# Classification of radial blow-up at the first critical exponent for the Lin-
Ni-Takagi problem in the ball

Denis Bonheure  Département de Mathématique, Université libre de Bruxelles,
Campus de la Plaine CP 213, Bd. du Triomphe, 1050 Bruxelles, Belgium
<EMAIL_ADDRESS><EMAIL_ADDRESS>,  Jean-Baptiste Casteras
CMAFcIO, Faculdade de Ciências da Universidade de Lisboa, Edificio C6, Piso 1,
Campo Grande 1749-016 Lisboa, Portugal<EMAIL_ADDRESS>and
Bruno Premoselli

###### Abstract.

We investigate the behaviour of radial solutions to the Lin-Ni-Takagi problem
in the ball $B_{R}\subset\mathbb{R}^{N}$ for $N\geq 3$:

$\left\\{\begin{aligned} -\triangle u_{p}+u_{p}&=|u_{p}|^{p-2}u_{p}&\textrm{
in }B_{R},\\\ \partial_{\nu}u_{p}&=0&\textrm{ on }\partial
B_{R},\end{aligned}\right.$

when $p$ is close to the first critical Sobolev exponent
$2^{*}=\frac{2N}{N-2}$. We obtain a complete classification of finite energy
radial smooth blowing up solutions to this problem. We describe the conditions
preventing blow-up as $p\to 2^{*}$, we give the necessary conditions in order
for blow-up to occur and we establish their sharpness by constructing examples
of blowing up sequences. Our approach allows for asymptotically supercritical
values of $p$. We show in particular that, if $p\geq 2^{\ast}$, finite-energy
radial solutions are precompact in $C^{2}(\overline{B_{R}})$ provided that
$N\geq 7$. Sufficient conditions are also given in smaller dimensions if
$p=2^{\ast}$. Finally we compare and interpret our results to the bifurcation
analysis of Bonheure, Grumiau and Troestler in Nonlinear Anal. 147 (2016).

B. P. was supported by the FNRS CdR grant J.0135.19 and the Fonds Thélam. D.B.
and B.P. are supported by an ARC Avancé 2020 at ULB. J.-B.C was supported by
FCT - Fundação para a Ciência e a Tecnologia, under the project:
UIDB/04561/2020. D.B. is Francqui Research Professor 2021-2024. He thanks the
Fondation Francqui for the support.

##  1\. Introduction

###  1.1. Statement of the results

Let $R>0$ and $B_{R}=B(0,R)$ be the ball in $\mathbb{R}^{N}$, $N\geq 3$. We
investigate the behaviour of families $(u_{p})_{p>2}$ of radial solutions to
the so-called Lin-Ni-Takagi problem in $B_{R}$:

(1.1) $\left\\{\begin{aligned} -\triangle
u_{p}+u_{p}&=|u_{p}|^{p-2}u_{p}&\textrm{ in }B_{R}\\\
\partial_{\nu}u_{p}&=0&\textrm{ in }\partial B_{R}.\end{aligned}\right.$

This equation can be derived from the Patlak-Keller-Segel system

(1.2) $\begin{cases}\dfrac{\partial u}{\partial t}=\Delta
u-D_{1}\nabla\cdot(u\nabla\phi(v))\quad&\text{in }B_{R},\\\\[11.38109pt]
\dfrac{\partial v}{\partial t}=D_{2}\Delta v-D_{3}v+D_{4}u\quad&\text{in
}B_{R},\\\\[5.69054pt] \partial_{\nu}u=\partial_{\nu}v=0\quad&\text{on
}\partial B_{R},\\\ u,v>0&\text{in }B_{R},\end{cases}$

where $D_{i}$, $i=1,\ldots,4$ are positive constants and $\phi$ is a smooth
strictly increasing function. This system has been introduced in [KS70] to
model chemotaxis, a situation where organisms move towards areas of high
concentration of the chemicals they secrete. In this model $u(x,t)$ represents
the concentration of the considered organisms and $v(x,t)$ the one of the
chemical released. A huge literature exists on the dynamics of this system,
concerning several fondamental questions such as the Cauchy theory, blow-up in
finite or infinite time and stability to cite a few of them. Since it is not
our purpose to analyse the time-depending solutions, we just refer to some
recent papers [SW19, DdPD+19, CGMN19, Win20] and the many citations therein to
trace the most important results in this direction. A large variety of highly
nontrivial equilibria have been detected since the seminal work [LNT88], see
for instance [MM02, MNW05, MM07, DRW12, AWZ13, dPMM14, PV15, BGT16, dPPV16,
BGNT16, BCN17a, BCN17b, BCR21]. This list of references is far from being
exhaustive. One can prove that stationary positive solutions satisfy

$\nabla\cdot(u\nabla(\log u-D_{1}\phi(v))=0,$

which, together with the boundary conditions, implies that
$u=Ce^{D_{1}\phi(v)}$ for some positive constant $C$. Taking $\phi(v)=\ln v$
one recovers (1.1) for a suitable choice of parameters $D_{i}$ and $C$. The
Lin-Ni-Takagi problem has been intensively studied these last decades. We
refer to [LNT88, LN88, AY91, BKP91, AY97, RW04, dPMP05, WWY10, WWY11, DRW12]
for a non-exhaustive list of existence and non-existence results for (1.1),
related to the so-called Lin-Ni conjecture, and mostly concerning the close-
to-critical case. As mentioned by several experts of the community, the role
of the many highlighted equilibria, as actually the role of any stationary
solution, in the dynamics of the parabolic problem (1.2), seems unclear up to
now. However, it is to expect that loss of compactness by bubbling in the
stationary frame plays a role in the dynamics of (1.2). This issue has been
investigated in other time-depending equations, as for instance in [LW08,
KST09, DdPW20, CdPM20, JM20, dPMW21a, dPMW21b] and the included citations.

We will be interested in this work in the blow-up behaviour of radial
solutions of (1.1) at the first critical exponent $p=2^{*}$, where we have let
$2^{*}=\frac{2N}{N-2}$. Our main goal in this paper is to provide a complete
description of the radial blow-up behaviour of (1.1) when $p\to 2^{*}$. This
means giving sufficient conditions preventing the blow-up; describing _a
priori_ necessary condition in order for blow-up to occur; and constructing,
when the necessary conditions are satisfied, radial blowing up sequences of
solutions of (1.1). Our investigations are motivated, among other things, by
the bifurcation analysis in [BGT16], for which we refer to Section 1.2 below.

For simplicity we will work, in the following, up to a subsequence: we let
$(p_{k})_{k\geq 0}$ be a sequence of positive real numbers converging towards
$2^{*}$ as $k\to+\infty$ and we let $(u_{k})_{k\geq 0}$ be a sequence of
_possibly sign-changing_ radial solutions of the following equation:

(1.3) $\left\\{\begin{aligned} -\triangle
u_{k}+u_{k}&=|u_{k}|^{p_{k}-2}u_{k}&\textrm{ in }B_{R},\\\
\partial_{\nu}u_{k}&=0&\textrm{ in }\partial B_{R},\end{aligned}\right.$

which is just (1.1) with $p=p_{k}$. We will always work with solutions that
are _energy-bounded_ in the following sense: we assume that there exists $C>0$
such that

(1.4) $\|u_{k}\|_{L^{\alpha_{k}}(B_{R})}\leq C,$

for all $k\geq 0$, where

$\alpha_{k}=\max\big{(}2^{*},\frac{N}{2}p_{k}-N\big{)}.$

If $p_{k}\leq 2^{*}$, then $\alpha_{k}=2^{*}$ and (1.4) implies that each
$u_{k}$ is smooth by the classical argument of [Tru68] (see also [BK79]).
Observe that in this case (1.4) is equivalent to a $H^{1}(B_{R})$ bound on the
sequence $(u_{k})_{k\geq 0}$. When $p_{k}>2^{*}$ we have $\alpha_{k}>p_{k}$
and thus assumption (1.4) is stronger than a uniform bound on the
$H^{1}(B_{R})$ norm. The exponent $\alpha_{k}$ appears however naturally since
it defines the precise norm which is preserved by the natural scaling that is
formally associated to (1.1) : given $p>2$, the $L^{\alpha}(\mathbb{R}^{N})$
norm of $\lambda^{\frac{2}{p-2}}u(\lambda x)$ is preserved if and only if
$\alpha=\frac{N}{2}(p-2)$. Such an $\alpha$ is also the exact exponent for
which a natural bootstrap procedure in (1.1) does not improve the
integrability of a solution. Observe also that when $p_{k}>2^{*}$ in (1.3),
the smoothness of the solutions is not guaranteed solely by (1.4) anymore. We
thus assume in addition, if $p_{k}>2^{*}$, that $u_{k}$ is smooth, by which we
mean $C^{2}(\overline{B_{R}})$, for any $k\geq 0$. Smooth radial solutions of
(1.1) for all $p$ have for instance been constructed in [MNW05, ST11, BNW12,
BST13, BGT16]. Singular solutions have been constructed in [Miy15, CF20] in
the supercritical case.

We say that a sequence of smooth, possibly sign-changing, radial solutions
$(u_{k})_{\geq 0}$ of (1.3) blows up if

$\lim_{k\to+\infty}\|u_{k}\|_{L^{\infty}(B_{R})}=+\infty.$

Since we deal with energy bounded sequences, Strauss lemma [Str77]
equivalently implies

$\lim_{k\to+\infty}|u_{k}(0)|=+\infty.$

Standard elliptic theory and (1.3) show that if $(u_{k})_{k\geq 0}$ does not
blow up, then it is precompact in the $C^{2}(\overline{B_{R}})$ topology. Our
first result is a precompactness theorem that states that if the $p_{k}$’s
approach the critical exponent $2^{*}$ from above, then an energy-bounded
sequence of solutions $(u_{k})_{k}$ of (1.3) never blows up in dimensions
$N\geq 7$.

######  Theorem 1.1 (Critical or supercritical case in dimensions $N\geq 7$).

Assume $N\geq 7$ and let $(p_{k})_{k\geq 0}$ be a sequence of positive real
numbers such that $p_{k}\geq 2^{*}$ for all $k\geq 0$ and
$\lim_{k\to+\infty}p_{k}=2^{*}$. Any sequence $(u_{k})_{k\geq 0}$ of
$C^{2}(\overline{B_{R}})$ radial solutions of (1.3) satisfying (1.4) is
precompact in $C^{2}(\overline{B_{R}})$.

By standard elliptic theory, Theorem 1.1 implies that $(u_{k})_{k\geq 0}$
strongly converges, up to a subsequence, in $C^{2}(\overline{B_{R}})$ to a
solution of (1.1) with $p=2^{*}$. By the result of [Ni83], radial solutions of
(2.2) when $p_{k}\geq 2^{*}$ have fixed sign (otherwise there would be a
Dirichlet solution on some smaller ball and this is impossible in a star-
shaped domain). In lower dimensions, we can cover the exactly critical case
$p_{k}=2^{*}$ for all $k\geq 0$.

######  Theorem 1.2 (Exactly critical case).

Let $p_{k}=2^{*}$ for all $k\geq 0$. Assume that $(u_{k})_{k\geq 0}$ is a
sequence of radial solutions of (1.3) that is uniformly bounded in
$L^{2^{*}}(B_{R})$.

