 0,\quad(-\Delta u)^{(1)}(r)\leqslant 0,\quad{\rm
and}\quad(\Delta^{2}u)^{(1)}(r)\leqslant 0\quad{\rm for}\quad r\in(0,R).$

Suppose that $u\in C^{1}(B^{*}_{1})$ does not have a removable singularity at
the origin. Then $u$ must be unbounded in $B_{1}$, and, in this case, also a
distribution solution to ($\mathcal{P}_{6,p,R}$) in the entire $B_{1}$. Since
$f_{p}(u)\in L^{n/(6-\delta)}(B_{1})$ for some $\delta\in(0,6)$, by bootstrap
argument, $u$ may be extended to a continuous weak solution to
($\mathcal{P}_{6,p,R}$), which is a contradiction.

First, we claim that $u(r)\rightarrow+\infty$ as $r\rightarrow 0$. In fact,
suppose by contradiction that $\liminf_{r\rightarrow 0}u(r)<+\infty$. Then
there exist two real sequences
$\\{m_{k}\\}_{k\in\mathbb{N}},\\{m_{k}\\}_{k\in\mathbb{N}}$ such that
$M_{k}>m_{k}\rightarrow 0$ as $k\rightarrow\infty$ and $u$ assumes local
maxima at $M_{k}$, local minima at $m_{k}$ for all $k\in\mathbb{N}$. Moreover
$u(M_{k})\rightarrow+\infty$ as $k\rightarrow\infty$ while
$\\{u(m_{k})\\}_{k\in\mathbb{N}}\subset\mathbb{R}$ remains bounded. Since
$-\Delta u\geqslant 0$, by the maximum principle, we get

$u(M_{k})\geqslant\min\\{u(m_{k}),u(m_{k+1})\\}\quad{\rm for\ all}\quad
k\in\mathbb{N},$

which contradicts the boundedness of
$\\{u(m_{k})\\}_{k\in\mathbb{N}}\subset\mathbb{R}$. Using the same argument,
it is also clear that $-\Delta u(r)\rightarrow+\infty$ and
$\Delta^{2}u(r)\rightarrow+\infty$ as $r\rightarrow 0$.

By integrating (4.3) on $(\rho,r)$, where $0<\rho\ll R$ small enough, we find

$r^{n-1}\psi_{j}^{(1)}(r)\leqslant\rho^{n-1}\psi_{j}^{(1)}(\rho)<0\quad{\rm
for}\quad j=0,1,2.$

This completes the proof of Step 1.

To continue with the proof, we need to introduce some auxiliary functions.
Namely, we define the functions
$\Psi_{0},\Psi_{1},\Psi_{2}:(0,R)\rightarrow\mathbb{R}$ given by

$\Psi_{0}(r)=r\psi_{0}(r),\quad\Psi_{1}(r)=r\psi_{1}(r)+(n-2)\psi^{(1)}_{1},\quad\Psi_{2}(r)=r\psi_{2}+(n-2)\psi^{(1)}_{2},$

which satisfy

$\begin{cases}-\Delta\Psi_{0}=\Psi_{1}\\\ -r^{-1}\Psi_{1}^{(1)}=\Psi_{2}\\\
-r^{-1}\Psi_{2}^{(1)}=(-\Delta)^{3}\Psi_{0}.\end{cases}$ (4.4)

Step 2. There exists $0<r_{0}\ll 1$ and $0<\rho<r_{0}$, it follows
$\lim_{r\rightarrow 0}r^{n-2(2-j)}\psi_{j}(r)=0$ and $\Psi_{j}(r)\geqslant 0$
in $r\in(0,\rho)$ for $j=0,1,2$.

Since $u\in L^{1}_{\rm loc}(B_{R})$ is a distribution solution to
($\mathcal{P}_{6,p,R}$) with $p=2_{\\#}$, it follows that
$\liminf_{r\rightarrow 0}r^{n-2(2-j)}\psi_{j}(r)=0$ for $j=0,1,2$.

We start with the case $j=2$. Indeed, let us denote by
$\psi_{0}^{*}(r^{-1})=r^{n-6}\psi_{0}(r)$ the sixth order Kelvin transform
$\psi_{0}=u$ with respect to $\partial B_{1}$. Hence, by [29, Lemma 3], we
find that $-\Delta^{3}\psi_{0}^{*}\geqslant 0$ in
$\mathcal{I}^{*}(B_{R}):=\mathbb{R}^{n}\setminus B^{*}_{1}$. Then, by Step 1,
$\psi_{0}^{*}$ is monotone near $+\infty$, which implies that
$r^{n-6}\psi_{0}(r)$ is monotone near the origin. Hence $\lim_{r\rightarrow
0}r^{n-6}\psi_{0}(r)>0$ exists and is positive for small $0<r\ll 1$.

To verify the case $j=1$, let us denote by
$\psi_{1}^{*}(r^{-1})(r^{-1})=r^{n-4}u(r)$ the fourth order Kelvin transform
of $\psi_{1}=\Delta u$ with respect to $\partial B_{1}$. Hence, by [29, Lemma
3], we find that $\Delta^{2}\psi_{1}^{*}\geqslant 0$ in
$\mathcal{I}^{*}(B_{R})$. Then, from Step 1, we conclude that $\psi_{1}^{*}$
is monotone near $+\infty$, which implies that $r^{n-4}\psi_{1}(r)$ is
monotone near the origin. Hence $\lim_{r\rightarrow 0}r^{n-4}\psi_{1}(r)>0$
exists and is positive for small $0<r\ll 1$.

Finally, for the case $j=0$, let us denote by
$\psi_{2}^{*}(r^{-1})=r^{n-2}u(r)$ the second order Kelvin transform of
$\psi_{2}=\Delta^{2}u$ with respect to $\partial B_{1}$. Hence, by [29, Lemma
3], we find that $-\Delta\psi_{2}^{*}\geqslant 0$ in $\mathcal{I}^{*}(B_{R})$.
Then, from Step 1, we conclude that $\psi_{2}^{*}$ is monotone near $+\infty$,
which implies that $r^{n-2}\psi_{2}(r)$ is monotone near the origin. Hence
$\lim_{r\rightarrow 0}r^{n-2}\psi_{2}(r)>0$ exists and is positive for small
$0<r\ll 1$.

The proof of Step 2 is concluded.

Step 3. $\lim_{r\rightarrow 0}r^{n-2}\Psi_{0}(r)=0$ and
$\Psi_{0}^{(1)}(r)+(n-2)\Psi_{0}(r)\geqslant 0$.

Finally, from the case $j=2$, it holds that $0\leqslant-
ru^{(1)}(r)\leqslant(n-6)u(r)$ in $(0,\rho)$. Thus, multiplying the last
inequality by $r^{n-2}$ and letting $r\rightarrow 0$, we get that
$\lim_{r\rightarrow 0}r^{n-1}u^{(1)}(r)=0$, and so

$\lim_{r\rightarrow 0}r^{n-2}\Psi_{0}(r)=\lim_{r\rightarrow
0}\left(r^{n-1}u^{(1)}(r)+(n-6)r^{n-2}u(r)\right)=0.$

Next, for $0<r_{0}\ll 1$ small given by Step 2, it follows from (4.4) that
$-\Delta\Psi_{0}\geqslant 0$ and $\Psi_{0}\geqslant 0$ in $B^{*}_{r_{0}}$.
Therefore, by considering $\Psi_{0}^{*}(r^{-1})=r^{n-2}\Psi_{0}(r)$, we obtain
$-\Delta\Psi_{0}^{*}\geqslant 0$ and $\Psi_{0}^{*}\geqslant 0$ in
$\mathcal{I}^{*}(B_{R})$. In addition, $\Psi_{0}^{*}(r^{-1})\rightarrow 0$.
Consequently, a direct application of the maximum principle, yields
$\Psi_{0}^{*}(r^{-1})\leqslant 0$ for all $r^{-1}>r_{0}^{-1}$, which in turn
proves completes the proof of Step 3.

Step 4. $\Psi_{0}^{(1)}(r)\leqslant 0$ for $0<r\ll 1$, then
$\Psi_{0}(r)\rightarrow+\infty$ as $r\rightarrow 0$.

We observe that from Step 3 it follows that
$-(r^{n-1}\Psi_{0}^{(1)})^{(1)}\geqslant 0$ for $r\in(0,\rho_{1})$, which by
integrating on $(\varepsilon,r)$, with $r\in(0,\rho_{1})$, yields

$-r^{n-1}\Psi_{0}^{(1)}(r)+\varepsilon^{n-1}\Psi_{0}^{(1)}(\varepsilon)\geqslant
0.$

Hence, by letting $\varepsilon\rightarrow 0$ in the last inequality we find
that $\Psi_{0}^{(1)}(r)\leqslant 0$ for $r\in(0,\rho_{1})$. At last, suppose
by contradiction that there exists $C>0$ such that $\Psi_{0}(r)\leqslant C$
for all $r>0$, which would imply that

$\Psi_{0}(r)=r^{7-n}(r^{n-6}u(r))^{(1)}\leqslant C.$

On the other hand, integrating the last inequality on $(0,r)$, we would obtain
$u(r)\leqslant C(n-6)^{-1}$, which is a contradiction of the origin is a non-
removable singularity and proves Step 4.

Step 5. $|\overline{u}(x)|\leqslant C_{0}(n)|x|^{6-n}(\ln|x|)^{\frac{6-n}{6}}$
for $0<|x|<r_{0}$.

Let us set

$\varphi_{0}(r)=r^{n-6}u(r),\quad\varphi_{1}(r)=r^{n-4}\varphi_{0}(r),\quad{\rm
and}\quad\varphi_{2}(r)=r^{n-2}\varphi_{1}(r).$

Whence, using Step 1, we get that $6\varphi_{0}^{(1)}(r)\geqslant 0$,
$\varphi_{1}^{(1)}(r)$ and $\varphi_{2}^{(2)}(r)\geqslant 0$ in $(0,r_{0})$.
Now, since $\Psi_{1}(r)=$ $r^{4-n}\varphi_{1}^{(1)}(r)$, one has from Step 2
the following holds

$-\left(r\Psi_{0}^{(1)}(r)+(n-2)\Psi_{0}(r)\right)^{(1)}=r^{4-n}\varphi_{1}^{(1)}(r),$

which by integrating on $(r,r_{0})$, yields

$-\left(r\Psi_{0}^{(1)}(r_{0})+(n-2)\Psi_{0}(r_{0})\right)+r\Psi_{0}^{(1)}(r)+(n-2)\Psi_{0}(r)\geqslant\varphi_{1}(r)\left(r^{4-n}-r_{0}^{4-n}\right).$

Now, since $\varphi_{0}(r)\rightarrow 0$ as $r\rightarrow 0$, one can find
$r_{1}\in(0,r_{0})$ such that

$\left(r\Psi_{0}^{(1)}(r_{0})+(n-2)\Psi_{0}(r_{0})\right)-\varphi_{1}(r)r_{0}^{4-n}\geqslant
0,\quad\text{ on }\quad(0,r_{1}),$

which, since from Step 5 implies $\Psi_{0}^{(1)}(r)\leqslant 0$ for $r$ small,
one can find $C>0$ such that

$\Psi_{0}(r)\geqslant Cr^{4-n}\varphi_{0}(r)\quad{\rm in}\quad(0,r_{1}),$

and so $\varphi_{0}^{(1)}(r)\geqslant Cr^{-1}\varphi_{1}(r)$. Hence, since
$\psi_{1}(r)\geqslant Cf_{\\#}(u(r))$, we get $\varphi_{1}(r)\geqslant
Cf_{\\#}(\varphi_{0}(r))$, which yields

$\varphi_{0}^{(1)}(r)\geqslant Cr^{-1}f_{\\#}(\varphi_{0}(r))\quad{\rm
for}\quad 0<r<r_{1}.$

Up to a rescaling, we may assume that the last inequality holds for
$r\in(0,1)$. Thus, by a direct integration, we obtain
$\varphi_{0}(r)^{-\frac{6}{n-6}}\geqslant C\ln r$ as $r\rightarrow 0$, which
in turn leads to

$u(r)r^{n-6}(\ln r)^{\frac{n-6}{6}}\leqslant C\quad{\rm as}\quad r\rightarrow
0.$

This completes the proof of the lemma. ∎

##  5\. Asymptotic radial symmetry

We prove the first part of Theorem 1 asserting that solutions to
($\mathcal{P}_{6,p,R}$) are radially symmetric about the origin. This symmetry
will later be used to convert the singular PDE into a non-singular ODE on the
cylinder.

Before that, we need to establish some preliminaries.

###  5.1. Kelvin transform

Later we will employ the moving spheres technique, which is based on the sixth
order Kelvin transform of a real valued function. To define the Kelvin
transform, we need to establish the concept of inversion about a sphere
$\partial B_{\mu}(x_{0})$, which is a map
$\mathcal{I}_{x_{0},\mu}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\setminus\\{x_{0}\\}$
given by $\mathcal{I}_{x_{0},\mu}(x)=x_{0}+K_{x_{0},\mu}(x)^{2}(x-x_{0})$,
where $K_{x_{0},\mu}(x)=\mu/|x-x_{0}|$.

######  Definition 5.1.

For any $u\in C^{6}(B_{R}^{*})$, let us consider the sixth order Kelvin
transform about the sphere with center at $x_{0}\in\mathbb{R}^{n}$ and radius
$\mu>0$ defined by

$u_{x_{0},\mu}(x)=K_{x_{0},\mu}(x)^{n-6}u\left(\mathcal{I}_{x_{0},\mu}(x)\right).$

######  Lemma 5.2.

If $u\in C^{6}(B_{R}^{*})$ is a solution to ($\mathcal{P}_{6,p,R}$), then
$u_{x_{0},\mu}\in C^{6}(B_{R}^{*}\setminus\\{x_{0}\\})$ is a solution to

$(-\Delta)^{3}u_{x_{0},\mu}=K_{x_{0},\mu}(x)^{(n-6)p-(n+6)}f_{p}(u_{x_{0},\mu})\quad{\rm
in}\quad B_{R}^{*}\setminus\\{x_{0}\\}.$

###### Proof.

It directly follows by using [33, Lemma 3.6]. ∎

###  5.2. Integral representation

Now we use a Green identity to transform the sixth order differential system
($\mathcal{P}_{6,p,R}$) into an integral system. In this way, we can avoid
using the classical form of the maximum principle, and a sliding method is
available [30, 21], which will be used to classify solutions. Besides, in this
setting is also possible to prove regularity through a barrier construction.

We start with the following result

######  Lemma 5.3.

Let $u\in C^{6}(B_{R}^{*})$ be a positive solution to ($\mathcal{P}_{6,p,R}$)
with $p\in(1,+\infty)$. Assume that $-\Delta u\geqslant 0$ and
$\Delta^{2}u\geqslant 0$. Then, $u\in L^{p}(B_{1})$. In particular, if
$p\in(2_{\\#},+\infty)$, then $u\in L^{1}(B_{1})$ is a distribution solution
to ($\mathcal{P}_{6,p,R}$), that is, for all positive $\phi\in
C^{\infty}_{c}(B_{1})$, one has

$\int_{B_{1}}u(-\Delta)^{3}{\phi}\mathrm{d}x=\int_{B_{1}}f_{p}(u){\phi}\mathrm{d}x.$

###### Proof.

See the proof of [34, Lemma 3.1] (see also [33, Theorem 3.7]). ∎

The next result uses the Green identity to convert ($\mathcal{P}_{6,p,R}$)
into an integral system. We divide this results in two cases, namely
$R<+\infty$ and $R=+\infty$.

######  Lemma 5.4.

Let $u\in C^{6}(B_{R}^{*})$ be a positive solution to ($\mathcal{P}_{6,p,R}$)
with $p\in(1,+\infty)$. Assume that $-\Delta u\geqslant 0$ and
$\Delta^{2}u\geqslant 0$.

  * (i)

If $R<+\infty$, then $($up to constant$)$ there exists $r_{0}>0$ such that

$u(x)=\int_{B_{r_{0}}}|x-y|^{6-n}{f}_{p}(u)\mathrm{d}y+\psi(x),$ (5.1)

where $\psi>0$ satisfies $(-\Delta)^{3}\psi=0$ in $B_{r_{0}}$. Moreover, one
can find a constant $C(\widetilde{r})>0$ such that
$\|\nabla\ln\psi\|_{C^{0}(B_{\widetilde{r}})}\leqslant C(\widetilde{r})$ for
all $0<\widetilde{r}<r_{0}$.

  * (ii)

If $R=+\infty$, then $($up to constant$)$, it follows

$u(x)=\int_{\mathbb{R}^{n}}|x-y|^{6-n}{f}_{p}(u)\mathrm{d}y.$

###### Proof.

For (i), the proof is a simple adaptation of [19, Lemma 2.3]. For (ii) see [9,
Theorem 4.3] (see also [8]). ∎

###  5.3. Sliding technique

Now we use the preliminary results to run an integral moving spheres
technique. We use an asymptotic moving spheres technique in the same spirit of
[13]. Although our proof are almost the same we include it here for the sake
fo completeness.

######  Proposition 5.5.

Let $u\in C^{6}(B_{R}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(1,2^{\\#}-1)$. Assume that $-\Delta
u\geqslant 0$ and $\Delta^{2}u\geqslant 0$. Then,

$u(x)=(1+\mathcal{O}(|x|))\overline{u}(x)\quad{\rm as}\quad x\rightarrow 0.$

###### Proof.

Initially, if the origin is a removable singularity, then the conclusion is
clear. Hence, we suppose that the origin is a non-removable singularity.

We divide the proof into some claims.

Claim 1: There exists small $0<\varepsilon\ll 1$ such that for any $z\in
B^{*}_{\varepsilon/2}$, it holds

$u_{z,r}\leqslant u\quad{\rm in}\quad
B_{1}\setminus\left(B_{r}(z)\cup\\{0\\}\right)\quad\mbox{for}\quad
0<r\leqslant|z|.$ (5.2)

Indeed, the proof follows almost the same lines as the one in [19, Lemma 3.2],
so we omit it.

In the next claim, we provide some estimates to be used later in the proof.

Claim 2: There exists $z\in B^{*}_{\varepsilon/2}$, $0<r<|z|$ and $\mu_{*}\gg
1$ large such that

$\frac{y_{\mu}}{|y_{\mu}|^{2}}-z=\left(\frac{r}{\left|y/|y|^{2}-z\right|}\right)^{2}\left(\frac{y}{|y|^{2}}-z\right)\quad{\rm
and}\quad\frac{|y_{\mu}|}{|y|}\leqslant\frac{1}{r}\left|\frac{y}{|y|^{2}}-z\right|\quad{\rm
for\ any}\quad\mu>\mu_{*}.$ (5.3)

Here $y_{\mu}=y+2(\mu-y\cdot{\bf e}){{\bf e}}$ is the reflection of $y$ about
the hyperplane $\partial H_{\mu}({\bf e})$, where $H_{\mu}({\bf
e})=\\{x:\langle x,{\bf e}\rangle>\mu\\}$ and ${\bf e}\in\mathbb{S}^{n-1}$. In
other words, ${y_{\mu}}{|y_{\mu}|^{-2}}$ is the reflection point of
${y}{|y|^{-2}}$ about $\partial B_{r}(z)$.

As matter of fact, choosing $r=|z|$, it involves an elementary computation, as
follows

$z=\frac{y}{|y|^{2}}+\frac{\left|y_{\mu}\right|^{2}}{|y|^{2}-\left|y_{\mu}\right|^{2}}\left(\frac{y}{|y|^{2}}-\frac{y_{\mu}}{\left|y_{\mu}\right|^{2}}\right)=\frac{\left(y-y_{\mu}\right)}{|y|^{2}-\left|y_{\mu}\right|^{2}}.$

Next, we establish a comparison involving the Kelvin transform of a component
solution with itself.

Claim 3: For any $\mu>\frac{1}{\varepsilon}$ and ${\bf e}\in\partial B_{1}$,
if $\langle x,{\bf e}\rangle>\mu$ and $|y_{\mu}|>1$, it holds
$u_{0,1}(y)\leqslant u_{0,1}\left(y_{\mu}\right).$

In fact, to prove the last inequality, let us note first that $y\in
B_{1/\varepsilon}$, if and only if, ${y}{|y|^{-2}}\in B_{\varepsilon}$. Now
given $y\in\mathbb{R}^{n}$ such that $\langle y,{\bf e}\rangle>\mu$,
$|y_{\mu}|>1$ and $0<r<|z|<{\varepsilon}/{2}$ satisfying (5.3). Let us define
$x={y}{|y|^{-2}}$ and $x_{z,r}={y_{\mu}}{|y_{\mu}|^{-2}}$. Then, since
$\langle y,{\bf e}\rangle>\mu>{\varepsilon}^{-1}$ and $\|y_{\mu}|>1$, we have
$x\in B_{r}(z)$ and $x_{z,r}\in B_{1}\setminus B_{r}(z)$. Hence, using (5.2)
and (5.3), we find

$u_{0,1}(y)\leqslant u_{0,1}(y_{\mu}),$

which proves the claim.

Ultimately, using Claim 3, we invoke [5, Theorem 6.1 and Corollary 6.2] to
find $C>0$, independent of $\varepsilon>0$, such that if
$|y|\geqslant|x|+C\varepsilon^{-1}$, it follows $u_{0,1}(y)\leqslant
u_{0,1}(x)$. Therefore, since $u_{0,1}$ is positive and satisfies $-\Delta
u_{0,1}\geqslant 0$ and $\Delta^{2}u_{0,1}\geqslant 0$, the last inequality
implies

$u_{0,1}(|x|)=\left(1+\mathcal{O}\left(\frac{1}{R}\right)\right)\left(\inf_{\partial
B_{R}}u_{0,1}\right)\quad{\rm as}\quad R\rightarrow+\infty,$

uniformly on $\partial B_{R}$, which in terms of $u$ implies the desired
asymptotic radial symmetry with respect to the origin, and the proof is
concluded. ∎

##  6\. The limiting Pohozaev levels

After proving the radial symmetry of singular solutions to
($\mathcal{P}_{6,p,R}$), we shall classify them in the blow-up and shrink-down
limit. The idea is to use a blow-up/shrink-down analysis, which comes from
tangent cone techniques from minimal hypersurface theory, and will be
described in the sequel.

For any $u$ solution to ($\mathcal{P}_{6,p,R}$) and $\lambda>0$, let us define
the following $\lambda$-rescaling solution given by

$\widehat{u}_{\lambda}(x):=\lambda^{\gamma_{p}}u(\lambda x),$ (6.1)

where we recall that $\gamma_{p}={6}{(p-1)^{-1}}$. Notice that the
$\lambda$-rescaled solution is still a positive solution to
($\mathcal{P}_{6,p,R}$) with $R=\lambda^{-1}$. Moreover, we get the following
scaling invariance

$\mathcal{P}_{\rm sph}(r,\widehat{u}_{\lambda})=\mathcal{P}_{\rm sph}(\lambda
r,u).$ (6.2)

This follows by directly using the inverse cylindrical transform as in Remark
3.5. Besides, by a blow-up (resp. shrink-down) solution $u_{0}$ (resp.
$u_{\infty}$) to ($\mathcal{P}_{6,p,R}$), we mean the limit
$u_{0}:=\lim_{\lambda\rightarrow 0}\widehat{u}_{\lambda}$ (resp.
$u_{\infty}:=\lim_{\lambda\rightarrow+\infty}\widehat{u}_{\lambda}$). In fact,
utilizing some a priori estimates and the compactness of the family
$\\{\widehat{u}_{\lambda}\\}_{\lambda>0}\subset C^{6,\alpha}_{\rm
loc}(\mathbb{R}^{n})$ for some $\alpha\in(0,1)$, these limits will be proven
to exist. Next, we study the limit Pohozaev functional both as $r\rightarrow
0$ (blow-up) and $r\rightarrow+\infty$ (shrink-down), this will give the
desired information about the asymptotic behavior for solutions to
($\mathcal{P}_{6,p,R}$).

Here is our main result of this section:

######  Proposition 6.1.

Let $u$ be a positive singular solution to ($\mathcal{P}_{6,p,\infty}$) with
$p\in(1,2^{\\#}-1)$. Assume that $u$ is homogeneous of degree $-\gamma_{p}$.

  * (a)

If $p\in(1,2_{\\#}]$, then $u\equiv 0$;

  * (b)

If $p\in(2_{\\#},2^{\\#}-1)$, then $\mathcal{P}_{\rm cyl}(r,u)$ converges both
as $r\rightarrow 0$ and $r\rightarrow+\infty$, namely

$\\{\mathcal{P}_{\rm sph}(0,u),\mathcal{P}_{\rm
sph}(+\infty,u)\\}=\\{-\omega_{n-1}\ell^{*}_{p},0\\},\quad\mbox{where}\quad\ell^{*}_{p}=\frac{p-1}{2(p+1)}K_{0}(n,p)^{\frac{p+1}{p-1}}.$
(6.3)

First, we prove that the invariance of the Pohozaev invariant is equivalent to
the homogeneity of the blow-up limit solutions to
($\mathcal{P}_{6,p,\infty}$). A similar result can also be found in [10],
where a different type of functional is considered.