  * •

If $N\not\in\\{3,6\\}$, then $(u_{k})_{k\geq 0}$ is precompact in
$C^{2}(\overline{B_{R}})$.

  * •

If $N=3$, there exists an exceptional radius $R^{*}>0$ such that if $R\neq
R^{*}$, then $(u_{k})_{k\geq 0}$ is precompact in $C^{2}(\overline{B_{R}})$.

  * •

If $N=6$ there exists as increasing sequence of exceptional radii
$(R_{\ell})_{\ell\geq 1}$, with $R_{\ell}\to+\infty$ and
$\liminf_{\ell\to+\infty}(R_{\ell}-R_{\ell-1})>0$, such that if
$R\not\in\\{R_{\ell}\\}_{\ell\geq 1}$ then $(u_{k})_{k\geq 0}$ is precompact
in $C^{2}(\overline{B_{R}})$.

The value of the exceptional radius $R^{*}$ in dimension $N=3$ is described in
Section 3.1. We believe that precompactness also holds for $R=R^{*}$ but it
would require an additional and tedious analysis to prove it, see Remark 3.2.
In dimension $N=6$, the values of $(R_{\ell})_{\ell\geq 1}$ are also
univocally determined. We refer again to the proof of Theorem 1.2 in Section
3.1. Under an additional assumption which seems numerically true, we are able
to fully prove precompactness, see Lemma 3.5 below and Remark 3.6 that
follows. This almost completely settles the exactly critical case
$p_{k}=2^{*}$ in (1.3) and leads to the following conjecture :

######  Conjecture 1.3 (Exactly critical case).

Any uniformly (energy) bounded sequence $(u_{k})_{k\geq 0}$ of radial
solutions of (1.3) with $p_{k}=2^{*}$ for all $k\geq 0$, is precompact in
$C^{2}(\overline{B_{R}})$.

The missing cases have just been mentioned : $N=3$ and $R=R^{*}$ in which case
the conjecture is supported by Remark 3.2 and $N=6$ and $R=R_{\ell}$ in which
cases the conjecture is supported by Lemma 3.5 below and Remark 3.6.

Theorems 1.1 and 1.2 are the main precompactness results of this paper. They
do not just follow from classical precompactness results for (possibly sign-
changing) solutions of equations similar to (1.1), such as [DS02]. There are
two main novelties in our approach here: first, we include _sign-changing_
solutions in our analysis, and second, we allow for asymptotically
_supercritical_ perturbations in (1.3). To our knowledge this seems to be the
first _a priori_ blow-up analysis for asymptotically supercritical solutions
of equations like (1.1). We prove Theorems 1.1 and 1.2 by contradiction.
Assuming that blow-up occurs we prove a sharp pointwise asymptotic description
of blowing up sequences of solutions (see Proposition 2.1 below). The _radial
symmetry_ forces blow-up to occur in the form of finite sums of positive
bubbles at the origin, with possibly alternating signs (Proposition 2.1 deals
also with the subcritical case and sign-changing solutions exist in that
case). This pointwise description is then used to rule out blow-up by means of
Pohozaev identities. The ideas that we use in the proofs have been recently
developed to prove compactness results for finite-energy sign-changing
solutions of critical elliptic equations: we refer for instance to [GMR,
PV22a, PV22b], and to [PV19] for compactness results without bound on the
energy.

The techniques that we use to prove Theorems 1.1 and 1.2 yield more generally
necessary conditions for blow-up for positive solutions. We start with the
subcritical case:

######  Theorem 1.4 (Blow-up in the subcritical case).

Assume that $p_{k}\leq 2^{*}$ for all $k\geq 0$,
$\lim_{k\to+\infty}p_{k}=2^{*}$ and let $(u_{k})_{k\geq 0}$ be a sequence of
_positive radial_ solutions of (1.3) satisfying (1.4) and denote by $u_{0}$
its weak limit. If there is blow-up, then

  * •

$u_{0}\equiv 0$ for $3\leq N\leq 5$ and $R\geq R^{*}$ when $N=3$,

  * •

$u_{0}(0)\leq\frac{1}{2}$ when $N=6$.

When $3\leq N\leq 6$, the blow-up occurs with a single positive bubble at $0$.

The value of the radius $R^{*}$ in dimension $N=3$ is as in Theorem 1.2. We
refer to Theorem 3.1 for the necessary mass condition explaining the role or
$R^{*}$. As a consequence of this result, a tower of bubbles, if it exist in
the subcritical case, can arise only in dimension $N\geq 7$.

Theorem 1.2 and Conjecture 1.3 state that a purely critical blow-up is not
possible. The missing cases are in dimension $3$ and $6$. In these dimensions
the direct counterpart of Theorem 1.4 implies that if there is blow-up, then
$u_{0}\equiv 0$ if $N=3$ and $R=R^{*}$, while $u_{0}(0)=\frac{1}{2}$ if $N=6$,
forcing $R$ to be in the set of critical points $\\{R_{\ell}\\}_{\ell\geq 1}$
of $u_{0}$ (those are fixed by the initial condition $u_{0}(0)=\frac{1}{2}$,
remembering that $u_{0}^{\prime}(0)=0$).

We finally cover the supercritical case $p_{k}>2^{*}$ for all $k\geq 0$.
Theorem 1.1 states that blow-up cannot happen in this case in dimension $N\geq
7$. In dimensions $3\leq N\leq 6$, a blow-up might occur, but we show that, at
least when $N=4,5,6$, the weak limit of a bubbling sequence is strictly
positive:

######  Theorem 1.5 (blow-up in the supercritical case in dimension $3\leq
N\leq 6$).

Assume $4\leq N\leq 6$. Let $p_{k}\geq 2^{*}$ for all $k\geq 0$ with
$\lim_{k\to+\infty}p_{k}=2^{*}$, $(u_{k})_{k\geq 0}$ be a sequence of
_positive radial_ $C^{2}(\overline{B_{R}})$ solutions of (1.3) satisfying
(1.4) and denote by $u_{0}$ its weak limit. If there is blow-up then $u_{0}>0$
and $u_{0}(0)\geq 1/2$ if $N=6$. If $N=3$ and $u_{0}=0$, then $R\leq R^{*}$.

When $N=3$ the weak limit need not be positive, as the examples in [RW04]
show, see Remark 4.6 below.

It is natural to investigate the sharpness of the conclusion of Theorems 1.1
to 1.5: that is, to understand if blowing up sequences of solutions do exist
when the necessary conditions of Theorems 1.4 and 1.5 are fulfilled. We pursue
this aims for positive solutions. We say that a sequence of positive radial
solutions $(u_{k})_{k\geq 0}$ to (1.3) is of the form $u_{0}+B$ if there
exists $u_{0}\in C^{2,\theta}(B_{R})$, $\theta\in(0,1)$, $u_{0}\geq 0$, and a
sequence of bubbles

$B_{\lambda_{k}}=[N(N-2)]^{\frac{N-2}{4}}\left(\dfrac{\lambda_{k}}{\lambda_{k}^{2}+|x|^{2}}\right)^{\frac{N-2}{2}}$

with $\lambda_{k}>0$, $\lambda_{k}\rightarrow 0$ as $k\to+\infty$, such that

(1.5) $u_{k}=u_{0}+B_{\lambda_{k}}+o(1)\quad\text{ in }H^{1}(B_{R}),$

as $k\rightarrow+\infty$. If $u_{0}\equiv 0$ in the previous definition we
just say that $(u_{k})_{k\geq 0}$ is of the form $B$.

We prove in Section 4 that positive radial blowing up solutions satisfying
(1.4) exist for $N\geq 4$ when $p_{k}\to 2^{*}$ from below and in all the
cases not covered by Theorem1.1 when $p_{k}\to 2^{*}$ from above. We start
with a subcritical bubbling sequence:

######  Theorem 1.6 (Type $B$ in the subcritical case in dimensions $N\geq
4$).

If $N\geq 4$, there exists a sequence $p_{k}\to 2^{*}$ with $p_{k}<2^{*}$ for
all $k\geq 0$ and a sequence $(u_{k})_{k\geq 0}$ of positive radial solutions
of (1.3) of type $B$.

Note that in the (sub)critical case a sequence of solutions of type $B$ or
$u_{0}+B$ is uniformly bounded in $L^{2^{*}}(B_{R})$. That a blowing up
sequence of solutions $(u_{k})_{k\geq 0}$ in the subcritical case has to be of
type $B$ in dimensions $N=4,5$ follows from Theorem 1.4. Theorem 1.6 does not
cover the case $N=3$. In this case blowing up solutions of type $B$ have been
constructed in [RW04]. We discuss the $N=3$ case in more details in Remark 4.6
below.

Still in the subcritical case, we also construct solutions of type $u_{0}+B$,
in dimension $N\geq 6$. Our constructions of solutions of type $u_{0}+B$
require that $u_{0}$ is a non-degenerate solution of (1.1) with the critical
exponent $p=2^{*}$. We recall that, given $h_{0}\in C^{0}(\overline{B_{R}})$,
a solution $u_{0}$ of

(1.6) $\left\\{\begin{aligned} -\triangle
u_{0}+h_{0}u_{0}&=|u_{0}|^{2^{*}-2}u_{0}&\textrm{ in }B_{R}\\\
\partial_{\nu}u_{0}&=0&\textrm{ in }\partial B_{R}.\end{aligned}\right.$

is _nondegenerate_ if, for any $\varphi\in H^{1}(B_{R})$,

(1.7) $\left\\{\begin{aligned}
-\triangle\varphi+\Big{(}h_{0}-(2^{*}-1)|u_{0}|^{2^{*}-2}\Big{)}\varphi&=0&\textrm{
in }B_{R}\\\ \partial_{\nu}\varphi&=0&\textrm{ in }\partial
B_{R}.\end{aligned}\right.$

implies $\varphi\equiv 0$.

######  Theorem 1.7 (Type $u_{0}+B$ in the subcritical case in dimensions
$N\geq 6$).

Let $N\geq 6$ and $u_{0}\in C^{2,\theta}$, $\theta\in(0,1)$, be a positive
nondegenerate radial solution of (1.1) with the critical exponent
$p=2^{\ast}$. If $N=6$, assume in addition that $u_{0}(0)<\frac{1}{2}$. There
exists a sequence $p_{k}$ with $\lim_{k\to+\infty}p_{k}=2^{\ast}$,
$p_{k}<2^{*}$ for all $k\geq 0$, and a sequence $(u_{k})_{k\geq 0}$ of
positive radial solutions of (1.3) of type $u_{0}+B$. Moreover, we have that

  * •

$u_{k}-1$ has $i+1$ zeros, if $u_{0}(0)<1$,

  * •

$u_{k}-1$ has $i$ zeros if $u_{0}(0)\geq 1$,

where $i$ is the number of zeros of $u_{0}-1$.

The assumption that $u_{0}$ is non-degenerate is not restrictive and is
generic in the choice of $R$ as Proposition 4.2 below shows. We mention that
in some cases non-degeneracy is not required to construct blowing up
solutions, see [RV15]. If $N=6$, solutions $u_{0}$ with $u_{0}(0)<\frac{1}{2}$
exist at least for $R>R_{1}$, see for instance [Ni83] and Remark 4.7. The
conditions $N\geq 6$ and $u_{0}(0)\leq\frac{1}{2}$ when $N=6$ are necessary in
view of Theorem 1.4.