######  Lemma 6.2.

Let $u$ be a positive singular solution to ($\mathcal{P}_{6,p,\infty}$) with
$p\in(2_{\\#},2^{\\#}-1)$. Then, $\mathcal{P}_{\rm sph}(r,\mathcal{U})$ is
constant, if and only if, for any $r\in(r_{1},r_{2})$ with $0<r_{1}\leqslant
r_{2}<R<+\infty$, $u$ is homogeneous of degree $-\gamma_{p}>0$ in
$B_{r_{2}}\setminus\bar{B}_{r_{1}}$, that is,

$u(x)=|x|^{-\gamma_{p}}u\left(\frac{x}{|x|}\right)\quad{\rm in}\quad
B_{r_{2}}\setminus\bar{B}_{r_{1}}.$

###### Proof.

Notice that if $p\neq 2^{\\#}-1$, then $K_{0}(n,p)\neq 0$. Thus, supposing
that $\mathcal{P}_{\rm sph}(r,u)$ is constant for $r_{1}<r<r_{2}$, together
with Remark 3.3 yields that $\partial_{\nu}u=\gamma_{p}r^{-1}u$ on $\partial
B_{r}$ for any $r_{1}<r<r_{2}$, where $\nu$ is the unit normal pointing
towards the origin. Therefore, $u$ is homogeneous of degree $-\gamma_{p}$ in
$B_{r_{2}}\setminus\bar{B}_{r_{1}}$, which concludes the proof. ∎

The following lemma provides an upper bound estimate for singular solutions to
($\mathcal{P}_{6,p,R}$).

######  Lemma 6.3.

Let $u$ be a positive singular solution to ($\mathcal{P}_{6,p,\infty}$) with
$p\in(1,2^{\\#}-1)$. Then,

$u(x)\leqslant\left(\frac{p-1}{2n}\right)^{-\frac{1}{p-1}}|x|^{-\gamma_{p}}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\\{0\\}.$

###### Proof.

By [26, Theorem 6.1], we know that $-\Delta u\geqslant 0$ and
$\Delta^{2}u\geqslant 0$ in $\mathbb{R}^{n}\setminus\\{0\\}$, which, by using
the extended maximum principle [15, Theorem 1], gives us

$\liminf_{x\rightarrow 0}u(x)>0.$

Considering $\varphi=u^{1-p}$, a direct computation, provides

$\Delta\varphi\geqslant\frac{p}{p-1}\frac{|\nabla\varphi|^{2}}{\varphi}+p-1\quad\mbox{in}\quad\mathbb{R}^{n}\setminus\\{0\\}.$

Thus, for any $r>0$, let us consider the auxiliary function
$\widetilde{\varphi}(x)=\varphi(x)-\frac{p-1}{2n}|x|^{2}$, which satisfies
$-\Delta\widetilde{\varphi}\leqslant 0$ in $B^{*}_{r}$. Furthermore, (6)
implies that $\widetilde{\varphi}$ is bounded close to the origin, and thus,
again, by the extended maximum principle, we find

$0\leqslant\limsup_{x\rightarrow
0}\widetilde{\varphi}(x)\leqslant\sup_{\partial
B_{r}}\widetilde{\varphi}=\sup_{\partial B_{r}}\varphi-\frac{p-1}{2n}r^{2},$

which yields

$\inf_{\partial
B_{r}}u\leqslant\left(\frac{p-1}{2n}\right)^{-\frac{1}{p-1}}r^{-\gamma_{p}}.$

Finally, a direct application of Proposition 5.5 finishes this proof. ∎

As a consequence of this uniform upper bound, we prove the compactness of the
family $\\{\widehat{u}_{\lambda}\\}_{\lambda>0}\subset C^{6,\alpha}_{\rm
loc}(\mathbb{R}^{n})$, for some $\alpha\in(0,1)$, which provides the existence
of both blow-up and shrink-down limits for the scaling family defined by
(6.1).

######  Lemma 6.4.

Let $u$ be a positive singular solution to ($\mathcal{P}_{6,p,\infty}$) with
$p\in(1,2^{\\#}-1)$. Then, $\\{\widehat{u}_{\lambda}\\}_{\lambda>0}\subset
C^{6,\alpha}_{\rm loc}(\mathbb{R}^{n})$ is uniformly bounded, for some
$\alpha\in(0,1)$.

###### Proof.

If the origin is a removable singularity of $\widehat{u}_{\lambda}$ for all
$\lambda>0$, according to Theorem A, $u$ is trivial and the conclusion
follows.

On the other hand, assuming that the origin is a removable singularity, Lemma
6.3 provides that $\\{\widehat{u}_{\lambda}\\}_{\lambda>0}$ is globally
bounded in $\mathbb{R}^{n}\setminus\\{0\\}$. Thus, we know that
$\\{\widehat{u}_{\lambda}\\}_{\lambda>0}$ is uniformly bounded in each compact
subset of $K\subset\mathbb{R}^{n}\setminus\\{0\\}$. Moreover, since for each
$\lambda>0$, the scaling $\widehat{u}_{\lambda}$ also satisfies
($\mathcal{P}_{6,p,\infty}$), it follows from standard elliptic estimates that
$\\{\widehat{u}_{\lambda}\\}_{\lambda>0}$ is uniformly bounded in
$C^{6,\alpha}(K)$, for some $\alpha\in(0,1)$, which concludes the proof. ∎

Recall that $\mathcal{P}_{\rm sph}(r,u)$ is the Pohozaev functional introduced
in Remark 3.3, which by Proposition 3.4 is monotonically nonincreasing in
$r>0$ when $p\in(1,2^{\\#}-1)$.

######  Lemma 6.5.

Let $u_{0}$ $($or $u_{\infty}$$)$ be a positive singular blow-up $($or shrink-
down$)$ solution to ($\mathcal{P}_{6,p,R}$) under the family of scalings
$\\{\widehat{u}_{\lambda}\\}_{\lambda>0}$. Then, $\mathcal{P}_{\rm
sph}(r,u_{0})\equiv\mathcal{P}_{\rm sph}(0,u)$ $($or $\mathcal{P}_{\rm
sph}(r,u_{\infty})\equiv\mathcal{P}_{\rm sph}(\infty,u)$$)$ is constant for
all $r>0$. In particular, both $u_{0}$ and $u_{\infty}$ are homogeneous of
degree $-\gamma_{p}$.

###### Proof.

Let $\\{\lambda_{k}\\}_{k\in\mathbb{N}}\subset(0,+\infty)$ be a blow-up
sequence such that $\lambda_{k}\rightarrow 0$, and
$u_{0}\in{C^{6,\alpha}(\mathbb{R}^{n}\setminus\\{0\\})}$ be its blow-up limit,
that is, $\widehat{u}_{\lambda_{k}}\rightarrow u_{0}$ in ${C_{\rm
loc}^{6,\alpha}(\mathbb{R}^{n}\setminus\\{0\\})}$ as $k\rightarrow+\infty$,
for some $\alpha\in(0,1)$. Now, using Proposition 3.4 and Lemma 6.4, one
concludes that there exists the limiting level $\mathcal{P}_{\rm
sph}(0,u):=\lim_{r\rightarrow 0}\mathcal{P}_{\rm sph}(r,u)$. Moreover, due to
the scaling invariance of the Pohozaev functional in (6.2), for any $r>0$, it
follows

$\mathcal{P}_{\rm sph}(r,u_{0})=\lim_{k\rightarrow+\infty}\mathcal{P}_{\rm
sph}(r,\widehat{u}_{\lambda_{k}})=\lim_{k\rightarrow+\infty}\mathcal{P}_{\rm
sph}(r\lambda_{k},u)=\mathcal{P}_{\rm sph}(0,u),$

which finishes the proof of the first assertion. Now, we can check that the
homogeneity follows from Lemma 6.2. Finally, notice that the same argument can
readily be employed, replacing the blow-up limit by the shrink-down limit, so
we omit it here. ∎

######  Lemma 6.6.

Let $u$ be a positive singular solution to ($\mathcal{P}_{6,p,\infty}$) with
$p\in(1,2^{\\#}-1)$. Assume that $u$ is homogeneous of degree $-\gamma_{p}$.

  * (a)

If $p\in(1,2_{\\#}]$, then $u\equiv 0$.

  * (b)

If $p\in(2_{\\#},2^{\\#}-1)$, then either $u\equiv 0$, or $u\equiv
K_{0}(n,p)^{\frac{1}{p-1}}|x|^{-\gamma_{p}}$.

###### Proof.

Since $u$ is homogeneous of degree $-\gamma_{p}$, the Emden–Fowler
transformation $v=\mathfrak{F}(u)$ given by (2.1) satisfies

$-\mathscr{L}_{\theta}v+f_{p}(v)=0\quad\mbox{on}\quad\mathbb{S}_{t}^{n-1},$
(6.4)

where
$\mathscr{L}_{\theta}:=\Delta_{\theta}^{3}+L_{0}(n,p)\Delta_{\theta}^{2}+J_{0}(n,p)\Delta_{\theta}v+K_{0}(n,p)$.
Now we divide the the proof into two cases:

Case 1: $p\in(1,2_{\\#}]$.

Initially, one can verify that $\mathscr{L}_{\theta}(v)\leqslant 0$ on
$\mathbb{S}_{t}^{n-1}$. Next, observe that $\mathscr{L}_{\theta}$ is the
composition of three elliptic operators. This, together with [18, Theorem
1.2], implies that $v\in C^{6}(\mathcal{C}_{T})$ does not attain any strict
local minimum on $\mathbb{S}_{t}^{n-1}$. Therefore, since
$\mathbb{S}_{t}^{n-1}$ is a compact manifold, it follows that $v$ is constant,
which yields $v\equiv v_{0}$. Nevertheless, using that $K_{0}(n,p)\leqslant
0$, any positive constant solution to (6.4) is trivial. By using the inverse
of the Emden–Fowler transformation, it holds that $u$ is trivial on $\partial
B_{1}$, which by the superharmonicity property, implies that $u$ is trivial in
the whole domain. This conclusion finishes the proof of the first case, and so
part (a) of the lemma follows.

Case 2: $p\in(2_{\\#},2^{\\#}-1)$.

Assume that $u$ is a nontrivial limit solution in the punctured space. Hence,
since each component of $u$ is positive and satisfies $-\Delta u\geqslant 0$
and $\Delta^{2}u\geqslant 0$, it quickly follows that $u>0$ in
$\mathbb{R}^{n}\setminus\\{0\\}$. By homogeneity, the origin is a non-
removable singularity of $u\in C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$. Hence,
by Proposition 5.5, $u$ is radially symmetric; thus, $u\equiv u_{0}$ is a
positive constant vector. Moreover, by (6.4), it holds
$u_{0}=K_{0}(n,p)^{\frac{1}{p-1}},$ which, by using the homogeneity of $u$ and
Lemma 6.2, finishes the proof of the second case, and so (b) holds. ∎

At last, we can prove the main result of this part.

###### Proof of Proposition 6.1.

Let $u_{0}$ and $u_{\infty}$ be, respectively, a blow-up and a shrink-down
limit of $u$. According to Lemma 7.8 both $u_{0}$ and $u_{\infty}$ are
homogeneous of degree $-\gamma_{p}$. In what follows, we divide the rest of
the proof into two cases:

Case 1: $p\in(1,2_{\\#}]$.

Here, it follows from Lemma 6.6 (a) that both $u_{0}$ and $u_{\infty}$ are
trivial, which, by Lemma 6.5, provides $\mathcal{P}_{\rm
sph}(0,u)=\mathcal{P}_{\rm sph}(+\infty,u)=0$. In addition, using the
monotonicity property of the Pohozaev functional, we find $\mathcal{P}_{\rm
sph}(r,u)=0$ for all $r>0$. Hence, by Lemma 6.2, $u$ is homogeneous of degree
$-\gamma_{p}$. Therefore, the proof of (a) of Proposition 6.1 is now an
immediate consequence of Lemma 6.6 (a).

Case 2: $p\in(2_{\\#},2^{\\#}-1)$.

Initially, by Lemmas 6.5 and 6.6 (b), any blow-up $u_{0}$ is either trivial or
has the form (1). If $u_{0}$ is trivial, then clearly $\mathcal{P}_{\rm
sph}(r,u_{0})=0$ for all $r>0$, which combined with Lemma 6.5 implies that
$\mathcal{P}_{\rm sph}(0,u)=0$. Otherwise, a simple computation shows
$\mathcal{P}_{\rm sph}(r,u_{0})=-\omega_{n-1}\ell^{*}_{p}$ for all $r>0$.
Therefore, using again Lemma 6.5, we have $\mathcal{P}_{\rm
sph}(0,u)=-\ell^{*}_{p}$. Since the converse trivially follows, we obtain that
$\mathcal{P}_{\rm sph}(0,u)\in$ $\\{-\omega_{n-1}\ell^{*}_{p},0\\}$. Moreover,
$\mathcal{P}_{\rm sph}(0,u)=0$, if and only if, all the blow-ups are trivial,
whereas $\mathcal{P}_{\rm sph}(0,u)=-\omega_{n-1}\ell^{*}_{p}$, if and only
if, all the blow-ups are of the form (1). In the case of shrink-down
$u_{\infty}$ solution, the strategy is similar, so we omit it. These
conclusions finish the proof of Case 2, and therefore the proposition holds. ∎

##  7\. Local asymptotic behavior

In this section, we present the proof of Theorem 1. First, we show an
asymptotic symmetry result, which permits us to migrate to an ODE setup.
Second, we prove some universal upper bound estimates, not depending on the
superharmonic assumption. However, we should emphasize that in the rest of the
argument, there is a significant change of behavior of radial solutions
($\mathcal{C}_{p,T}$) for distinct values of the power $p\in(1,2^{\\#}-1]$.
This difference occurs due to the change of sign of the coefficients in the
tri-Laplacian written in cylindrical coordinates. These signs control the
Lyapunov stability of the solutions to linearized operator around a limit
blow-up solution, and so the asymptotic behavior of the local solutions near
the isolated singularity.

We divide our argument into three subsections, where we prove, respectively,
the local behavior near the isolated singularity for the situations:
$p\in(1,2^{\\#}-1)$ in Subsection 7.1, $p=2_{\\#}$ in Subsection 7.2, and
$p\in(2_{\\#},2^{\\#}-1)$ in Subsection 7.3.

###  7.1. Serrin–Lions case

We prove Theorem 1 (a). The asymptotic analysis for this case is
straightforward. We are based in the approach given by [13].

In the sequel, we aim to prove the following proposition

######  Proposition 7.1.

Let $u\in C^{6}(B_{R}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(1,2_{\\#})$. Assume that $-\Delta
u\geqslant 0$ and $\Delta^{2}u\geqslant 0$. Then, there exist $C_{1},C_{2}>0$
$($depending on $u$$)$ such that $C_{1}|x|^{6-n}\leqslant u(x)\leqslant
C_{2}|x|^{6-n}$ for $0<|x|\ll 1$, or equivalently, $u(x)\simeq|x|^{6-n}$ as
$x\rightarrow 0$.

First, we prove an upper bound estimate based on a Green identity from Lemma
5.4 (i).

######  Lemma 7.2.

Let $u\in C^{6}(B_{R}^{*})$ be a positive solution to ($\mathcal{P}_{6,p,R}$)
with $p\in(1,2_{\\#})$. Assume that $-\Delta u\geqslant 0$ and
$\Delta^{2}u\geqslant 0$. Then, there exists $C_{2}>0$, depending only on $u$,
such that $u(x)\leqslant C_{2}|x|^{4-n}$ as $x\rightarrow 0$.

###### Proof.

Initially, by Lemma 5.3, we have that $u\in L^{p}(B_{1})$. Moreover, since
$p\in(1,2_{\\#})$ and $u\in C^{6}(B_{R}^{*})$ satisfies the Harnack inequality
in Lemma 4.4, it follows

$u(x)=\mathrm{o}\left(|x|^{-\gamma_{p}}\right)\quad{\rm as}\quad x\rightarrow
0,$

which by $n-6<\gamma_{p}$, implies that for any $n-6<q<\gamma_{p}$, there
exists $0<r_{q}<1$ depending only on $n$ $p$, and $q$ such that

$u(x)<|x|^{-q}\quad\mbox{in}\quad B^{*}_{r_{q}},$

where in the last claim we have used a blow-up argument. Now taking $r_{q}>0$
as before, and using Lemma 5.3 again, we get that $(-\Delta)^{3}u=f_{p}(u)\in
L^{1}(B_{1})$. Thus, using (5.1), we decompose

$u(x)=|x|^{6-n}-\int_{B_{r_{q}}}|x-y|^{6-n}(-\Delta)^{3}u(y)\mathrm{d}y+\psi(x)\quad\mbox{in}\quad
B^{*}_{r_{q}},$ (7.1)

where $\psi\in C^{\infty}(B_{1})$ is such that $(-\Delta)^{3}\psi=0$ in
$B_{r_{q}}$. Nevertheless, using (5.1) that there exists $C_{q}>0$, depending
only on $n$, $p$, and $q$ such that

$\left|\int_{B_{r_{q}}}|x-y|^{6-n}(-\Delta)^{3}u(y)\mathrm{d}y\right|\leqslant\int_{B_{r_{q}}}|x-y|^{6-n}|y|^{-pq}\mathrm{d}y\leqslant
C_{q}|x|^{6-n}.$

Hence, fixing $n-6<q<\gamma_{p}$ and choosing suitable $r_{q}>0$ and $C_{q}>0$
on the last inequality, the proof follows directly from (7.1). ∎

Second, we give a sufficient condition to classify whether the origin is a
removable singularity or non-removable singularity.

######  Lemma 7.3.

Let $u\in C^{6}(B_{R}^{*})$ be a positive solution to ($\mathcal{P}_{6,p,R}$)
with $p\in(1,2_{\\#})$. Assume that $-\Delta u\geqslant 0$ and
$\Delta^{2}u\geqslant 0$. If

$u(x)=\mathrm{o}\left(|x|^{6-n}\right)\quad{\rm as}\quad x\rightarrow 0,$
(7.2)

then, the origin is a removable singularity.

###### Proof.

By (7.2), we get $u\in L^{q}(B_{1})$ for any $q\in[1,2_{\\#})$. Moreover,
since $p\in(1,2_{\\#})$ and $|(-\Delta)^{3}u|\leqslant|u|^{p}$, it follows
$(-\Delta)^{3}u\in L^{q/p}(B_{1})$ for any $q\in[1,2_{\\#})$. Whence, we can
use standard elliptic theory combined with a bootstrap argument to find $u\in
W^{5,Q}(B_{1})$ for any $Q\in(1,+\infty)$. In particular, it holds from
Morrey’s embedding that $u\in C^{4,\alpha}(B_{1})$ for any $\alpha\in(0,1)$.
Therefore, $u\in C^{6}(B_{R})$, that is, it must have a removable singularity
at the origin. ∎

Now we are in a position to prove our main result of this part.

###### Proof of Proposition 7.1.

Suppose by contradiction that $u\in C^{6}(B_{R}^{*})$ has a non-removable
singularity at the origin, that is, $u\in C^{6}(B_{R})$. Then, using Lemma
7.3, we get that $u$ does not satisfy (7.2), that is, there exists $\rho>0$
and $\\{r_{k}\\}_{k\in\mathbb{N}}$ such that $r_{k}\rightarrow 0$ as
$k\rightarrow+\infty$ satisfying

$\sup_{\partial B_{r_{k}}}u\geqslant\rho r_{k}^{6-n}.$

On the other hand, by the Harnack inequality in Lemma 4.4, there exists
$c_{1}>0$ satisfying $\inf_{\partial B_{r_{k}}}u\geqslant c_{1}\rho
r_{k}^{6-n}$, where $c_{1}>0$ depends only on $n$, and $p$. Taking $0<\rho\ll
1$ smaller to ensure that there exists $c_{2}\rho\leqslant\inf_{\partial
B_{1/2}}u$, it follows from the maximum principle that

$u(x)\geqslant c_{2}\rho|x|^{6-n}\quad\mbox{in}\quad B^{*}_{1/2},$

which proves the asymptotic lower bound estimate in this case, and together
with Lemma 7.2, the proof of the proposition is concluded. ∎

###  7.2. Gidas–Spruck case

The objective of this subsection is to prove Theorem 1 (c). Our strategy is
based on the monotonicity formula for the Pohozaev functional in cylindrical
coordinates (see Proposition 3.4), which relies on the strategy given in [34,
18, 25]. More precisely, we show that the local models near the origin are the
limit blow-up solutions, whose limits are provided by its image under the
action of the spherical Pohozaev functional. Finally, to prove the
removability of the singularity theorem, we use a technique relying on the
regularity lifting method from [7].

We will prove the result below

######  Proposition 7.4.

Let $u\in C^{6}(B_{R}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(2_{\\#},2^{\\#}-1)$. Then,

$u(x)=(1+\mathrm{o}(1))K_{0}(n,p)^{\frac{1}{p-1}}|x|^{-\gamma_{p}}.$

Now we use this rescaled family
$\\{\widehat{u}_{\lambda}\\}_{\lambda>0}\subset C^{6,\alpha}(B^{*}_{1})$, for
some $\alpha\in(0,1)$, to obtain the blow-up limit for
($\mathcal{P}_{6,p,R}$). This allows us to study the limiting values for the
Pohozaev functional, by using the classification results from Theorem A.

######  Lemma 7.5.

Let $u\in C^{6}(B_{R}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(2_{\\#},2^{\\#}-1)$ and
$v={\mathfrak{F}}(u)$ be its autonomous Emden–Fowler transformation given by
(2.2). Then, ${\mathcal{P}}_{\rm cyl}(-\infty,v)\in\\{-\ell^{*}_{p},0\\}$,
where $\ell^{*}_{p}$ is given by (6.3). Moreover, it follows

  * (i)

${\mathcal{P}}_{\rm cyl}(-\infty,v)=0$, if and only if,

$u(x)=\mathrm{o}\left(|x|^{-\gamma_{p}}\right)\quad{\rm as}\quad x\rightarrow
0.$ (7.3)

  * (ii)

${\mathcal{P}}_{\rm cyl}(-\infty,v)=-\ell^{*}_{p}$, if and only if,

$u(x)=(1+\mathrm{o}(1))K_{0}(n,p)^{\frac{1}{p-1}}|x|^{-\gamma_{p}}\quad{\rm
as}\quad x\rightarrow 0.$

###### Proof.

Initially, by Lemma 4.2, for any $K\subset B_{1/2\lambda}$ compact subset, the
family $\\{\widehat{u}_{\lambda}\\}_{\lambda>0}\subset
C^{6,\alpha}(B^{*}_{1})$ is uniformly bounded, for some $\alpha\in(0,1)$.
Then, by standard elliptic theory, there exists a positive function $u_{0}\in
C^{6,\alpha}(\mathbb{R}^{n}\setminus\\{0\\})$, such that, up to a subsequence,
we have that $\|\widehat{u}-u_{0}\|_{C_{\rm
loc}^{6,\alpha}(\mathbb{R}^{n}\setminus\\{0\\})}$ as $\lambda\rightarrow 0$,
where $u_{0}$ satisfies the blow-up limit system ($\mathcal{P}_{6,p,R}$).
Moreover, by [18, Theorem 1.2], we know that $u_{0}\in
C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$ is satisfies $-\Delta u_{0}\geqslant 0$
and $\Delta^{2}u_{0}\geqslant 0$ in $\mathbb{R}^{n}\setminus\\{0\\}$, which,
by the maximum principle, yields that either $u_{0}\equiv 0$ or $u_{0}>0$ in
$\mathbb{R}^{n}\backslash\\{0\\}$. Therefore, by Theorem A, the blow-up limit
$u_{0}$ is radially symmetric with respect to the origin. Furthermore, by the
scaling invariance of the Pohozaev functional, we get

$\mathcal{P}_{\rm sph}(r,u_{0})=\lim_{\lambda\rightarrow 0}\mathcal{P}_{\rm
sph}(r,\widehat{u}_{\lambda})=\lim_{\lambda\rightarrow 0}\mathcal{P}_{\rm
sph}(\lambda r,u_{0})=\mathcal{P}_{\rm sph}(0,u_{0}).$ (7.4)

In addition, since $v_{0}=\mathfrak{F}(u_{0})$ satisfies
($\mathcal{C}_{p,T}$), by (7.4), we get that $\mathcal{P}_{\rm
cyl}(t,v_{0})=\mathcal{P}_{\rm sph}(r,u_{0})$ is a constant. Consequently, by
the monotonicity formula in Proposition 3.4, we get

$\frac{\mathrm{d}}{\mathrm{d}t}\mathcal{P}_{\rm
cyl}(t,v_{0})=\left[-K_{5}(n,p){v_{0}^{(3)}}^{2}+K_{3}(n,p){v_{0}^{(2)}}^{2}-K_{1}(n,p){v_{0}^{(1)}}^{2}\right]\equiv
0.$

Moreover, since $K_{5}(n,p),K_{1}(n,p)>0$ and $K_{3}(n,p)<0$, we find that
$v^{(1)}_{0}\equiv 0$ in $\mathbb{R}$, and so $v_{0}$ is constant, which can
be directly computed, namely either $v_{0}=0$ or
$v_{0}=K_{0}(n,p)^{\frac{1}{p-1}}$. Moreover, by (7.4), it follows
$\mathcal{P}_{\rm cyl}(0,v_{0})\in\\{-\ell^{*}_{p},0\\}$ and $\mathcal{P}_{\rm
sph}(0,u_{0})\in\\{-\omega_{n-1}\ell^{*}_{p},0\\}$.