Our last constructive result of blow-up regards the supercritical case. As
shown by Theorems 1.1 and 1.5, only dimensions $3\leq N\leq 6$ allow for
supercritical bubbling and solutions cannot be of type $B$ in dimension $4\leq
N\leq 6$.

######  Theorem 1.8 (Type $u_{0}+B$ in the supercritical case in dimensions
$3\leq N\leq 6$).

Let $3\leq N\leq 6$ and $u_{0}\in C^{2,\theta}$, $\theta\in(0,1)$, be a
positive nondegenerate radial solution of (1.1) with the critical exponent
$p=2^{\ast}$. If $N=6$, assume in addition that $u_{0}(0)>\frac{1}{2}$. Then
there exists a sequence $p_{k}$ with $\lim_{k\to+\infty}p_{k}=2^{\ast}$,
$p_{k}>2^{*}$ for all $k\geq 0$, and a sequence $(u_{k})_{k\geq 0}$ of
positive radial solutions of (1.3) of type $u_{0}+B$. Moreover, we have that

  * •

$u_{k}-1$ has $i+1$ zeros, if $u_{0}(0)<1$,

  * •

$u_{k}-1$ has $i$ zeros if $u_{0}(0)\geq 1$,

where $i$ is the number of zeros of $u_{0}-1$.

If $N=6$, non constant solutions $u_{0}$ with $u_{0}(0)>\frac{1}{2}$ exist for
any radius $R>0$, see [Ni83, AY91, BGT16] and Remark 4.7. Observe also that
$u_{0}\equiv 1$ can be chosen in Theorems 1.7 and 1.8. In that case, the
nondegeneracy condition is even explicit : $R$ should be such that
$\frac{4}{N-2}$ is not a radial eigenvalue of $-\Delta$ on the ball $B_{R}$
with Neumann boundary condition, i.e. only a discrete set of radii is
forbidden.

When $N=3$ and $p_{k}>2^{*}=6$ blowing up solutions may be of type $B$ and
have for instance been constructed in [RW04].

The Figure 1 summarizes the classification. It shows the classification is
almost complete.

N |  $p_{k}<2^{*}$ |  $p_{k}=2^{*}$ |  $p_{k}>2^{*}$
---|---|---|---
$3$ |  $\bullet$ no blow-up if $R<R^{*}$ |  no blow-up1 |
|  $\bullet$ single bubble only |  |  $\bullet$ towers might exist
|  $\bullet$ only type $B$ is possible |  |  $\bullet$ no type $B$ if $R>R^{*}$
|  _$\bullet$ type $B$ occurs for large $R$_ |  |  _$\bullet$ type $B$ occurs for small $R$_
|  |  |  $\bullet$ type $u_{0}+B$ with $u_{0}>0$ occurs
$4,5$ |  $\bullet$ single bubble only |  no blow-up |  $\bullet$ towers might exist
|  $\bullet$ only type $B$ is possible |  |  $\bullet$ no type $B$
|  $\bullet$ type $B$ occurs |  |  $\bullet$ type $u_{0}+B$ with $u_{0}>0$ occurs
6 |  $\bullet$ single bubble only |  no blow-up2 |  $\bullet$ towers might exist
|  $\bullet$ only type $u_{0}+B$ with $u_{0}(0)\leq 1/2$ is possible |  |  $\bullet$ only type $u_{0}+B$ $u_{0}(0)\geq 1/2$ is possible
|  $\bullet$ type $u_{0}+B$ with $u_{0}(0)<1/2$ occurs |  |  $\bullet$ type $u_{0}+B$ with $u_{0}(0)>1/2$ occurs
$\geq 7$ |  $\bullet$ towers might exist |  no blow-up |  no blow-up
|  $\bullet$ type $B$ occurs |  |
|  $\bullet$ type $u_{0}+B$ with $u_{0}>0$ occurs |  |

1 see Remark 3.2 \- 2 see Remark 3.6

Figure 1. All the assertions are proved in the paper except those two in
italic which are proved in [RW04]. We also underlined the cases where we
expect towers of bubbles.

In dimension $N=3$, one expects blow-up occurs as soon as $R>R^{*}$ in the
subcritical case and $R<R^{*}$ in the supercritical case. This would require a
quantitative analysis of the conditions in [RW04]. As already mentioned, the
case $R=R^{*}$ remains open (also in the sub- and super-critical cases). The
classification is complete in dimensions $N=4,5$ and $N\geq 7$. The missing
piece in dimension $N=6$ concerns the critical case. Finally, we mention that
we did not address in this paper the construction of towers of bubbles when
they can exist. This part of the classification will be the subject of further
works. We also emphasize that our focus in this paper has been put on the
classification of _radial_ blow-up solutions of (1.3) at the critical Sobolev
exponent. The techniques used in this paper have however been successfully
used to prove precompactness results even in the non-radial case: [Pre21,
PV22a, PV22b]. Let us also mention that equations like (1.1) are sensitive to
perturbations: when the linear operator in the left-hand side is replaced by
$-\triangle$ the situation can be quite different, see for instance [SnT22] or
[PSnT22].

The proof of Theorems 1.6 to 1.8 goes through a Lyapunov-Schmidt reduction and
follows closely [MPV09] and [RV13] (see also [BC14]). We only construct
positive solutions in Theorems 1.6 to 1.8 since this is enough to prove the
sharpness of Theorems 1.1 and 1.2. It is however very likely that the approach
we develop in Section 4 will yield sign-changing solutions of (1.3) having the
form $u_{0}-B$ under suitable conditions in the subcritical case. Towers of
bubbles for (1.3) are also likely to occur, with or without alternating signs
(see Lemma 3.4 below where we prove necessary conditions for towers of
bubbles). For critical elliptic equations such as (1.3) towers of bubbles have
been constructed in [MPV09, PV13, MPV17, Pre22]. The references [MPV09, RV13]
concern a similar problem on a manifold without boundary (and are not
restricted to the radial case) but the techniques straightforwardly adapt to
open sets with Neumann boundary conditions and therefore to (1.3).

###  1.2. Radial bifurcation analysis of (1.1)

We conclude this introduction by describing how Theorems 1.1 to 1.8 complement
the radial bifurcation analysis of [BGT16] around the critical exponent
$p=2^{*}$. In [BGT16] the authors have studied radial positive solutions of
problem (1.1) bifurcating from the trivial solution $1$ as the parameter $p$
varies. Let $\lambda_{i}^{rad}$ be the $i$-th eigenvalue of the operator
$-\Delta+1$ in $B_{R}$ with Neumann boundary condition on radial solutions.
Among other things, the authors have shown in [BGT16] that for every $i\geq
2$, $(2+\lambda_{i}^{rad},1)$ is a bifurcation point for problem (1.1). If
$\mathcal{B}_{i}$ denotes the continuum of solutions that branches out of
$(2+\lambda_{i}^{rad},1)$ they also proved that

  * (i)

the branches $\mathcal{B}_{i}$ are unbounded and do not intersect; close to
$(2+\lambda_{i}^{rad},1)$, $\mathcal{B}_{i}$ is a $C^{1}$-curve;

  * (ii)

if $u\in\mathcal{B}_{i}$ then $u>0$;

  * (iii)

each branch consists of two connected components: the lower branch
$\mathcal{B}_{i}^{-}$, along which $u(0)<1$, and the upper branch
$\mathcal{B}_{i}^{+}$, along which $u(0)>1$;

  * (iv)

if $u\in\mathcal{B}_{i}$ then $u-1$ has exactly $i-1$ zeros, $u^{\prime}$ has
exactly $i-2$ zeros and each zero of $u^{\prime}$ lies between two zeros of
$u-1$;

  * (v)

the functions satisfying $u(0)<1$ are uniformly bounded in the $C^{1}$-norm.

Figure 2. Bifurcation diagram in dimension $N=4$. This figure is taken from
[BGT16] with the courtesy of the authors.

The numerical simulations of [BGT16] seem to indicate that the upper branches
$\mathcal{B}_{i}^{+}$ can be split into two categories: those having a
vertical asymptote at $p=2^{*}$, that we shall call “critical”, and those that
have vertical asymptotes at supercritical exponents $p_{i}>2^{*}$, that we
shall call “supercritical” : see e.g. Figure $1$ of this paper and Figure $13$
and Figure $16$ of [BGT16].

Theorems 1.6 to 1.8 provide explicit examples of solutions that, we believe,
describe the critical upper branches constructed in [BGT16]. For instance, as
Theorems 1.6 shows, positive radial solutions of type $B$ exist for all
dimensions $N\geq 4$ in the subcritical case. These solutions take the value
$1$ only once and satisfy $u(0)>1$ when $p$ is close to $2^{*}$. They are good
candidates to belong to the second upper branch $\mathcal{B}^{+}_{2}$ but a
proof of this fact seems quite difficult. Similarly, the branches
$\mathcal{B}^{+}_{i}$, for $i\geq 3$, are made of functions that take $i+1$
times the value $1$ so it is natural to believe that they correspond to the
solutions of the form $u_{0}^{i}+B$ constructed in Theorems 1.7 and 1.8, where
$u_{0}^{i}$ is a solution of (1.6) which takes $i$ (if $u^{i}(0)<1$) or $i+1$
(if $u^{i}(0)\geq 1$) times the value $1$. A weak limit $u_{0}^{i}$ that
satisfies $u_{0}^{i}(0)<1$ and that takes $i$ times the value $1$ is for
instance obtained by taking the intersection of the lower branch
$\mathcal{B}^{-}_{i+1}$ with the vertical line $p=2^{\ast}$. This always
occurs for the $i\geq 1$ satisfying $2+\lambda^{i}_{rad}<2^{\ast}$. Note that,
for fixed $i\geq 1$, $\lambda^{i}_{rad}$ goes to $0$ as $R\to+\infty$. Up to
assuming that $R$ is big enough we can thus generically find weak solutions
$u_{0}^{i}$ that oscillate around $1$ a fixed arbitrary number of times.

Thorem 1.1 and Theorem 1.2 also provide informations on the behaviour of the
critical branches $\mathcal{B}^{+}_{i}$. We now explain how this sheds new
light on the numerical simulations of [BGT16]. Consider Figure $1$ of this
paper, which covers dimension $N=4$. We consider the two branches
$\mathcal{B}_{3}^{+}$ and $\mathcal{B}_{4}^{+}$ (those starting from
$2+\lambda_{rad}^{3}$ and $2+\lambda_{rad}^{4}$). These branches seem to blow
up from the left of the vertical line $p=2^{*}$. _We assume that this is the
case._ If we moreover assume that they have bounded $H^{1}(B_{R})$ energy
contradicts Theorem 1.4: this Theorem indeed claims, since $N=4$, that
solutions along $\mathcal{B}_{3}^{+}$ and $\mathcal{B}_{4}^{+}$ blow up with a
single bubble at $0$. But then Proposition 2.1 below would show that these
solutions only take the value $1$ once, which is not the case. There are two
possible explanations for this phenomenon. The first is that the solutions on
the branches $\mathcal{B}_{3}^{+}$ and $\mathcal{B}_{4}^{+}$ in the figure all
have finite energy and actually cross the line $p=2^{\ast}$ for very large
values of $u(0)$ that are not in the figure; and they then blow up from the
right of the vertical line $p=2^{*}$ according to Theorem 1.8, or may even
blow up at a larger supercritical exponent $p_{*}>2^{*}$. The second is that
the solutions along $\mathcal{B}_{3}^{+}$ and $\mathcal{B}_{4}^{+}$ do blow up
from the left of $p=2^{*}$ but have infinite energy as $u(0)\to+\infty$ along
the branch. The latter situation could correspond, for instance, to an
infinite tower of bubbles centered at $0$ glued to a non-zero weak limit
$u_{0}$ as $p\to 2^{*}$, $p\leq 2^{*}$. Note however that infinite tower of
bubbles as $p\to 2^{*}$ are not known to exist in the context of (1.1) yet. In
the same vein, Theorem 1.2 implies that the finite-energy (in the sense of
(1.4)) branches that blow up along the vertical line $p=2^{*}$ can only
oscillate around it a finite number of times (dimension $3$ and $6$ are not
fully covered by Theorem 1.2). If Conjecture 1.3 holds, it completely rules
out a snaking behaviour of branches of finite-energy solutions around
$p=2^{*}$.