Finally, if $\mathcal{P}_{\rm sph}(0,u_{0})=0$, then, by uniqueness of the
limit $u_{0}\equiv 0$. Whence, we conclude that
$\|\widehat{u}_{\lambda}\|_{C^{6,\alpha}(K)}\rightarrow 0$ for any sequence of
$\lambda\rightarrow 0$, for some $\alpha\in(0,1)$, which straightforwardly
provides (7.3). Otherwise, we have

$u_{0}\equiv K_{0}(n,p)^{\frac{1}{p-1}}|x|^{-\gamma_{p}},$

which proves (ii) of this lemma and finishes the proof. ∎

Next, we use the last lemma to prove the removable singularity theorem. Our
proof is based on regularity lifting methods combined with the De
Giorgi–Nash–Moser iteration technique.

######  Lemma 7.6.

Let $u\in C^{6}(B_{R}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p\in(2_{\\#},2^{\\#}-1)$. If

$u(x)=\mathrm{o}\left(|x|^{-\gamma_{p}}\right)\quad{\rm as}\quad x\rightarrow
0,$

then the origin is a removable singularity.

###### Proof.

Without loss of generality, let us consider $R=1$. The proof will be divided
into some claims.

Claim 1: If $u(x)=\mathrm{o}(|x|^{-\gamma_{p}})$ as $x\rightarrow 0$, then

$\int_{B_{1/2}}u^{n\gamma_{p}^{-1}}\mathrm{d}x<+\infty.$ (7.5)

In fact, let us consider $\phi(|x|)=|x|^{\zeta_{p}}$, where
$\zeta_{p}=-2\gamma_{p}(2^{\\#}-1)(n-2_{\\#})(p-1)^{-1}$. Then, a direct
computation, provides

$\Delta^{3}\phi(x)=\zeta_{p}(\zeta_{p}-2)(\zeta_{p}-3)(\zeta_{p}+n-2)(\zeta_{p}+n-4)(\zeta_{p}+n-6)|x|^{\zeta_{p}-6}\quad{\rm
in}\quad\mathbb{R}^{n}\setminus\\{0\\},$

which, since $\zeta_{p}+n-6=\gamma_{p}>0$, it follows that

$A_{p}:=\zeta_{p}(\zeta_{p}-2)(\zeta_{p}-3)(\zeta_{p}+n-2)(\zeta_{p}+n-4)(\zeta_{p}+n-6)>0.$

Thus, we can write

$\frac{(-\Delta)^{3}\phi}{\phi}=\frac{A_{p}}{|x|^{6}}\quad\mbox{in}\quad\mathbb{R}^{n}\setminus\\{0\\}.$
(7.6)

For any $0<\varepsilon\ll 1$, let us consider $\eta_{\varepsilon}\in
C^{\infty}\left(\mathbb{R}^{n}\right)$ with
$0\leqslant\eta_{\varepsilon}\leqslant 1$ a cut-off function satisfying

$\eta_{\varepsilon}(x)=\begin{cases}0,&\mbox{for}\
\varepsilon\leqslant|x|\leqslant 1/2\\\ 1,&\mbox{for}\
|x|\leqslant\varepsilon/2\ \mbox{or}\ |x|\geqslant 3/4,\end{cases}$ (7.7)

and $|D^{(j)}\eta_{\varepsilon}(x)|\leqslant C\varepsilon^{-j}$ for
$j=0,1,2,3,4,5,6$. Defining $\xi_{\varepsilon}=\eta_{\varepsilon}\phi$,
multiplying ($\mathcal{P}_{6,p,R}$) by $\xi_{\varepsilon}$ and integrating by
parts in $B_{1}$, we obtain

$\int_{B_{1}}\eta_{\varepsilon}u\phi\left(\frac{(-\Delta)^{3}\phi}{\phi}-f_{p}(u)\right)\mathrm{d}x=-\int_{B_{1}}u\mathfrak{T}\left(\eta_{\varepsilon},\phi\right)\mathrm{d}x,$
(7.8)

where $\mathfrak{T}_{\varepsilon}:C^{\infty}_{c}(B_{1})\rightarrow
C^{\infty}_{c}(B_{1})$ is defined by

$\mathfrak{T}_{\varepsilon}(\phi)=\mathfrak{T}\left(\eta_{\varepsilon},\phi\right)=6\nabla\eta_{\varepsilon}\nabla\Delta^{2}\phi-15\Delta\eta_{\varepsilon}\Delta^{2}\phi+20\nabla\Delta\eta_{\varepsilon}\nabla\Delta\phi-15\Delta^{2}\eta_{\varepsilon}\Delta\phi+6\nabla\Delta^{2}\eta_{\varepsilon}\nabla\phi-\phi\Delta^{3}\eta_{\varepsilon}.$

Using Lemma 4.2 combined with the estimates on the cut-off function (7.7) and
its derivatives, there exist $c_{1},c_{2}>0$, independent of $\varepsilon$,
satisfying the following estimates,

$\left|\int_{B_{1}}u\mathfrak{T}_{\varepsilon}(\phi)\mathrm{d}x\right|\leqslant
c_{1}+c_{2}\varepsilon^{n}\varepsilon^{\zeta_{p}-6}\varepsilon^{-\gamma_{p}}<+\infty,$

which implies that the right-hand side of (7.8) is uniformly bounded. In
addition, assumption (7.5) yields that $u^{p-1}(x)=\mathrm{o}(1)|x|^{-6}$ as
$x\rightarrow 0$, which together with (7.6) and (7.9) provides that there
exists $C>0$ satisfying

$\int_{B_{1}}\eta_{\varepsilon}u(x)|x|^{\zeta_{p}-6}\mathrm{d}x\leqslant C.$
(7.9)

Therefore, by Lemma 4.2, it holds

$\displaystyle\int_{\\{\varepsilon\leqslant|x|\leqslant{1}/{2}\\}}u(x)^{n\gamma_{p}^{-1}}\mathrm{d}x\leqslant
C\int_{\\{\varepsilon\leqslant|x|\leqslant{1}/{2}\\}}u(x)|x|^{\zeta_{p}-6}\mathrm{d}x\leqslant
C\int_{B_{1}}\eta_{\varepsilon}u(x)|x|^{\zeta_{p}-6}\mathrm{d}x<+\infty,$
(7.10)

where the last inequality comes from (7.9). Finally, passing to the limit as
$\varepsilon\rightarrow 0$ in (7.10), the proof of Claim 1 follows by applying
the dominated convergence theorem.

Claim 2: If (7.5) holds, then $u\in L^{q}(B_{1})$ for all $q>2^{\\#}$.

Indeed, by Lemma 5.4 (i), there exist a Green function with homogeneous
Dirichlet boundary conditions $G_{3}(x,y)$ and $\psi\in L^{\infty}_{\rm
loc}(B_{1/2})$ with $\Delta^{2}\psi=0$ and $\Delta\psi=0$ such that

$u(x)=\int_{B_{1}}G_{2}(x,y)\Delta^{3}u\mathrm{d}x+\psi(x)\quad\mbox{in}\quad
B_{1/2}.$

More precisely, $G_{3}(x,y)$ is a distributional solution to the Dirichlet
problem

$\begin{cases}\Delta^{3}G_{2}(x,y)=\delta_{x}(y)&\mbox{in}\quad B_{1/2}\\\
G_{3}(x,y)=\partial_{\nu}G_{3}(x,y)=\Delta G_{3}(x,y)=0&\mbox{on}\quad\partial
B_{1/2},\end{cases}$

and there exists positive constant $C_{n}>0$ such that

$0<G_{3}(x,y)\leqslant\widetilde{G}_{3}(|x-y|):=C_{n}|x-y|^{6-n}\quad\mbox{for}\quad
x,y\in B_{1/2}\quad\mbox{and}\quad|x-y|>0,$

where $\widetilde{G}_{3}(x,y)=C_{n}|x-y|^{6-n}$ is the fundamental solution to
$\Delta^{3}$ in $\mathbb{R}^{n}$. Recall that $u\in C^{6}(B_{1}^{*})$
satisfies

$(-\Delta)^{3}u=V(x)u\quad\mbox{in}\quad B_{1}^{*},$ (7.11)

where $V(x)=u^{p-1}$. Moreover, using (7.5), we find that $V\in
L^{n/6}(B_{1/2})$.

Let us consider $Z=C^{\infty}_{c}(B_{1/4})$, $X=L^{2^{\\#}}(B_{1/4})$ and
$Y=L^{p}(B_{1/4})$ for $q\in(2^{\\#},+\infty)$ as in [7, Theorem 3.3.1].
Hence, it is well-defined the following inverse operator

$(Tu)(x)=\int_{B_{1/4}}\widetilde{G}_{3}(x,y)u(y)\mathrm{d}y.$

We also consider the operator $T_{M}:=\widetilde{G}_{3}\ast V_{M}$, which
applied in both sides of (7.11), provides $u=T_{M}u+\widetilde{T}_{M}u$, where

$(T_{M}u)(x)=\int_{B_{1/4}}\widetilde{G}_{3}(x,y)V_{M}(y)u(y)\mathrm{d}y\quad\text{and}\quad(\widetilde{T}_{M}u)(x)=\int_{B_{1/4}}\widetilde{G}_{3}(x,y)\widetilde{V}_{M}(y)u(y)\mathrm{d}y.$

Here, for $M>0$, we define $\widetilde{V}_{M}(x)=V(x)-V_{M}(x)$, where

$V_{M}(x)=\begin{cases}V(x),\ {\rm if}\ |V(x)|\geqslant M,\\\ 0,\ {\rm
otherwise}.\end{cases}$

Now we can run the regularity lifting method, which is divided into two steps.

Step 1: For $q\in(2_{\\#},+\infty)$, there exists $M\gg 1$ large such that
$T_{M}:L^{q}(B_{1/4})\rightarrow L^{q}(B_{1/4})$ is a contraction.

In fact, for any $q\in(2_{\\#},+\infty)$, there exists $m\in(1,n/6)$ such that
$q=nm/(n-6m)$. Then, by the Hardy–Littlewood–Sobolev and Hölder inequalities
[22], for any $u\in L^{q}(\mathbb{R}^{n})$, we get

$\|T_{M}u\|_{L^{q}(B_{1/4})}\leqslant\|\widetilde{G}_{3}\ast
V_{M}u\|_{L^{q}(B_{1/4})}\leqslant
C\|V_{M}\|_{L^{{n}/{6}}(B_{1/4})}\|u\|_{L^{q}(B_{1/4})}.$

Since $V_{M}\in L^{n/6}(B_{1/4})$ it is possible to choose a large $M\gg 1$
satisfying $\|V_{M}\|_{L^{{n}/{6}}(B_{1/4})}<{1}/{2C}$. Therefore, we arrive
at

$\|T_{M}u\|_{L^{q}(B_{1/4})}\leqslant{1}/{2}\|u\|_{L^{q}(B_{1/4})},$

which implies that $T_{M}$ is a contraction.

Step 2: For any $q\in(2_{\\#},+\infty)$, it follows that
$\widetilde{T}_{M}u\in L^{q}(B_{1/4})$.

Indeed, for any $q\in(2_{\\#},+\infty)$, we pick $1<m<n/6$ satisfying
$q=nm/(n-6m)$. Since $\widetilde{V}_{M}$ is bounded, we get

$\|\widetilde{T}_{M}\|_{L^{q}(B_{1/4})}=\|\widetilde{G}_{3}\ast\widetilde{V}_{M}u\|_{L^{q}(B_{1/4})}\leqslant
C\|\widetilde{V}_{M}u\|_{L^{m}(B_{1/4})}\leqslant C\|u\|_{L^{m}(B_{1/4})}.$

However, using (7.5), we have that $u\in L^{q}(B_{1/4})$ for
$q\in(1,n\gamma_{p}^{-1})$. Besides, $q=(p-2)n\gamma_{p}^{-1}$ when
$m=n\gamma_{p}^{-1}$. Thus, we obtain that $u\in L^{q}(B_{1/4})$ for

$\begin{cases}1<q<\infty,&{\rm if}\ p\geqslant 2,\\\
1<q\leqslant(2-p)^{-1}n\gamma_{p}^{-1},&{\rm if}\ 1<p<2.\end{cases}$

Now we can repeat the argument for $m=(p-2)n\gamma_{p}^{-1}$ to get that $u\in
L^{q}(B_{1/4})$ for

$\begin{cases}1<q<\infty,&{\rm if}\ p\geqslant 2,\\\
1<q\leqslant(2-p)^{-1}n\gamma_{p}^{-1},&{\rm if}\ 1<p<2.\end{cases}$

Therefore, by proceeding inductively as in [34, Lemma 3.8], the proof of the
claim follows. Ultimately, combining Steps 1 and 2, we can apply [7, Theorem
3.3.1] to show that $u\in L^{p}(B_{1/4})$ for all $p>2^{\\#}$. In particular,
the proof of the claim is finished.

Now, by the Morrey’s embedding theorem, it follows that $u\in
C^{0,\alpha}(B_{1/4})$, for some $\alpha\in(0,1)$. Finally using Schauder
estimates, one gets that $u\in C^{6,\alpha}(B_{1/4})$. In particular, the
singularity at the origin is removable, which concludes the proof of the
lemma. ∎

###### Proof of Proposition 7.4.

Suppose by contradiction that $u\in C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$ has
a non-removable singularity at the origin, then by Lemma 7.6, $u$ does not
satisfy (7.3). Therefore, the proof follows as a consequence of Lemma 7.5. ∎

###  7.3. Aviles case

Finally, we prove Theorem 1 (b). The asymptotic analysis for the lower
critical exponent, $p=2_{\\#}$ exhibits its subtlety. First, since
$\gamma(2_{\\#})=n-6$, one would expect the singular solutions to
($\mathcal{P}_{6,p,R}$) to have the same behavior as the fundamental solution
to the tri-Laplacian near the origin; thus, the isolated singularity would be
removable.

Our objective is to prove the proposition below

######  Proposition 7.7.

Let $u\in C^{6}(B_{R}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p=2_{\\#}$. Assume that $-\Delta u\geqslant 0$
and $\Delta^{2}u\geqslant 0$. Then,

$u(x)=(1+\mathrm{o}(1))\widehat{K}_{0}(n)^{\frac{n-6}{6}}|x|^{6-n}(\ln|x|)^{\frac{6-n}{6}},$

where $\widehat{K}_{0}(n):=-\lim_{t\rightarrow-\infty}t\widetilde{K}_{0}(n,t)$
is given by (2).

As in the autonomous case, we use the limiting energy levels
$\widetilde{\mathcal{P}}_{\rm cyl}(-\infty,w)$ to classify the local behavior
near the isolated singularity.

######  Lemma 7.8.

Let $u\in C^{6}(B_{R}^{*})$ be a positive singular solution to
($\mathcal{P}_{6,p,R}$) with $p=2_{\\#}$ and $w=\widetilde{\mathfrak{F}}(v)$
be its nonautonomous Emden–Fowler transformation given by (2.2). Assume that
$-\Delta u\geqslant 0$ and $\Delta^{2}u\geqslant 0$. Then,
$\widetilde{\mathcal{P}}_{\rm cyl}(-\infty,w)\in\\{-\ell^{*}_{\\#},0\\}$,
where $-\ell^{*}_{\\#}=\frac{3}{2(n-3)}\widehat{K}_{0}(n)^{\frac{n-3}{3}}$.
Moreover, it follows

  * (i)

$\widetilde{\mathcal{P}}_{\rm cyl}(-\infty,w)=0$ if and only if,

$u(x)=\mathrm{o}\left(|x|^{6-n}(\ln|x|)^{\frac{6-n}{6}}\right)\quad{\rm
as}\quad x\rightarrow 0.$ (7.12)

  * (ii)

$\widetilde{\mathcal{P}}_{\rm cyl}(-\infty,w)=\ell^{*}_{\\#}$ if and only if,

$u(x)=(1+\mathrm{o}(1))\widehat{K}_{0}(n)^{\frac{n-6}{6}}|x|^{6-n}(\ln|x|)^{\frac{6-n}{6}}\quad{\rm
as}\quad x\rightarrow 0.$

###### Proof.

First, combining (3.2) with Proposition 3.10 and Lemma 3.9, we find

$\widetilde{\mathcal{P}}_{\rm
cyl}(-\infty,w)=\lim_{t\rightarrow-\infty}\int_{\mathbb{S}_{t}^{n-1}}\left(\frac{n-6}{2(n-3)}|w|^{2_{\\#}+1}+\widehat{K}_{0}(n)|w|^{2}\right)\mathrm{d}\theta.$

Furthermore, by (3.2), we see that for any $\\{t_{k}\\}_{k\in\mathbb{N}}$ such
that $t_{k}\rightarrow+\infty$ as $k\rightarrow-\infty$, it follows that
$\\{w(t_{k},\theta)\\}_{k\in\mathbb{N}}$ converges to a limit, which is
independent of $\theta\in\mathbb{S}_{t}^{n-1}$. Hence, up to subsequence,
there exists $w_{0}\in\mathbb{R}$ such that $w(t_{k},\theta)\rightarrow w_{0}$
uniformly on $\theta\in\mathbb{S}_{t}^{n-1}$ , which gives us

$\widetilde{\mathcal{P}}_{\rm
cyl}(-\infty,w)=\frac{n-6}{2(n-3)}|w_{0}|^{2_{\\#}+1}+\widehat{K}_{0}(n)|w_{0}|^{2}.$
(7.13)

Thus, since the right-hand side of the last equation has at most three
nonnegative roots, the limit $w_{0}\in\mathbb{R}$, under the uniform
convergence of $w(t,\theta)$ on $\mathbb{S}_{t}^{n-1}$ as
$t\rightarrow-\infty$, is unique. Finally, taking the inner product of
($\widetilde{\mathcal{C}}_{T}$) with $w$, integrating both sides over
$(-\infty,t_{0},)\times\mathbb{S}^{n-1}$, and using (3.2),(3.3) and Lemma 3.9,
it follows

$\left|\int_{-\infty}^{t_{0}}\frac{1}{t}\int_{\mathbb{S}_{t}^{n-1}}\left(\widehat{K}_{0}(n)-|w|^{2_{\\#}-1}\right)|w|^{2}\mathrm{d}\theta\mathrm{d}t\right|<+\infty.$

Now since $\lim_{t\rightarrow-\infty}w(t,\theta)=w_{0}$ uniformly on
$\mathbb{S}_{t}^{n-1}$, we get either $w_{0}=0$ or
$w_{0}=\widehat{K}_{0}(n)^{\frac{n-6}{6}}$, which by substituting into (7.13),
implies that either $\widetilde{\mathcal{P}}_{\rm cyl}(-\infty,w)=0$ if and
only if, $w_{0}=0$, or $\widetilde{\mathcal{P}}_{\rm
cyl}(-\infty,w)=\ell^{*}_{\\#}$, otherwise. The proof trivially follows by
applying the inverse $\widetilde{\mathfrak{F}}^{-1}$ of the nonautonomous
cylindrical transform. ∎

Now we are left to show that, if Lemma 7.8 (i) holds, then the singularity at
the origin is removable. Here, we are based on the barriers construction in
[21] (see also [19]), which is available due to the integral representation
(5.1).

######  Lemma 7.9.

Let $u\in C^{6}(B_{R}^{*})$ be a positive solution to ($\mathcal{P}_{6,p,R}$)
with $p=2_{\\#}$. Assume that $-\Delta u\geqslant 0$ and $\Delta^{2}u\geqslant
0$. If

$u(x)=\mathrm{o}\left(|x|^{6-n}(\ln|x|)^{\frac{6-n}{6}}\right)\quad{\rm
as}\quad x\rightarrow 0,$

then, the origin is a removable singularity.

###### Proof.

For any $\delta>0$, we choose $0<\rho\ll 1$ such that
$u(x)\leqslant\delta|x|^{-\gamma_{p}}$ in $B^{*}_{\rho}$. Fixing
$\varepsilon>0$, $\kappa\in\left(0,\gamma_{p}\right)$ and $M\gg 1$ to be
chosen later, we define

$\varsigma(x)=\begin{cases}{M|x|^{-\kappa}+\varepsilon|x|^{6-n-\kappa},}&\mbox{if}\
{0<|x|<\rho},\\\ {u(x)},&\mbox{if}\ {\rho<|x|<2}.\end{cases}$

Notice that for every $0<\kappa<n-6$ and $0<|x|<2$, there exists $C>0$ such
that

$\displaystyle\int_{\mathbb{R}^{n}}{|x-y|^{6-n}|y|^{-6-\kappa}}\mathrm{d}y=|x|^{6-n}\int_{\mathbb{R}^{n}}{\left||x|^{-1}x-|x|^{-1}y\right|^{6-n}|y|^{\\-\kappa-6}}\mathrm{d}y\leqslant
C\left(\frac{1}{n-6-\kappa}+\frac{1}{\kappa}+1\right)|x|^{-\kappa},$

which, for $0<|x|<2$ and $0<\delta\ll 1$, yields

$\displaystyle\int_{B_{\rho}}{u^{2_{\\#}-1}(y)\varsigma(y)}{|x-y|^{6-n}}\mathrm{d}y\leqslant\delta^{2_{\\#}-1}\int_{\mathbb{R}^{n}}{\varsigma(y)}{|x-y|^{n-6}|y|^{-6}}\mathrm{d}y\leqslant
C\delta^{2_{\\#}-1}\varsigma(x)<\frac{1}{2}\varsigma(x).$

Moreover, for $0<|x|<\rho$ and $\bar{x}=\rho x|x|^{-1}$, we get

$\displaystyle\int_{B_{2}\setminus
B_{\rho}}{u^{2_{\\#}-1}(y)\varsigma(y)}{|x-y|^{6-n}}\mathrm{d}y=\int_{B_{2}\setminus
B_{\rho}}\frac{|\bar{x}-y|^{n-6}}{|x-y|^{n-6}}\frac{u^{2_{\\#}}(y)}{|\bar{x}-y|^{n-6}}\mathrm{d}y\leqslant
2^{n-6}\max_{\partial B_{\rho}}u.$

The last inequality implies that for $0<|x|<\tau$ and
$M\geqslant\max_{\partial B_{\rho}}u$,

$\psi(x)+\int_{B_{2}}\frac{u^{2_{\\#}-1}(y)\varsigma(y)}{|x-y|^{6-n}}\mathrm{d}y\leqslant\psi(x)+2^{n-6}\max_{\partial
B_{\rho}}u+\frac{1}{2}\varsigma(x)<\varsigma(x).$

We show that $\varsigma$ can be taken as a barrier for any $u$. Namely, we
claim that $u(x)\leqslant\varsigma(x)$ in $B^{*}_{\rho}$. In fact, suppose by
contradiction that the conclusion is not true. Then, since
$u(x)\leqslant\delta|x|^{-\gamma_{p}}$ in $B^{*}_{\rho}$, by the definition of
$\varsigma$, there exists $\widetilde{\tau}\in(0,\rho)$, depending on
$\varepsilon$, such that $\varsigma\geqslant u$ in $B^{*}_{{\rho}}$ and
$\varsigma>u$ close to the boundary $\partial B_{\rho}$. Let us consider
$\bar{\tau}:=\inf\\{\tau>1:\tau\psi>u\ \mbox{in}\ B^{*}_{\rho}\\}$. Then, we
have that $\bar{\tau}\in(1,+\infty)$ and there exists $\bar{x}\in
B_{\rho}\setminus\bar{B}_{\widetilde{\tau}}$ such that
$\bar{\tau}\varsigma(\bar{x})=u(\bar{x})$ and, for $0<|x|<\tau$, it follows

$\bar{\tau}\varsigma(x)\geqslant\int_{B_{2}}{u^{2_{\\#}-1}(y)\bar{\tau}\varsigma(y)}{|x-y|^{6-n}}\mathrm{d}y+\bar{\tau}\psi(x)\geqslant\int_{B_{2}}{u^{2_{\\#}-1}(y)\bar{\tau}\varsigma(y)}{|x-y|^{6-n}}\mathrm{d}y+\psi(x),$

which gives us

$\bar{\tau}\varsigma(x)-u(x)\geqslant\int_{B_{2}}{u^{2_{\\#}-1}(y)(\bar{\tau}\varsigma(y)-u(y))}{|x-y|^{6-n}}\mathrm{d}y.$

Finally, by evaluating the last inequality at $\bar{x}\in
B_{\rho}\setminus\bar{B}_{\widetilde{\tau}}$, we get a contradiction.

At last, we find $u(x)\leqslant\varsigma(x)\leqslant
M|x|^{-\kappa}+\varepsilon|x|^{6-n-\kappa}$ in $B^{*}_{\rho}$, which yields
that $u^{2_{\\#}-1}\in L^{p}(B^{*}_{\rho})$ for some $p>{n}/{6}$. Hence,
standard elliptic regularity concludes the proof of the lemma. ∎

Ultimately, the proof of the main result in this section is merely a
consequence of the last results.