We also expect the supercritical branches to correspond to the solutions that
have been constructed in [CF20] when $p>2^{\ast}$. These are positive radial
solutions blowing up at the origin and intersecting an arbitrary number of
time the value $1$. These solutions were proved to have infinite Morse index
if their vertical asymptote $p$ satisfies $2^{\ast}-1<p<p_{JL}$, where
$p_{JL}$ is the Joseph and Lundgren exponent (see [JL73]) and have finite
Morse index if $p>p_{JL}$. This suggests that the branches with vertical
asymptote between $2^{\ast}-1$ and $p_{JL}$ oscillates infinitely many times
around the vertical line $p$, a phenomenon which is also observed numerically
see Figure $1$. We refer to [BCF20] for similar results related to the Keller-
Segel equation. In this case, the Joseph and Lundgren exponent is replaced by
a condition on the dimension of $B_{R}$.

The previous considerations describe the behaviour of the branches
$\mathcal{B}_{i}$ in the direction where $u(0)$ blows up. It is also natural
to investigate what happens on the other end of $\mathcal{B}_{i}$, i.e. when
$p\rightarrow\infty$. We believe that the solutions corresponding to the lower
branches $\mathcal{B}_{i}^{-}$ are the solutions that have been constructed in
[BGNT16]. There the authors prove, when $p\rightarrow\infty$, the existence of
radial positive solutions concentrating on an arbitrary number of spheres and
thus intersecting an arbitrary number of times the value $1$ and whose value
at the origin is strictly less than $1$ as $p\rightarrow\infty$. Let us
finally mention that analogous results have also been obtained for the Keller-
Segel equation in [BCN17b], where the polynomial nonlinearity is replaced by
$e^{\mu(u-1)}$ and the bifurcation analysis is done with respect to the
parameter $\mu>0$.

###  1.3. Organization of the paper

In Section 2 we prove a sharp pointwise asymptotic description of finite-
energy solutions of (1.3), namely Proposition 2.1. This is, to our knowledge,
the first sharp a priori asymptotic pointwise description of smooth solutions
of (1.3) when $p_{k}>2^{*}$, $p_{k}\to 2^{*}$. We apply it to obtain necessary
condition for the existence of blowing up solutions of (1.3) in Section 3 in a
general setting (Theorem 3.1). We prove Theorems 1.1 to 1.5 as a consequence.
We finally prove Theorems 1.6 to 1.8 in Section 4.

##  2\. A priori analysis of finite-energy solutions of (1.3)

Let $N\geq 3$ and let $(p_{k})_{k\geq 0}$ be a sequence of positive numbers,
$p_{k}\geq 2$, converging to $2^{*}=\frac{2N}{N-2}$ as $k\to+\infty$. For
$R>0$ we will denote by $B_{R}\subset\mathbb{R}^{N}$ the euclidean ball
centered at $0$ and of radius $R$. Throughout this section we let
$(u_{k})_{k\geq 0}$ be a sequence of possibly sign-changing radial solutions
of (1.3) in $C^{2}(\overline{B_{R}})$ satisfying (1.4). A consequence of (1.3)
and (1.4) is that $u_{k}$ weakly converges in $H^{1}(B_{R})$ and
$L^{2^{*}}(B_{R})$ towards some weak limit $u_{0}$, that solves (1.1) with
$p=2^{*}$ and is hence at least in $C^{2}(\overline{B_{R}})$.

We assume throughout this section that the sequence $(u_{k})_{k\geq 0}$ blows
up, that is

(2.1) $\lim_{k\to+\infty}\|u_{k}\|_{L^{\infty}(B_{R})}=+\infty$

up to a subsequence. For any $k\geq 0$, since $u_{k}$ is radial, it satisfies
the ODE

(2.2) $\left\\{\begin{aligned}
-u_{k}^{\prime\prime}(r)-\frac{N-1}{r}u_{k}^{\prime}(r)+u_{k}(r)&=|u_{k}(r)|^{p_{k}-2}u_{k}(r)&\textrm{
in }[0,R]\\\ u_{k}^{\prime}(0)=u_{k}^{\prime}(R)&=0.&\end{aligned}\right.$

Since the energy

(2.3) $r\mapsto
E_{k}(r):=\frac{|u_{k}^{\prime}(r)|^{2}}{2}+\frac{|u_{k}(r)|^{p_{k}}}{p_{k}}-\frac{|u_{k}(r)|^{2}}{2}$

is nonincreasing in $[0,R]$, $\max_{B_{R}}|u_{k}|=|u_{k}(0)|$, hence
$|u_{k}(0)|\to+\infty$ as $k\to+\infty$ by (2.1). Let $1\leq\ell\in\mathbb{N}$
and $(\mu_{1,k})_{k\geq 0}$, $\dots$, $(\mu_{\ell,k})_{k\geq 0}$ be $\ell$
sequences of positive numbers converging to $0$ as $k\to+\infty$. For $1\leq
i\leq\ell$, we adopt the notations

(2.4)
$B_{i,k}(r)=\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}\left(\frac{\mu_{i,k}}{\mu_{i,k}^{2}+r^{2}}\right)^{\frac{N-2}{2}}\quad\textrm{
and
}\quad\tilde{B}_{i,k}(r)=\mu_{i,k}^{\frac{N-2}{2}-\frac{2}{p_{k}-2}}B_{i,k}(r),$
(2.5)
$B_{0}(r)=\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}\Big{(}1+r^{2}\Big{)}^{1-\frac{N}{2}},$

where $r\geq 0$. It is well-known that

$\int_{\mathbb{R}^{N}}B_{0}^{2^{*}}dx=K_{N}^{-N},$

where $K_{N}$ is the sharp constant for the embedding of
$\dot{H}^{1}(\mathbb{R}^{N})$ (the homogeneous Sobolev space) into
$L^{2^{\ast}}(\mathbb{R}^{N})$ given by

(2.6) $K_{N}=\sqrt{\frac{4}{N(N-2)\omega_{N}^{\frac{2}{N}}}}.$

Since $p_{k}\to 2^{*}$, observe that $\frac{N-2}{2}-\frac{2}{p_{k}-2}\to 0$ as
$k\to+\infty$, and $\tilde{B}_{i,k}$ has to be understood as a perturbation of
$B_{i,k}$ that takes into account the asymptotic criticality of (2.2). The
main result of this section is the following, sharp pointwise asymptotic
description of $u_{k}$:

######  Proposition 2.1.

Assume that $(u_{k})_{k\geq 0}$ satisfies (2.1) (2.2) and (1.4) and let
$u_{0}$ be its weak limit in $H^{1}(B_{R})$. There exist a sequence
$(\varepsilon_{k})_{k\geq 0}$ converging to $0$, $C>0$, an integer $L\geq 1$,
$L$ sequences $(\mu_{1,k})_{k\geq 0},\dots,(\mu_{L,k})_{k\geq 0}$ of positive
numbers satisfying $\mu_{i,k}=o(\mu_{i+1,k})$ for any $1\leq i\leq L-1$ and a
family $(\kappa_{i})_{1\leq i\leq L}\in\\{\pm 1\\}^{L}$ such that

$\left|u_{k}(r)-u_{0}(r)-\sum_{i=1}^{L}\kappa_{i}\tilde{B}_{i,k}(r)\right|\leq\varepsilon_{k}\Big{(}\|u_{0}\|_{L^{\infty}(B_{R})}+\sum_{i=1}^{L}\tilde{B}_{i,k}(r)\Big{)}+C\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}}$

for any $k\geq 0$ and every $r\in[0,R]$.

Proposition 2.1 implies that for any sequence $(r_{k})_{k\geq 0}$ of points of
$[0,R]$, we have

$u_{k}(r_{k})=u_{0}(r_{k})+\sum_{i=1}^{L}\kappa_{i}\tilde{B}_{i,k}(r_{k})+o\Big{(}\|u_{0}\|_{L^{\infty}(B_{R})}+\sum_{i=1}^{L}\tilde{B}_{i,k}(r_{k})\Big{)}+O(\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}})$

as $k\to+\infty$. In other words, $u_{k}$ looks like, at a pointwise level and
in strong spaces, the sum of alternating positive bubbles centred at $0$ and
concentrating at different rates.

We prove Proposition 2.1 in the general context of possibly _sign-changing_
solutions, and not just positive ones. Pointwise descriptions of finite-energy
blowing up _positive_ sequences of solutions of equations like (1.3) have been
known to hold for a while. A general result is available in [DHR04], and other
possible references are [LZ99, Dru03, Heb14]. Specific reference for the Lin-
Ni-Takagi problem are [DRW12, Thi16]. Very recently, in [Pre21], this type of
results has been extended to the general setting of possibly _sign-changing_
solutions, and applications to the precompactness of sign-changing solutions
of Schrödinger-Yamabe type equation on manifolds have been given in [PV22b,
PV22a]. An analogous result for sign-changing solutions of Hardy-Schrödinger
type equations is in [GMR].

The proof of Proposition 2.1 goes through an iterative construction of
bubbling scales. The radiality of $u_{k}$ rules out non-trivial sign-changing
bubbles and forces $u_{k}$ to blow up, at the origin, as a sum of positive
bubbles with possibly alternating signs. This observation allows us to adapt
in the context of (1.3) the proof of [GMR], which is our main inspiration.
Note that our proof does not assume the existence of a Struwe-type
decomposition [Str84] for solutions of (1.3): for this reason it also works in
the asymptotically supercritical case $p_{k}\geq 2^{*}$ (when (1.4) is
satisfied).

The remaining part of the section is dedicated to the proof of Proposition
2.1. The proof goes through several steps. The arguments in what follows are
written using the notations from (1.3) since many of them would adapt to the
non-radial case. We however explicitly mention when radiality is crucially
used.

Step $1$: A global scale-invariant estimate. We claim that there exists $C>0$
such that, for all $k\geq 0$ and $r\in[0,R]$,

(2.7) $r^{\frac{2}{p_{k}-2}}|u_{k}(r)|\leq C.$

In the case where $p_{k}\leq 2^{*}$ for all $k\geq 0$, the estimate (2.7)
follows directly from Strauss’s lemma [Str77] in $H^{1}(B_{R})$, i.e.