###### Proof of Proposition 7.7.

Suppose that $u\in C^{6}(\mathbb{R}^{n}\setminus\\{0\\})$ is a positive
singular solution to ($\mathcal{P}_{6,p,R}$) with $p=2_{\\#}$, then by Lemma
7.9, $u$ does not satisfy (7.12). Therefore, the proof follows as a
consequence of Lemma 7.8. ∎

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. He also wishes to express gratitude
to Professor João Marcos do Ó for his constant support and several valuable
conversations.

##  Appendix A Tri-Laplacian coefficients

In this appendix, we show the explicit formula for the coefficients of the
tri-Laplacian in autonomous and nonautonomous Emden–Fowler coordinates,
respectively.

###  A.1. Polar coordinates

We consider the cylinder $\mathcal{C}_{R}:=(0,R)\times\mathbb{S}^{n-1}$ and
$(-\Delta)^{3}_{\rm sph}$ the tri-Laplacian written in spherical (polar)
coordinates given by

$\displaystyle\Delta^{3}_{\rm sph}$
$\displaystyle={r^{-6}}\partial_{r}^{(6)}+M_{5}(n,r)\partial_{r}^{(5)}+M_{4}(n,r)\partial_{r}^{(4)}+M_{3}(n,r)\partial_{r}^{(3)}+M_{2}(n,r)\partial_{r}^{(2)}+M_{1}(n,r)\partial_{r}$
$\displaystyle+2r^{-2}\partial^{(4)}_{r}\Delta_{\sigma}+N_{3}(n,r)\partial^{(3)}_{r}\Delta_{\sigma}+N_{2}(n,r)\partial^{(2)}_{r}\Delta_{\sigma}+N_{1}(n,r)\partial_{r}\Delta_{\sigma}+N_{0}(n,r)\Delta_{\sigma}$
$\displaystyle+3r^{-4}\partial^{(2)}_{r}\Delta^{2}_{\sigma}+O_{1}(n,r)\partial_{r}\Delta^{2}_{\sigma}+O_{0}(n,r)\Delta^{2}_{\sigma}+{r^{-6}}\Delta^{3}_{\sigma},$

where $\Delta_{\sigma}$ denotes the Laplace–Beltrami operator in
$\mathbb{S}^{n-1}$ and

$\displaystyle M_{5}(n,r)$ $\displaystyle=3(n-1)r^{-1}$ $\displaystyle
M_{4}(n,r)$ $\displaystyle=3(n-1)(n-3)r^{-2}$ $\displaystyle M_{3}(n,r)$
$\displaystyle=(n-1)(n-3)(n-8)r^{-3}$ (A.1) $\displaystyle M_{2}(n,r)$
$\displaystyle=-3(n-1)(n-3)(n-5)r^{-4}$ $\displaystyle M_{1}(n,r)$
$\displaystyle=3(n-1)(n-3)(n-5)r^{-5}$

and

$\displaystyle N_{3}(n,r)$ $\displaystyle=2(n-7)r^{-3}$ $\displaystyle
N_{2}(n,r)$ $\displaystyle=2(n^{2}-n-3)r^{-4}$ $\displaystyle N_{1}(n,r)$
$\displaystyle=6(7n-23)r^{-5}$ $\displaystyle N_{0}(n,r)$
$\displaystyle=8(n-1)(n-5)r^{-6}$ (A.2) $\displaystyle O_{1}(n,r)$
$\displaystyle=3(n-5)r^{-5}$ $\displaystyle O_{0}(n,r)$
$\displaystyle=-2(3n-16)r^{-6}$

are its coefficients in this coordinate system.

###  A.2. Autonomous case

Let us recall the sixth order autonomous Emden–Fowler change of variables
$($or cylindrical logarithm coordinates$)$ given by

$v(t,\theta)=r^{\gamma_{p}}u(r,\sigma),\quad\mbox{where}\quad t=\ln r\quad{\rm
and}\quad\sigma=\theta=x|x|^{-1}.$

Using this change of variables and performing a lengthy computation, we arrive
at the following sixth order operator on the cylinder,

$\displaystyle P^{3}_{\rm cyl}$
$\displaystyle=\partial_{t}^{(6)}+K_{5}(n,p)\partial_{t}^{(5)}+K_{4}(n,p)\partial_{t}^{(4)}+K_{3}(n,p)\partial_{t}^{(3)}+K_{2}(n,p)\partial_{t}^{(2)}+K_{1}(n,p)\partial_{t}+K_{0}(n,p)$
$\displaystyle+2\partial^{(4)}_{t}\Delta_{\theta}+J_{3}(n,p)\partial^{(3)}_{t}\Delta_{\theta}+J_{2}(n,p)\partial^{(2)}_{t}\Delta_{\theta}+J_{1}(n,p)\partial_{t}\Delta_{\theta}+J_{0}(n,p)\Delta_{\theta}$
$\displaystyle+3\partial^{(2)}_{t}\Delta^{2}_{\theta}+L_{1}(n,p)\partial_{t}\Delta^{2}_{\theta}+L_{0}(n,p)\Delta^{2}_{\theta}+\Delta^{3}_{\theta},$

where

$\displaystyle K_{0}(n,p)$
$\displaystyle=-24(p-1)^{-6}(p+2)(2p+1)(np-6p-n)(np-4p-2-n)(np-2p-n-4)$
$\displaystyle K_{1}(n,p)$
$\displaystyle=4(p-1)^{-5}(np-6p-n-6)(2n^{2}p^{4}-12np^{4}+16p^{4}+10n^{2}p^{3}-108np^{3}+224p^{3}$
$\displaystyle-15n^{2}p^{2}-36np^{2}+492p^{2}-8n^{2}p+120np+224p+11n^{2}+36n+16)$
(A.3) $\displaystyle K_{2}(n,p)$
$\displaystyle=-2(p-1)^{-4}(3n^{3}p^{4}-48n^{2}p^{4}+228np^{4}-320p^{4}-3n^{3}p^{3}-78n^{2}p^{3}+996np^{3}-2392p^{3}-9n^{3}p^{2}$
$\displaystyle+198n^{2}p^{2}+180np^{2}-4296p^{2}+15n^{3}p+30n^{2}p-1020np-2392p-6n^{3}-102n^{2}-384n-320)$
$\displaystyle K_{3}(n,p)$
$\displaystyle=-(p-1)^{-3}(np-6p-6-n)(n^{2}p^{2}-24np^{2}+68p^{2}-2n^{2}p-12np+224p+n^{2}+36n+68)$
$\displaystyle K_{4}(n,p)$
$\displaystyle=(p-1)^{-2}(3n^{2}p^{2}-42np^{2}+124p^{2}-6n^{2}p-6np+292p+3n^{2}+48n+124)$
$\displaystyle K_{5}(n,p)$ $\displaystyle=3(p-1)^{-1}(np-6p-6-n)$

and

$\displaystyle J_{0}(n,p)$
$\displaystyle=4(p-1)^{-4}(2n^{2}p^{4}-12np^{4}+10p^{4}-5n^{2}p^{3}-24np^{3}+218p^{3}+21n^{2}p^{2}+72np^{2}-192p^{2}$
$\displaystyle-35n^{2}p-132np+1094p+17n^{2}+96n-482)$ $\displaystyle
J_{1}(n,p)$
$\displaystyle=-2(p-1)^{-3}(n^{2}p^{3}-24np^{3}+86p^{3}+9n^{2}p^{2}+24np^{2}+90p^{2}-21n^{2}p-84np+966p$
$\displaystyle+11n^{2}+84n-278)$ $\displaystyle J_{2}(n,p)$
$\displaystyle=2(p-1)^{-2}(n^{2}p^{2}-4np^{2}+29p^{2}-2n^{2}p-10np+176p+n^{2}+14n+11)$
(A.4) $\displaystyle J_{3}(n,p)$ $\displaystyle=2(p-1)^{-1}(np-13p-n-11)$
$\displaystyle L_{0}(n,p)$
$\displaystyle=-(p-1)^{-2}(3np^{2}16p^{2}-3np+22p+6n+16)$ $\displaystyle
L_{1}(n,p)$ $\displaystyle=3(p-1)^{-1}(np-6p-6-n).$

###  A.3. Nonautonomous case

Let us recall the sixth order nonautonomous Emden–Fowler change of variables
$($or cylindrical logarithm coordinates$)$ given by

$w(t,\theta)=r^{6-n}(\ln r)^{\frac{6-n}{6}}u(r,\sigma),\quad{\rm where}\quad
t=\ln r\quad{\rm and}\quad\sigma=\theta=x|x|^{-1}.$

Using this coordinate system and performing a lengthy computation, we arrive
at the following sixth order nonautonomous operator PDE on the cylinder

$\displaystyle\widetilde{P}^{3}_{\rm cyl}$
$\displaystyle=\partial_{t}^{(6)}+\widetilde{K}_{5}(n,t)\partial_{t}^{(5)}+\widetilde{K}_{4}(n,t)\partial_{t}^{(4)}+\widetilde{K}_{3}(n,t)\partial_{t}^{(3)}+\widetilde{K}_{2}(n,t)\partial_{t}^{(2)}+\widetilde{K}_{1}(n,p)\partial_{t}+\widetilde{K}_{0}(n,t)$
$\displaystyle+2\partial^{(4)}_{t}\Delta_{\theta}+\widetilde{J}_{3}(n,t)\partial^{(3)}_{t}\Delta_{\theta}+\widetilde{J}_{2}(n,t)\partial^{(2)}_{t}\Delta_{\theta}+\widetilde{J}_{1}(n,t)\partial_{t}\Delta_{\theta}+\widetilde{J}_{0}(n,t)\Delta_{\theta}$
$\displaystyle+3\partial^{(2)}_{t}\Delta^{2}_{\theta}+\widetilde{L}_{1}(n,t)\partial_{t}\Delta^{2}_{\theta}+\widetilde{L}_{0}(n,t)\Delta^{2}_{\theta}+\Delta^{3}_{\theta},$

where

$\displaystyle\widetilde{K}_{0}(n,t)$
$\displaystyle=\frac{4(n-6)(n^{3}-12n^{2}+44n-48)}{3t}-\frac{(n-6)n(3n^{3}-48n^{2}+228n-320)}{18t^{2}}$
$\displaystyle+\frac{(n-6)n(n^{4}-24n^{3}+32n^{2}+864n-2448)}{216t^{3}}+\frac{(n-6)n(3n^{4}+12n^{3}-416n^{2}-792n+8928)}{1296t^{4}}$
$\displaystyle+\frac{(n-6)n(n^{4}+30n^{3}+180n^{2}-1080n-7776)}{2592t^{5}}+\frac{(n-6)n(n^{4}+60n^{3}+1260n^{2}+10800n+31104)}{46656t^{6}}$
$\displaystyle\widetilde{K}_{1}(n,t)$
$\displaystyle=-8(n^{3}-12n^{2}+44n-48)+\frac{2(n^{4}-66n^{3}+516n^{2}-1688n^{3}+1920)}{3t}$
$\displaystyle-\frac{(n-6)^{2}n(n^{2}-24n+68)}{12t^{2}}-\frac{n(3n^{4}-42n^{3}+16n^{2}+1512n-4464)}{54t^{3}}$
(A.5)
$\displaystyle-\frac{5(n-6)^{2}n(n^{2}+18n+72)}{432t^{4}}-\frac{n(n^{4}+30n^{3}+180n^{2}-1080n-7776)}{1296t^{5}}$
$\displaystyle\widetilde{K}_{2}(n,t)$
$\displaystyle=-(6n^{3}-96n^{2}+456n-640)+\frac{(n-6)^{2}(n^{2}-24n+68)}{2t}$
$\displaystyle+\frac{n(3n^{3}-60n^{2}+376n-744)}{6t^{2}}+\frac{5(n-6)^{2}n(n+6)}{36t^{3}}+\frac{5n(n^{3}+12n^{2}-36n-432)}{432t^{4}}$
$\displaystyle\widetilde{K}_{3}(n,t)$
$\displaystyle=-(n^{3}-30n^{2}+212n-408)-\frac{(6n^{3}-120n^{2}+752n-3496)}{3t}-\frac{5(n-6)^{2}n}{6t^{2}}-\frac{5n(n^{2}-36)}{54t^{3}}$
$\displaystyle\widetilde{K}_{4}(n,t)$
$\displaystyle=(3n^{2}-42n+124)+\frac{5(n-6)^{2}}{2t}+\frac{5n(n-6)}{12t^{2}}$
$\displaystyle\widetilde{K}_{5}(n,t)$ $\displaystyle=-3(n-6)+\frac{n-6}{t}$

and

$\displaystyle\widetilde{J}_{0}(n,t)$
$\displaystyle=2(n^{4}-8n^{3}-39n^{2}+470n-964)+\frac{(3n^{4}-34n^{3}+34n^{2}+650n-1668)}{3t}$
$\displaystyle+\frac{n(4n^{3}-43n^{2}+125n-66)}{18t^{2}}+\frac{n(n^{3}-11n^{2}-108n+396)}{108t^{3}}$
$\displaystyle+\frac{n(n^{3}+12n^{2}-36n-432)}{648t^{4}}$
$\displaystyle\widetilde{J}_{1}(n,t)$
$\displaystyle=-(6n^{3}-32n^{2}-124n+556)+\frac{2(4n^{3}-43n^{2}+125n-66)}{3t}$
$\displaystyle-\frac{n(3n^{2}-29n+66)}{6t^{2}}-\frac{n(n^{2}-36)}{27t^{3}}$
$\displaystyle\widetilde{J}_{2}(n,t)$
$\displaystyle=(8n^{2}-38n+22)+\frac{(3n^{2}-29n+66)}{t}+\frac{n(n-6)}{3t^{2}}$
(A.6) $\displaystyle\widetilde{J}_{3}(n,t)$
$\displaystyle=-\frac{4(n-6)}{3}-(6n-22)t$
$\displaystyle\widetilde{L}_{0}(n,t)$
$\displaystyle=-\frac{(72n-384)}{12}+\frac{(n-6)^{2}}{2t}+\frac{n(n-6)}{12t^{2}}$
$\displaystyle\widetilde{L}_{1}(n,t)$ $\displaystyle=-3(n-6)-\frac{n-6}{t}.$