$r^{\frac{2}{p_{k}-2}}|u_{k}(r)|\leq
R^{\frac{2p_{k}-N(p_{k}-2)}{2(p_{k}-2)}}r^{\frac{N-2}{2}}|u_{k}(r)|\leq C.$

When $p_{k}<2^{*}$, the power $\frac{2}{p_{k}-2}$ is clearly not sharp. In the
supercritical case $p_{k}>2^{*}$, we need a preliminary step. The assumption
(1.4) implies $(u_{k})_{k}$ is bounded in $W^{2,s}(B_{R})$ with
$s=\frac{N(p_{k}-2)}{2p_{k}-2}$ (observe this exponent is strictly bigger than
$1$). By Sobolev embeddings, $(u_{k})_{k}$ is therefore bounded in
$W^{1,t}(B_{R})$ with $t=\frac{Ns}{N-s}=\frac{N(p_{k}-2)}{p_{k}}$. Now,
Strauss’s radial lemma in $W^{1,t}(B_{R})$, see for instance [Lio82,
Corollaire II.1], implies

$r^{\frac{N-t}{t}}|u_{k}(r)|=r^{\frac{2}{p_{k}-2}}|u_{k}(r)|\leq C.$

In the case $p_{k}>2^{*}$, the power $\frac{2}{p_{k}-2}$ is probably sharp.

A first consequence of (1.3), (2.7) and standard elliptic theory is that

(2.8) $u_{k}\to u_{0}\quad\text{ in }C^{2}([\delta,R]),\text{ for any fixed
}\delta>0$

as $k\to+\infty$. Hence blow-up only occurs at $0$. We now iteratively
construct the blow-up scales of $(u_{k})_{k\geq 0}$. If $\ell\geq 1$ is an
integer we consider the following property that we denote by $(H_{\ell})$:

($H_{\ell}$) $\begin{array}[]{l}\textrm{there exist }\ell\textrm{ sequences
}(\mu_{1,k})_{k\geq 0},\dots,(\mu_{\ell,k})_{k\geq 0}\textrm{ of positive
numbers }\textrm{ satisfying: }\\\\[5.69054pt] \quad\bullet\
\mu_{i,k}=o(\mu_{i+1,k})\text{ as }k\to+\infty\text{ for any }1\leq
i\leq\ell-1\text{ and }\lim_{k\to+\infty}\mu_{\ell,k}=0\\\\[5.69054pt]
\quad\bullet\
\mu_{i,k}^{\frac{2}{p_{k}-2}}u_{k}\big{(}\mu_{i,k}\,\cdot)\to\kappa_{i}B_{0}\text{
in }C^{2}_{loc}(\mathbb{R}^{N}\backslash\\{0\\})\text{ for some
}\kappa_{i}\in\\{-1,1\\}\\\\[5.69054pt] \text{ as }k\to+\infty\text{ for any
}1\leq i\leq\ell.\\\ \end{array}$

Step $2$: $(H_{1})$ holds true. By (2.1) and (2.3),
$\max_{B_{R}}|u_{k}|=|u_{k}(0)|\to+\infty$ as $k\to+\infty$. We can thus let

(2.9)
$\mu_{1,k}=\big{(}N(N-2)\big{)}^{\frac{(N-2)(p_{k}-2)}{8}}|u_{k}(0)|^{-\frac{p_{k}-2}{2}},$

and $\mu_{1,k}\to 0$ as $k\to+\infty$. The sequence
$v_{1,k}(x):=\mu_{1,k}^{\frac{2}{p_{k}-2}}u_{k}\big{(}\mu_{1,k}x\big{)}$,
defined for $x\in B_{\frac{R}{\mu_{1,k}}}$ then satisfies
$|v_{1,k}(0)|=\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}$,
$\|v_{1,k}\|_{L^{\infty}(B_{\frac{R}{\mu_{1,k}}})}\leq\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}$
and, by (1.3),

$-\triangle v_{1,k}+\mu_{1,k}^{2}v_{1,k}=v_{1,k}^{p_{k}-1}\quad\text{ in
}B_{\frac{R}{\mu_{1,k}}}.$

By standard elliptic theory, and since $u_{k}$ is radial, $v_{1,k}$ converges
in $C^{2}_{loc}(\mathbb{R}^{N})$, up to a subsequence, towards a radial
solution $v_{1}$ of $-\triangle v_{1}=|v_{1}|^{2^{*}-2}v_{1}$ in
$\mathbb{R}^{N}$ with $|v_{1}(0)|=\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}$ and
$\|v_{1}\|_{{}^{\infty}(\mathbb{R}^{N})}\leq\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}$.
By the Pohozaev identity [Poh65] there do not exist radial sign-changing
solutions of this equation in $\mathbb{R}^{N}$. Hence $v_{1}$ is of constant
sign, and by the classification result of [CGS89] there exists
$\kappa_{1}\in\\{\pm 1\\}$ such that $v_{1}(r)=\kappa_{1}B_{0}(r)$, where
$B_{0}$ is as in (2.5). Since $v_{1,k}\to B_{0}$ in
$C^{2}_{loc}(\mathbb{R}^{N})$ this proves $(H_{1})$.

Step $3$: the induction property.

Step $3.1$. We claim that the following holds true: assume that $(H_{\ell})$
is true for some $\ell\geq 1$. Then either

(2.10)
$\lim_{M\to+\infty}\limsup_{k\to+\infty}\max_{[M\mu_{\ell,k},R]}r^{\frac{2}{p_{k}-2}}|u_{k}(r)-u_{0}(r)|=0$

or $(H_{\ell+1})$ is true.

###### Proof of (2.10).

We proceed by contradiction and, up to passing to a subsequence, assume that
there exists $s_{k}\in(0,R]$ with $s_{k}>>\mu_{\ell,k}$ and

(2.11) $s_{k}^{\frac{2}{p_{k}-2}}|u_{k}(s_{k})-u_{0}(s_{k})|\to a>0$

as $k\to+\infty$. Remark that the right-hand side of (2.11) is always finite
by (2.7). By (2.8) $s_{k}\to 0$, thus
$s_{k}^{\frac{2}{p_{k}-2}}|u_{k}(s_{k})|\to a$ and $|u_{k}(s_{k})|\to+\infty$.
For all $r\in[0,\frac{R}{s_{k}}]$, we let

$w_{k}(r)=s_{k}^{\frac{2}{p_{k}-2}}u_{k}\big{(}s_{k}r\big{)}.$

By (1.3) and (2.7) $w_{k}$ satisfies $-\triangle
w_{k}+s_{k}^{2}w_{k}=|w_{k}|^{p_{k}-2}w_{k}$ in $B_{\frac{R}{s_{k}}}$ and
$|w_{k}(r)|\leq Cr^{-\frac{2}{p_{k}-2}}$ for any $0<r\leq\frac{R}{s_{k}}$. By
standard elliptic theory $w_{k}$ thus converges, in
$C^{2}_{loc}(\mathbb{R}^{N}\backslash\\{0\\})$, towards a function $w_{0}$
satisfying $|w_{0}(1)|=a>0$ and

$-\triangle w_{0}=|w_{0}|^{2^{*}-2}w_{0}\quad\text{ in
}\mathbb{R}^{N}\backslash\\{0\\}.$

Using (1.4) we have, for $k\geq 0$ and for all $0<\delta<1$ fixed,

$\displaystyle\int_{B_{\frac{1}{\delta}}\backslash
B_{\delta}}|w_{0}|^{2^{*}}dx$
$\displaystyle=\int_{B_{\frac{1}{\delta}}\backslash
B_{\delta}}|w_{k}|^{\alpha_{k}}dx+o(1)$
$\displaystyle=s_{k}^{\frac{2\alpha_{k}}{p_{k}-2}-N}\int_{B_{\frac{\nu_{\ell+1,k}}{\delta}}\backslash
B_{\delta\nu_{\ell+1,k}}}|u_{k}|^{\alpha_{k}}dx+o(1)$ $\displaystyle\leq
Cs_{k}^{\frac{2\alpha_{k}}{p_{k}-2}-N}+o(1).$

As explained in the proof of Step $1$ we have
$s_{k}^{\frac{2\alpha_{k}}{p_{k}-2}-N}\leq 1$ so that, after passing to the
limit as $k\to+\infty$ and then as $\delta\to 0$, $w_{0}\in
L^{2^{*}}(\mathbb{R}^{N})$. Classical removable singularity results (see e.g.
Lemma $2.1$ in [CGS89]) show that $w_{0}$ satisfies $-\triangle
w_{0}=|w_{0}|^{2^{*}-2}w_{0}$ in $\mathbb{R}^{N}$ and is thus at least of
class $C^{2}$. As in the proof of Step $2$, since $u_{k}$ is radial $w_{0}$ is
also radial and in particular does not change sign. Since $|w_{0}(1)|=a>0$ the
classification result of [CGS89] again shows that

$w_{0}(r)=\kappa_{\ell+1}\frac{\lambda^{\frac{N-2}{2}}}{\big{(}\lambda^{2}+\frac{r^{2}}{N(N-2)}\big{)}^{\frac{N-2}{2}}}$

for some $\lambda>0$ depending on $a$ and some $\kappa_{\ell+1}\in\\{\pm
1\\}$. It remains to let

$\mu_{\ell+1,k}=\big{(}N(N-2)\big{)}^{\frac{1}{2}}\lambda s_{k}.$

Since $s_{k}>>\mu_{\ell,k}$ and $s_{k}=o(1)$ this proves that $(H_{\ell+1})$
is satisfied for the sequences $(\mu_{1,k})_{k\geq
0},\dots,(\mu_{\ell,k})_{k\geq 0}$ and $(s_{k})_{k\geq 0}$. ∎

Step $3.2$. We claim that the following holds true: assume that $(H_{\ell})$
is true for some $\ell\geq 1$ and let $\delta>0$. Then either

(2.12)
$\lim_{M\to+\infty}\limsup_{k\to+\infty}\max_{[M\mu_{i,k},\delta\mu_{i+1,k}]}r^{\frac{2}{p_{k}-2}}|u_{k}(r)-\tilde{B}_{i+1,k}(r)|=0$

for all $1\leq i\leq\ell-1$ or $(H_{\ell+1})$ is true.