Here, the real-valued functions $\mathfrak{p}_{j}(n,\cdot):(-\infty,\ln
R)\rightarrow\mathbb{R}$ for $j=0,1,2,3,4,5$ are given by

$\displaystyle\mathfrak{p}_{5}(n,t)$ $\displaystyle:=-\frac{3}{2}(n-6),$
$\displaystyle\mathfrak{p}_{4}(n,t)$
$\displaystyle:=-\frac{5}{2}{n(n-6)}t^{-2}+\frac{1}{2}(3n^{2}-42n+124),$
$\displaystyle\mathfrak{p}_{3}(n,t)$
$\displaystyle:=\frac{5}{12}n(n^{2}-36)t^{-3}+\frac{5}{12}n(n-6)t^{-2}-\frac{1}{2}(n^{3}-30n^{2}+212n-408),$
$\displaystyle\mathfrak{p}_{2}(n,t)$
$\displaystyle:=-\frac{5}{288}n(n-6)t^{-4}-\frac{5}{36}(n-6)^{2}n(n+6)t^{-3}-\frac{1}{12}n(3n^{3}-60n^{2}+376n-744)t^{-2}$
$\displaystyle-(3n^{3}-48n^{2}+228n-320),$
$\displaystyle\mathfrak{p}_{1}(n,t)$
$\displaystyle:=\frac{(n-6)}{864}\left[(64512-44928n+10368n^{2}-864n^{3})t^{-5}\right.$
(A.7)
$\displaystyle+(553392n+170532n^{2}+15732n^{3}+447n^{4})t^{-4}+(9744n-2228n^{2}-324n^{3}+53n^{4})t^{-3}$
$\displaystyle+(-8736n+6608n^{2}-1272n^{3}+60n^{4})t^{-2}+(-14688n+7632n^{2}-1080n^{3}+36n^{4})t^{-1}$
$\displaystyle\left.-(864n^{3}-10368n^{2}+44928n-64512)+(3456n^{2}-20736n+27648)t\right],$
$\displaystyle\mathfrak{p}_{0}(n,t)$
$\displaystyle:=-\frac{5}{93312}n(n-6)(n^{4}+60n^{3}+1260n^{2}+10800n+31104)t^{-6}$
$\displaystyle+\frac{1}{1296}n(n-6)(n^{4}+30n^{3}+180n^{2}-1080n-7776)t^{-5}$
$\displaystyle+\frac{5}{864}n(n-6)(3n^{4}+12n^{3}-416n^{2}-792n+8928)t^{-4}$
$\displaystyle+\frac{5}{216}n(n-6)(n^{4}-24n^{3}+32n^{2}+864n-2448)t^{-3}+\frac{1}{36}(3n^{3}-48n^{2}+228n-320)t^{-2}.$

###  A.4. Upper critical case

When $p=2^{\\#}-1$, it follows

$\displaystyle K^{\\#}_{0}(n)=-2^{-8}(n-6)^{2}(n-2)^{2}(n+2)^{2}$
$\displaystyle K^{\\#}_{1}(n)=0$ $\displaystyle
K^{\\#}_{2}(n)=2^{-4}(3n^{4}-24n^{3}+72n^{2}-96n+304)$ $\displaystyle
K^{\\#}_{3}(n)=0$ $\displaystyle K^{\\#}_{4}(n)=-2^{-2}(3n^{2}-12n+44)$
$\displaystyle K^{\\#}_{5}(n)=0$ $\displaystyle
J^{\\#}_{0}(n)=2^{-3}(3n^{4}-18n^{3}-192n^{2}+1864n-3952)$ (A.8)
$\displaystyle J^{\\#}_{1}(n)=-2^{-1}(3n^{3}+3n^{2}-244n+620)$ $\displaystyle
J^{\\#}_{2}(n)=2n^{2}+13n-68$ $\displaystyle J^{\\#}_{3}(n)=-2(n+1)$
$\displaystyle L^{\\#}_{0}(n)=-2^{-2}(3n^{2}-12n-20)$ $\displaystyle
L^{\\#}_{1}(n)=0.$

###  A.5. Lower critical case

When $p=2_{\\#}$, it follows

$\displaystyle K_{0,\\#}(n)=0$ $\displaystyle K_{1,\\#}(n)=-8(n-6)(n-4)(n-2)$
$\displaystyle K_{2,\\#}(n)=-2(3n^{3}-48n^{2}+228n-320)$ $\displaystyle
K_{3,\\#}(n)=-(n-6)(n^{2}-24n+68)$ $\displaystyle K_{4,\\#}(n)=3n^{2}-42n+124$
$\displaystyle K_{5,\\#}(n)=-3(n-6)$ $\displaystyle
J_{0,\\#}(n)=2(n^{4}-8n^{3}-39n^{2}+470n-964)$ (A.9) $\displaystyle
J_{1,\\#}(n)=-2(3n^{3}-16n^{2}-62n+278)$ $\displaystyle
J_{2,\\#}(n)=2(4n^{2}-19n+11)$ $\displaystyle J_{3,\\#}(n)=-2(3n-11)$
$\displaystyle L_{0,\\#}(n)=-2(3n-16)$ $\displaystyle L_{1,\\#}(n)=-3(n-6).$

## References

  * [1] J. H. Andrade and J. M. do Ó, Qualitative properties for solutions to conformally invariant fourth order critical systems, _arXiv:2002.12491 [math.AP]_ (2020).
  * [2] J. H. Andrade and J. M. do Ó, Qualitative properties for solutions to subcritical fourth order systems, _to appear in Nonlinearity_ (2022).
  * [3] J. H. Andrade and J. Wei, Classification for positive singular solutions to critical sixth order equations, _arXiv:2210.04376 [math.AP]_ (2022).
  * [4] P. Aviles, Local behavior of solutions of some elliptic equations, _Comm. Math. Phys._ 108 (1987) 177–192.
  * [5] L. A. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, _Comm. Pure Appl. Math._ 42 (1989) 271–297.
  * [6] R. Caju, J. M. do Ó and A. Santos, Qualitative properties of positive singular solutions to nonlinear elliptic systems with critical growth, _Ann. Inst. H. Poincaré Anal. Non Linéaire_ 36 (2019) 1575–1601.
  * [7] W. Chen and C. Li, A necessary and sufficient condition for the Nirenberg problem, _Comm. Pure Appl. Math._ 48 (1995) 657–667.
  * [8] W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations, _Comm. Partial Differential Equations_ 30 (2005) 59–65.
  * [9] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, _Comm. Pure Appl. Math._ 59 (2006) 330–343.
  * [10] J. Dávila, L. Dupaigne, K. Wang and J. Wei, A monotonicity formula and a Liouville-type theorem for a fourth order supercritical problem, _Adv. Math._ 258 (2014) 240–285.
  * [11] W. H. Fleming, On the oriented Plateau problem, _Rend. Circ. Mat. Palermo (2)_ 11 (1962) 69–90.
  * [12] R. L. Frank and T. König, Classification of positive singular solutions to a nonlinear biharmonic equation with critical exponent, _Anal. PDE_ 12 (2019) 1101–1113.
  * [13] M. Ghergu, S. Kim and H. Shahgholian, Isolated singularities for semilinear elliptic systems with power-law nonlinearity, _Anal. PDE_ 13 (2020) 701–739.
  * [14] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, _Comm. Pure Appl. Math._ 34 (1981) 525–598.
  * [15] D. Gilbarg and N. S. Trudinger, _Elliptic partial differential equations of second order_ , Classics in Mathematics, Springer-Verlag, Berlin (2001), reprint of the 1998 edition.
  * [16] Z. Guo, X. Huang, L. Wang and J. Wei, On Delaunay solutions of a biharmonic elliptic equation with critical exponent, _J. Anal. Math._ 140 (2020) 371–394.
  * [17] Z. Guo, J. Wei and F. Zhou, Singular radial entire solutions and weak solutions with prescribed singular set for a biharmonic equation, _J. Differential Equations_ 263 (2017) 1188–1224.
  * [18] X. Huang, Y. Li and H. Yang, Super polyharmonic property and asymptotic behavior of solutions to the higher order Hardy-Hénon equation near isolated singularities, _arXiv:2210.04619 [math.AP]_ (2022).
  * [19] T. Jin and J. Xiong, Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities, _Ann. Inst. H. Poincaré Anal. Non Linéaire_ (2020).
  * [20] N. Korevaar, R. Mazzeo, F. Pacard and R. Schoen, Refined asymptotics for constant scalar curvature metrics with isolated singularities, _Invent. Math._ 135 (1999) 233–272.
  * [21] Y. Li, Remark on some conformally invariant integral equations: the method of moving spheres, _J. Eur. Math. Soc. (JEMS)_ 6 (2004) 153–180.
  * [22] E. H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, _Ann. of Math. (2)_ 118 (1983) 349–374.
  * [23] P.-L. Lions, Isolated singularities in semilinear problems, _J. Differential Equations_ 38 (1980) 441–450.
  * [24] S. Luo, J. Wei and W. Zou, Decomposition of polyharmonic operator and classification of homogeneous stable solutions, _Proc. Amer. Math. Soc._ 149 (2021) 2957–2968.
  * [25] S. Luo, J. Wei and W. Zou, Monotonicity Formula and Classification of Stable Solutions to Polyharmonic Lane–Emden Equations, _Int. Math. Res. Not. IMRN_ (2021), rnab212.
  * [26] Q. A. Ngô and D. Ye, Existence and non-existence results for the higher order Hardy-Hénon equations revisited, _J. Math. Pures Appl. (9)_ 163 (2022) 265–298.
  * [27] J. Serrin, Local behavior of solutions of quasi-linear equations, _Acta Math._ 111 (1964) 247–302.
  * [28] L. Simon, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, _Ann. of Math. (2)_ 118 (1983) 525–571.
  * [29] R. Soranzo, Isolated singularities of positive solutions of a superlinear biharmonic equation, _Potential Anal._ 6 (1997) 57–85.
  * [30] L. Sun and J. Xiong, Classification theorems for solutions of higher order boundary conformally invariant problems, I, _J. Funct. Anal._ 271 (2016) 3727–3764.
  * [31] L. Véron, Comportement asymptotique des solutions d’équations elliptiques semi-linéaires dans ${\bf R}^{N}$, _Ann. Mat. Pura Appl. (4)_ 127 (1981) 25–50.
  * [32] J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations, _Math. Ann._ 313 (1999) 207–228.
  * [33] X. Xu, Uniqueness theorem for the entire positive solutions of biharmonic equations in $\mathbb{R}^{n}$, _Proc. Roy. Soc. Edinburgh Sect. A_ 130 (2000) 651–670.
  * [34] H. Yang, Asymptotic behavior of positive solutions to a nonlinear biharmonic equation near isolated singularities, _Calc. Var. Partial Differential Equations_ 59 (2020) 130.
  * [35] H. Yang and W. Zou, On isolated singularities of fractional semi-linear elliptic equations, _Ann. Inst. H. Poincaré Anal. Non Linéaire_ (2020).