###### Proof of (2.12)..

We proceed again by contradiction and assume that there exists
$i\in\\{1,\dots,\ell-1\\}$ and a sequence $(s_{k})_{k\geq 0}$ with
$\frac{s_{k}}{\mu_{i,k}}\to+\infty$ and $s_{k}\leq\delta\mu_{i+1,k}$ such
that, up to a subsequence,

(2.13)
$\lim_{k\to+\infty}s_{k}^{\frac{2}{p_{k}-2}}|u_{k}(s_{k})-\tilde{B}_{i+1,k}(s_{k})|=a>0.$

Note that the limit is finite by (2.4) and (2.7). By the second bullet in
$(H_{\ell})$ we also have $s_{k}=o(\mu_{i+1,k})$, and hence (2.13) becomes,
with (2.4),

$\lim_{k\to+\infty}s_{k}^{\frac{2}{p_{k}-2}}|u_{k}(s_{k})|=a>0.$

For all $r\in[0,\frac{R}{s_{k}}]$, we let

$w_{k}(r)=s_{k}^{\frac{2}{p_{k}-2}}u_{k}\big{(}s_{k}r\big{)}.$

Mimicking the proof of (2.10) shows that $(H_{\ell+1})$ is then satisfied for
the sequences $(\mu_{1,k})_{k\geq 0},\dots,(\mu_{i,k})_{k\geq 0}$,
$\big{(}\big{(}N(N-2)\big{)}^{\frac{1}{2}}\lambda s_{k}\big{)}_{k\geq 0}$,
$(\mu_{i+1,k})_{k\geq 0},\dots,(\mu_{\ell,k})_{k\geq 0}$, where $\lambda$ is
some positive number that depends on $a$. ∎

Step $4$: the number of concentration points is finite. We let

$L=\max\\{\ell\geq 1\text{ such that }(H_{\ell})\text{ holds true}\\}.$

We claim that $L<+\infty$ and that, for any fixed $\delta>0$,

(2.14)
$\displaystyle\lim_{M\to+\infty}\limsup_{k\to+\infty}\max_{[M\mu_{i,k},\delta\mu_{i+1,k}]}r^{\frac{2}{p_{k}-2}}|u_{k}(r)-\tilde{B}_{i+1,k}(r)|=0,\quad
1\leq i\leq L-1$
$\displaystyle\lim_{M\to+\infty}\limsup_{k\to+\infty}\max_{[M\mu_{L,k},R]}r^{\frac{2}{p_{k}-2}}|u_{k}(r)-u_{0}(r)|=0.$

###### Proof of the fact that $L<+\infty$ and of (2.14)..

Let $\ell\geq 1$ for which $(H_{\ell})$ is true and let $0<\delta<1$. Then by
$(H_{\ell})$ we can write that

$\displaystyle\int_{B_{R}}|u_{k}|^{\alpha_{k}}dx$
$\displaystyle\geq\sum_{i=1}^{\ell}\int_{B_{\frac{\mu_{i,k}}{\delta}}\backslash
B_{\delta\mu_{i,k}}}|u_{k}|^{\alpha_{k}}dx$
$\displaystyle=\sum_{i=1}^{\ell}\mu_{i,k}^{N-\frac{2\alpha_{k}}{p_{k}-2}}\Big{(}\int_{B_{\frac{1}{\delta}}\backslash
B_{\delta}}B_{0}^{2^{*}}dx+o(1)\Big{)}$
$\displaystyle=\sum_{i=1}^{\ell}\mu_{i,k}^{N-\frac{2\alpha_{k}}{p_{k}-2}}\Big{(}K_{N}^{-N}+\varepsilon_{\delta}+o(1)\Big{)},$

where $\varepsilon_{\delta}\to 0$ as $\delta\to 0$. As before,
$N-\frac{2\alpha_{k}}{p_{k}-2}\leq 0$, so that
$\mu_{i,k}^{N-\frac{2\alpha_{k}}{p_{k}-2}}\geq 1$ for all $1\leq i\leq\ell$.
Passing to the limit as $k\to+\infty$ and as $\delta\to 0$ we thus obtain

$\ell\leq CK_{N}^{N}$

where $C$ is the bound on the energy in (1.4). This shows that $L<\infty$.
Equation (2.14) then follows from the maximality of $L$, (2.10) and (2.12). ∎

In what follows, for any $0<\delta<R$ fixed, we let

$\eta_{k}(\delta)=\max_{r\in[\delta,R]}|u_{k}(r)|.$

Step $5$: Interpolation inequalities.

In this step we obtain a first set of pointwise estimates on $|u_{k}|$ that
interpolate between (2.10) and (2.12).

Step $5.1$: interpolation between $\tilde{B}_{L,k}$ and $u_{0}$. Let $L$ be as
in Step $4$. We claim that for any $0<\varepsilon<\frac{1}{2}$ fixed there
exists $\delta_{\varepsilon}$ >0 and $C_{\varepsilon}>0$ such that, for any
$r\in[\mu_{L,k},R]$,

(2.15) $|u_{k}(r)|\leq
C_{\varepsilon}\Big{(}\mu_{L,k}^{\frac{2}{p_{k}-2}(1-2\varepsilon)}r^{-\frac{4(1-\varepsilon)}{p_{k}-2}}+\eta_{k}(\delta_{\varepsilon})r^{-\frac{4\varepsilon}{p_{k}-2}}\Big{)}.$

###### Proof of (2.15).

Let $0<\varepsilon<\frac{1}{2}$ and $0<\delta<R$ be fixed. For
$r\in[\mu_{L,k},R]$ we let

$\Phi_{k}^{\varepsilon}(r)=\mu_{L,k}^{\frac{2}{p_{k}-2}(1-2\varepsilon)}G(r)^{\frac{4(1-\varepsilon)}{(N-2)(p_{k}-2)}}+\eta_{k}(\delta)G(r)^{\frac{4\varepsilon}{(N-2)(p_{k}-2)}},$

where, for $r=|x|$ we have let $G(r)=\frac{1}{(N-2)\omega_{N-1}}|x|^{2-N}$,
where $\omega_{N-1}$ is the area of $\mathbb{S}^{N-1}\subset\mathbb{R}^{N}$.
We let $s_{k}\in[\mu_{L,k},R]$ be such that

$\frac{u_{k}(s_{k})}{\Phi_{k}^{\varepsilon}(s_{k})}=\max_{s\in[\mu_{L,k},R]}\frac{u_{k}(s)}{\Phi_{k}^{\varepsilon}(s)}.$

We first assume that, up to a subsequence,

(2.16) $\mu_{L,k}=o(s_{k})\quad\text{ and }\quad s_{k}\leq\delta$

as $k\to+\infty$. Let $x_{k}\in B_{R}\backslash B_{\mu_{L,k}}$ be such that
$s_{k}=|x_{k}|$. By definition of $s_{k}$ we have

(2.17) $\frac{-\triangle
u_{k}(x_{k})}{u_{k}(x_{k})}\geq\frac{-\triangle\Phi_{k}^{\varepsilon}(x_{k})}{\Phi_{k}^{\varepsilon}(x_{k})}.$

On the one side, straightforward computations show that

$s_{k}^{2}\frac{-\triangle\Phi_{k}^{\varepsilon}(x_{k})}{\Phi_{k}^{\varepsilon}(x_{k})}\to(n-2)^{2}\varepsilon(1-\varepsilon)>0$

as $k\to+\infty$. On the other side, (1.3), (2.14) and (2.16) show, since
$s_{k}\to 0$, that

$s_{k}^{2}\frac{-\triangle
u_{k}(x_{k})}{u_{k}(x_{k})}\leq\delta^{2}\|u_{0}\|_{L^{\infty}(B_{R})}^{\frac{4}{N-2}}+o(1)$

as $k\to+\infty$. If $u_{0}\equiv 0$ we have a contradiction. If
$u_{0}\not\equiv 0$ we can now choose $\delta=\delta_{\varepsilon}$ so that
$\delta_{\varepsilon}^{2}\|u\|_{L^{\infty}(B_{R})}^{\frac{4}{N-2}}\leq\frac{1}{2}\varepsilon(1-\varepsilon)$.
We then get a contradiction with (2.17), which shows that (2.16) cannot
happen.

Thus, we either have $\frac{s_{k}}{\mu_{L,k}}\to C\geq 1$ or
$s_{k}\geq\delta_{\varepsilon}$, for some $\delta_{\varepsilon}>0$, up to a
subsequence. In the first case (2.15) follows from the second bullet in
$(H_{L})$, while in the second case (2.15) follows from the definition of
$\eta_{k}(\delta_{\varepsilon})$. This shows that

$u_{k}(r)\leq
C_{\varepsilon}\Big{(}\mu_{L,k}^{\frac{2}{p_{k}-2}(1-2\varepsilon)}r^{-\frac{4(1-\varepsilon)}{p_{k}-2}}+\eta_{k}(\delta_{\varepsilon})r^{-\frac{4\varepsilon}{p_{k}-2}}\Big{)}$

for any $r\in[\mu_{L,k},R]$. Since $-u_{k}$ also satisfies (1.4), (2.1) and
(2.2) the same inequality remains true also for $-u_{k}$. Hence (2.15) holds
true. ∎

Step $5.2$: interpolation between $\tilde{B}_{i,k}$ and $\tilde{B}_{i+1,k}$.
Let $L$ be as in Step $4$ and $i\in\\{1,\dots,L-1\\}$. We claim that for any
$0<\varepsilon<\frac{1}{2}$ there exists a constant $C_{\varepsilon}>0$ such
that, for any $r\in[\mu_{i,k},\mu_{i+1,k}]$,

(2.18) $u_{k}(r)\leq
C_{\varepsilon}\Big{(}\mu_{i,k}^{\frac{2}{p_{k}-2}(1-2\varepsilon)}r^{-(1-\varepsilon)\frac{4}{p_{k}-2}}+\mu_{i+1,k}^{-\frac{2}{p_{k}-2}}\left(\frac{\mu_{i+1,k}}{r}\right)^{\frac{4\varepsilon}{p_{k}-2}}\Big{)}.$

Remark that, when $r=\mu_{i+1,k}$, $u_{k}$ is of order
$\mu_{i+1,k}^{-\frac{2}{p_{k}-2}}$ by the second bullet in $(H_{L})$. Estimate
(2.18) is therefore the direct analogue of (2.15) where $\eta_{k}(\delta)$ has
been formally replaced by $\mu_{i+1,k}^{-\frac{2}{p_{k}-2}}$ (and
$r^{-\frac{4\varepsilon}{p_{k}-2}}$ has been replaced by
$\mu_{i+1,k}^{\frac{4\varepsilon}{p_{k}-2}}r^{-\frac{4\varepsilon}{p_{k}-2}}$
to attain the value $\mu_{i+1,k}^{-\frac{2}{p_{k}-2}}$ at the correct scale).

###### Proof of (2.18).

The proof is very similar to the proof of (2.15). Let $i\in\\{1,\dots,L-1\\}$
and $0<\varepsilon<\frac{1}{2}$ be fixed. For $r\in[\mu_{i,k},\mu_{i+1,k}]$ we
let

$\Phi_{i,k}^{\varepsilon}(r)=\mu_{i,k}^{\frac{2}{p_{k}-2}(1-2\varepsilon)}G(r)^{\frac{4(1-\varepsilon)}{(N-2)(p_{k}-2)}}+\mu_{i+1,k}^{-\frac{2(1-2\varepsilon)}{p_{k}-2}}G(r)^{\frac{4\varepsilon}{(N-2)(p_{k}-2)}},$

where $G(r)$ is as in the proof of (2.15). We let $s_{k}\in[\mu_{L,k},R]$ be
such that

$\frac{u_{k}(s_{k})}{\Phi_{i,k}^{\varepsilon}(s_{k})}=\max_{s\in[\mu_{L,k},R]}\frac{u_{k}(s)}{\Phi_{i,k}^{\varepsilon}(s)}.$

We first assume that, up to a subsequence, $\mu_{i,k}=o(s_{k})$ and
$s_{k}=o(\mu_{i+1,k})$ as $k\to+\infty$. Let $x_{k}$ be such that
$s_{k}=|x_{k}|$. By definition of $s_{k}$ we have

$\frac{-\triangle
u_{k}(x_{k})}{u_{k}(x_{k})}\geq\frac{-\triangle\Phi_{i,k}^{\varepsilon}(x_{k})}{\Phi_{i,k}^{\varepsilon}(x_{k})}.$

Straightforward computations show again that

$s_{k}^{2}\frac{-\triangle\Phi_{i,k}^{\varepsilon}(x_{k})}{\Phi_{i,k}^{\varepsilon}(x_{k})}\to(n-2)^{2}\varepsilon(1-\varepsilon)>0$

as $k\to+\infty$, while (1.3) and (2.14) again show that

$s_{k}^{2}\frac{-\triangle u_{k}(x_{k})}{u_{k}(x_{k})}=o(1)$

as $k\to+\infty$, a contradiction. Thus, we either have
$\frac{s_{k}}{\mu_{i,k}}\to C\geq 1$ or $\frac{s_{k}}{\mu_{i+1,k}}\to C\leq 1$
up to a subsequence. In both cases we obtain that

$\frac{u_{k}(s_{k})}{\Phi_{i,k}^{\varepsilon}(s_{k})}\leq C_{\varepsilon}$

for some positive constant $C_{\varepsilon}$. The inequality on $-u_{k}$ is
obtained as in the proof of (2.15) and (2.18) follows. ∎

Step $6$: improved pointwise estimates. Let $0<\delta<\frac{R}{2}$ be fixed.
We prove that there exists $C_{\delta}>0$ such that, for any $k\geq 0$ and
$r\in[0,R]$,

(2.19) $\big{|}u_{k}(r)\big{|}\leq
C_{\delta}\Big{(}\sum_{i=1}^{L}\tilde{B}_{i,k}(r)+\eta_{k}(\delta)\Big{)}$

holds, where $\tilde{B}_{i,k}$ is as in (2.4).

###### Proof of (2.19).

Let $G_{1}$ be the Green’s function of $-\triangle+1$ in $B_{R}$ with Neumann
boundary condition. Recall that there exists $C>0$ such that
$\big{|}G_{1}(x,y)\big{|}\leq C|x-y|^{2-n}$ for any $x,y\in B_{R},x\neq y$
(see the arguments in [Roba]). Let $(s_{k})_{k\geq 0},s_{k}=|x_{k}|$, be any
sequence of points in $[0,R]$. First, if $s_{k}\in[R/2,R]$, (2.19) follows
from the definition of $\eta_{k}(\delta)$. We can thus assume that
$s_{k}\in[0,R/2]$ for all $k\geq 0$. A representation formula for $u_{k}$ at
$s_{k}$ (or $x_{k}$) then gives, thanks to (1.3) and $(H_{L})$,

(2.20) $\displaystyle u_{k}(r_{k})$
$\displaystyle=\int_{B_{\mu_{1,k}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy$
$\displaystyle+\sum_{i=2}^{L}\int_{B_{\mu_{i,k}}\backslash
B_{\mu_{i-1,k}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy$
$\displaystyle+\int_{B_{R}\backslash
B_{\mu_{L,k}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy.$

Straightforward computations with the second bullet of $(H_{L})$ and (2.18)
show that

(2.21)
$\left|\int_{B_{\mu_{1,k}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(y)dy\right|\leq
C\tilde{B}_{1,k}(s_{k})$

(see e.g. [Heb14], Proposition $6.1$) and that, for $2\leq i\leq L-1$,

(2.22) $\left|\int_{B_{\mu_{i,k}}\backslash
B_{\mu_{i-1,k}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(y)dy\right|\leq
C\Big{(}\tilde{B}_{i-1,k}(s_{k})+\tilde{B}_{i,k}(s_{k})\Big{)}$

for some $C>0$ independent of $k$. Similarly (2.15) shows, with
straightforward computations, that

(2.23) $\left|\int_{B_{R}\backslash
B_{\mu_{L,k}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(y)dy\right|\leq
C\Big{(}\tilde{B}_{L,k}(s_{k})+\eta_{k}(\delta)^{p_{k}-1}\Big{)}$

for some $C>0$ independent of $k$. Since $\eta_{k}(\delta)^{p_{k}-1}\leq
C\eta_{k}(\delta)$ by (2.8), plugging (2.21), (2.22) and (2.23) in (2.20)
proves (2.19). ∎

We are now in position to conclude the proof of Proposition 2.1.

###### End of the proof of Proposition 2.1.

Let $0<\delta<\frac{R}{2}$ be fixed. We first prove that, if $u_{0}\equiv 0$,
we have

(2.24) $\eta_{k}(\delta)\leq
C_{\delta}^{\prime}\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}}$

for some $C_{\delta}^{\prime}>0$. We proceed by contradiction and assume that,
up to a subsequence,

$\eta_{k}(\delta)>>\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}}$

as $k\to+\infty$, and let $\tilde{u}_{k}=\frac{u_{k}}{\eta_{k}(\delta)}$. We
then have $\eta_{k}(\delta)>>\mu_{i,k}^{N-2-\frac{2}{p_{k}-2}}$ for any $1\leq
i\leq L$ and, by (2.19) and the expression of $\tilde{B}_{i,k}$ in (2.4), we
have

(2.25) $\|\tilde{u}_{k}\|_{L^{\infty}(B_{R}\backslash B_{\sigma})}\leq
C_{\delta}+o(1)$

for any fixed $0<\sigma<R$. Since $u_{0}\equiv 0$ we have, by (2.8),
$\eta_{k}(\delta)=o(1)$ as $k\to+\infty$. Therefore, by (1.3) and standard
elliptic theory, $\tilde{u}_{k}\to\tilde{u}_{0}$ in
$C^{2}_{loc}(B_{R}\backslash\\{0\\})$ where $\tilde{u}_{0}$ satisfies

$\left\\{\begin{aligned} -\triangle\tilde{u}_{0}+\tilde{u}_{0}&=0&\textrm{ in
}B_{R}\backslash\\{0\\}\\\ \partial_{\nu}\tilde{u}_{0}&=0&\textrm{ in
}\partial B_{R}\end{aligned}\right.$

Passing (2.25) to the limit shows that $|\tilde{u}_{0}|\leq C_{\delta}$ in
$B_{R}\backslash\\{0\\}$, so the singularity at $0$ is removable. Hence
$\tilde{u}_{0}$ is smooth in $B_{R}$ and satisfies

$\left\\{\begin{aligned} -\triangle\tilde{u}_{0}+\tilde{u}_{0}&=0&\textrm{ in
}B_{R}\\\ \partial_{\nu}\tilde{u}_{0}&=0&\textrm{ in }\partial
B_{R}\end{aligned}\right..$

Integrating by parts yields $\tilde{u}_{0}\equiv 0$, which is a contradiction
since, by definition, $|\tilde{u}_{0}(s_{0})|=1$ for some
$s_{0}\in[\delta,R]$. This proves (2.24)

If now $u_{0}\not\equiv 0$ in $B_{R}$, we have
$\eta_{k}(\delta)\leq\|u_{0}\|_{L^{\infty}(B_{R})}+o(1)$ by (2.8). Coming back
to (2.19) with (2.24) we have thus shown that, whatever $u_{0}$ is, there
exists $C>0$ such that, for any $k\geq 0$,

(2.26) $\big{|}u_{k}(r)\big{|}\leq
C\Big{(}\sum_{i=1}^{L}\tilde{B}_{i,k}(r)+\|u_{0}\|_{L^{\infty}(B_{R})}+\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}}\Big{)}$

holds for any $r\in[0,R]$. We now improve this estimate into an optimal
expansion of $u_{k}$. We let again $s_{k}=|x_{k}|$ be a sequence in $[0,R]$.
If $s_{k}\not\to 0$, then

(2.27)
$u_{k}(s_{k})=u_{0}(s_{k})+o\big{(}\|u_{0}\|_{L^{\infty}(B_{R})}\big{)}+O(\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}})$

by (2.8) and (2.26), which proves Proposition 2.1 when $s_{k}\not\to 0$. We
thus assume that $s_{k}=o(1)$ as $k\to+\infty$. Let $0<\sigma<1$. Note first
that

(2.28)
$\int_{B_{\frac{\mu_{L,k}}{\sigma}}}G_{1}(x_{k},y)|u_{0}|^{2^{*}-2}(y)u_{0}(|y|)dy=o\big{(}\|u_{0}\|_{L^{\infty}(B_{R})}\big{)}$

as $k\to+\infty$. This is obvious if $u_{0}\equiv 0$ and, if $u_{0}\not\equiv
0$, it follows since $\mu_{L,k}=o(1)$. Taking the difference of two
representation formulae, one for $u_{k}$ and one for $u_{0}$, we then obtain
with (2.28) that

(2.29) $\displaystyle u_{k}(s_{k})-u_{0}(s_{k})$
$\displaystyle=\int_{B_{\frac{\mu_{1,k}}{\sigma}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy+o\big{(}\|u_{0}\|_{L^{\infty}(B_{R})}\big{)}$
$\displaystyle+\sum_{i=2}^{L}\int_{B_{\frac{\mu_{i,k}}{\sigma}}\backslash
B_{\frac{\mu_{i-1,k}}{\sigma}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy$
$\displaystyle+\int_{B_{R}\backslash
B_{\frac{\mu_{L,k}}{\sigma}}}G_{1}(x_{k},y)\Big{(}|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)-|u_{0}(|y|)|^{2^{*}-2}u_{0}(|y|)\Big{)}dy.$

Using (2.26) and the dominated convergence theorem, straightforward
computations show that

(2.30)
$\displaystyle\int_{B_{\frac{\mu_{1,k}}{\sigma}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy$
$\displaystyle=\big{(}1+O(\varepsilon_{\sigma})\big{)}\tilde{B}_{1,k}(s_{k})$

and that, for any $1\leq i\leq L$,

(2.31) $\displaystyle\int_{B_{\frac{\mu_{i,k}}{\sigma}}\backslash
B_{\frac{\mu_{i-1,k}}{\sigma}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy$
$\displaystyle=\int_{B_{\sigma\mu_{i,k}}\backslash
B_{\frac{\mu_{i-1,k}}{\sigma}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy$
$\displaystyle+\int_{B_{\frac{\mu_{i,k}}{\sigma}}\backslash
B_{\sigma\mu_{i,k}}}G_{1}(x_{k},y)|u_{k}(|y|)|^{p_{k}-2}u_{k}(|y|)dy$
$\displaystyle=\big{(}1+O(\varepsilon_{\sigma})\big{)}\tilde{B}_{i,k}(s_{k})+O\big{(}\varepsilon_{\sigma}\tilde{B}_{i-1,k}(s_{k})\big{)}+O\big{(}\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}}\delta_{iL}\big{)}$

hold as $k\to+\infty$ (see e.g. Proposition $6.1$ and lemmatas $7.6$ and $7.7$
in [Heb14]). In (2.30) and (2.31) $\varepsilon_{\sigma}$ denotes a quantity
that satisfies $\lim_{\sigma\to 0}\varepsilon_{\sigma}=0$ and the notation
$\delta_{iL}$ in (2.31) means that the last term only appears if $i=L$. We now
estimate the last integral in (2.29). Assume first that $u_{0}\not\equiv 0$.
Then, using (2.8) and (2.26), we have

(2.32) $\displaystyle\int_{B_{R}\backslash
B_{\frac{\mu_{L,k}}{\sigma}}}G_{1}(x_{k},\cdot)\Big{(}|u_{k}|^{p_{k}-2}u_{k}-|u_{0}|^{2^{*}-2}u_{0}\Big{)}dy$
$\displaystyle=\int_{B_{\sigma}\backslash
B_{\frac{\mu_{L,k}}{\sigma}}}G_{1}(x_{k},\cdot)\Big{(}|u_{k}|^{p_{k}-2}u_{k}-|u_{0}|^{2^{*}-2}u_{0}\Big{)}dy+o(\|u_{0}\|_{L^{\infty}(B_{R})})$
$\displaystyle=O\big{(}\varepsilon_{\sigma}\tilde{B}_{L,k}(s_{k})\big{)}+o(\|u_{0}\|_{L^{\infty}(B_{R})})$

as $k\to+\infty$. Assume finally that $u_{0}\equiv 0$. Then, with (2.26),
straightforward computations show that

(2.33) $\int_{B_{R}\backslash
B_{\frac{\mu_{L,k}}{\sigma}}}G_{1}(x_{k},\cdot)\big{(}|u_{k}|^{p_{k}-2}u_{k}-|u_{0}|^{2^{*}-2}u_{0}\big{)}dy=O\big{(}\varepsilon_{\sigma}\tilde{B}_{L,k}(s_{k})\big{)}$

as $k\to+\infty$. Plugging (2.30), (2.31), (2.32) and (2.33) in (2.29) shows
that for any $0<\sigma<1$ fixed and any $k\geq 0$ large enough we have

$u_{k}(s_{k})=u_{0}(s_{k})+\big{(}1+O(\varepsilon_{\sigma})\big{)}\sum_{i=1}^{L}\tilde{B}_{i,k}(s_{k})+O\big{(}\varepsilon_{\sigma}\|u_{0}\|_{L^{\infty}(B_{R})}\big{)}+O(\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}}),$

where $\lim_{\sigma\to 0}\varepsilon_{\sigma}=0$. In other words, and up to
passing to a subsequence, there is a constant $C>0$ and a sequence
$(\varepsilon_{k})_{k\geq 0}$ of positive numbers converging to $0$ as
$k\to+\infty$ such that

$\Big{|}u_{k}(s_{k})-u_{0}(s_{k})-\sum_{i=1}^{L}\tilde{B}_{i,k}(s_{k})\Big{|}\leq\varepsilon_{k}\Big{(}\sum_{i=1}^{L}\tilde{B}_{i,k}(s_{k})+\|u_{0}\|_{L^{\infty}(B_{R})}\Big{)}+C\mu_{L,k}^{N-2-\frac{2}{p_{k}-2}}.$

That the sequence $(\varepsilon_{k})_{k\geq 0}$ and the constant $C$ can be
made independent of the choice of $(s_{k})_{k\geq 0}$, and thus only dependent
on the choice of the subsequence in $(u_{k})_{k\geq 0}$, follows from standard
arguments (see e.g. [Heb14], proof of Proposition $7.3$). Together with (2.27)
this concludes the proof of Proposition 2.1. ∎

Pointwise estimates on the derivatives of $u_{k}$ are also obtained thanks to
Proposition 2.1. Differentiating the representation formula (2.29) with
respect to $s$ and using Proposition 2.1 we obtain that there exists a
constant $C>0$ such that, up to passing to a subsequence for $(u_{k})_{k\geq
0}$,

(2.34) $|u_{k}^{\prime}(r)|\leq
C\Bigg{(}\sum_{i=1}^{L}\frac{\mu_{i,k}^{N-2-\frac{2}{p_{k}-2}}}{\big{(}\mu_{i,k}+r\big{)}^{N-1}}+\|u_{0}\|_{L^{\infty}(B_{R})}\Bigg{)}\quad\text{
holds for any }r\in[0,R].$

The result of Proposition 2.1 is robust and adapts to more general settings.
Let for instance $h_{0}$ be a smooth radial function in $B_{R}$ such that
$-\triangle+h_{0}$ has no kernel on $H^{1}$ functions with Neumann boundary
condition. By this we mean that there does not exists any non-zero $u\in
H^{1}(B_{R})$ satisfying

(2.35) $\left\\{\begin{aligned} -\triangle u+h_{0}u&=0&\textrm{ in }B_{R}\\\
\partial_{\nu}u&=0&\textrm{ in }\partial B_{R}\end{aligned}\right.$

in a weak sense. Such an operator $-\triangle+h_{0}$ then admits a uniquely
defined radial Green’s function $G_{h_{0}}$ in $B_{R}$ (see e.g. the arguments
in [Roba]). In particular, it comes with a representation formula as in
(2.29). A careful inspection of the proof of Proposition 2.1 then shows that
the exact statement of Proposition 2.1 remains true if $-\triangle+1$ in (1.3)
is replaced by any operator $-\triangle+h_{0}$ with no kernel, for a smooth
radial function $h_{0}$ in $[0,R]$.

If $-\triangle+h_{0}$ has a kernel in the sense of (2.35) and $(u_{k})_{k\geq
0}$ is a sequence of possibly sign-changing radial and
$C^{2}(\overline{B_{R}})$ solutions of

$\left\\{\begin{aligned} -\triangle
u_{k}+h_{0}u_{k}&=|u_{k}|^{p_{k}-2}u_{k}&\textrm{ in }B_{R}\\\
\partial_{\nu}u_{k}&=0&\textrm{ in }\partial B_{R}\end{aligned}\right.$

satisfying (1.4) and (2.1), the proof of Proposition 2.1 again adapts and
shows that $u_{k}$ satisfies

$\left|u_{k}(r)-u_{0}(r)-\sum_{i=1}^{L}\kappa_{i}\tilde{B}_{i,k}(r)\right|\leq\varepsilon_{k}\Big{(}1+\sum_{i=1}^{L}\tilde{B}_{i,k}(r)\Big{)}$

for some $C>0$, $L\geq 1$ and $L$ sequences $(\mu_{1,k})_{k\geq
0},\dots,(\mu_{L,k})_{k\geq 0}$ satisfying $\mu_{i,k}=o(\mu_{i+1,k})$ for
$1\leq i\leq L-1$ and a family $(\kappa_{i})_{1\leq i\leq L}\in\\{\pm
1\\}^{L}$. The kernel of $-\triangle+h_{0}$ prevents in general to improve the
term $\varepsilon_{k}$ in the right-hand side, even when $u_{0}\equiv 0$. We
refer to [DRW12] for a detailed analysis in this case and to [Pre21] for a
general proof of pointwise estimates even in the presence of a kernel.

##  3\. Necessary conditions for blow-up and proof of Thorem 1.1 and Theorem
1.2

In this section we apply Proposition 2.1 to precisely describe the blow-up
configurations that radial finite-energy solutions of (1.3) can develop. We
first provide a complete description of blow-up configurations for positive
solutions of (1.3), which allows us to prove Theorems 1.2 to 1.5. We then
apply the same techniques to prove Theorem 1.1. The main tool for this is the
classical Pohozaev identity that we now recall. Let $(u_{k})_{k\geq 0}$ be a
sequence of smooth possibly sign-changing solutions of (1.3) and let
$(\delta_{k})_{k\geq 0}$ be a sequence of real numbers with $0<\delta_{k}\leq
R$. Multiplying (2.2) by $ru_{k}^{\prime}(r)+\frac{N-2}{2}u_{k}(r)$ and
integrating over $B_{\delta_{k}}$ yields, after a few integration by parts
(see e.g. [Heb14], proposition $6.2$), the following identity:

(3.1) $\displaystyle\int_{B_{\delta_{k}}}u_{k}^{2}dx$
$\displaystyle+\frac{(N-2)^{2}}{4N}(p_{k}-2^{*})\int_{B_{\delta_{k}}}|u_{k}|^{p_{k}}dx$
$\displaystyle=\omega_{N-1}\delta_{k}^{N-1}\Bigg{(}-\frac{\delta_{k}}{2}(u_{k}^{\prime}(\delta_{k}))^{2}-\frac{N-2}{2}u_{k}(\delta_{k})u_{k}^{\prime}(\delta_{k})$
$\displaystyle+\frac{\delta_{k}}{2}(u_{k}(\delta_{k}))^{2}-\frac{\delta_{k}}{p_{k}}|u_{k}(\delta_{k})|^{p_{k}}\Bigg{)}.$

This identity will be used repeatedly in the following to obtain necessary
conditions on the blow-up behaviour of sequences of solutions of (1.3).

###  3.1. Classification of positive blow-up configurations

We first recall the definition of the _mass_ of a Green’s function. Let
$h_{0}$ be a smooth radial function in $B_{R}$ such that $-\triangle+h_{0}$
has no kernel (see (2.35)). Let $G_{h_{0}}$ be the Green’s function of
$-\triangle+h_{0}$ with Neumann boundary condition on $B_{R}$, which is then
uniquely defined. The iterative construction of $G_{h_{0}}$ (see [RW04] when
$N=3$ but also, in the Dirichlet case and when $N\geq 4$, [LZ99, Aub82, Robb,
Roba]) show that $x\mapsto G_{h_{0}}(0,x)$ is radial and expands as

(3.2)
$G_{h_{0}}(0,r)=\frac{1}{(N-2)\omega_{N-1}}r^{2-N}+\left\\{\begin{aligned}
&H_{h_{0}}+O(r^{1-\alpha})&\textrm{ if }N=3\\\
&\alpha_{4}\ln\frac{1}{r}+O(1)&\textrm{ if }N=4\\\
&\alpha_{N}r^{4-N}+O(r^{4+\alpha-N})&\textrm{ if }N\geq 5\\\
\end{aligned}\right.$

for some $0<\alpha<1$, where $H_{h_{0}}\in\mathbb{R}$ and $\alpha_{N}\not=0$
for $N\geq 4$. If $h_{0}$ vanishes at a certain order at $0$ these expansions
can be improved (see [RV]). Assume for instance that
$h_{0}(0)=h_{0}^{\prime}(0)=0$. Then

(3.3)
$G_{h_{0}}(0,r)=\frac{1}{(N-2)\omega_{N-1}}r^{2-N}+\left\\{\begin{aligned}
&H_{h_{0}}+O(r^{1-\alpha})&\textrm{ if }N=3,4,5\\\
&-c_{0}\ln\frac{1}{r}+O(1)&\textrm{ if }N=6\\\
&\alpha_{N}r^{6-N}+O(r^{6+\alpha-N})&\textrm{ if }N\geq 7,\\\
\end{aligned}\right.$

where $c_{0}=\frac{h_{0}^{\prime\prime}(0)}{32\omega_{5}}$ and for some
constant $\alpha_{N}\neq 0$. If we have in addition
$h_{0}^{\prime\prime}(0)=0$, the expansion in the six-dimensional case also
becomes

$G_{h_{0}}(0,r)=\frac{1}{4\omega_{5}}r^{-4}+H_{h_{0}}+O(r^{1-\alpha})\quad\textrm{
as }r\to 0.$

We say that $H_{h_{0}}$ in expansions (3.2) – (3.3) is the _mass_ of $G$ at
$0$. Remark that our sign convention is different than the one in [RW04]:
using the notations of [RW04], our mass $H_{1}$ of $G_{1}$ defined in (3.2)
reads as

(3.4) $H_{1}=-(1+H_{1}(0,0))$

where $H_{1}$ is defined in [RW04, Equation (1.2)].

The following Theorem completely describes the blow-up behaviour for positive
radial solutions of (1.3):

######  Theorem 3.1.

Let $(u_{k})_{k\geq 0}$ be a sequence of _positive radial_ solutions of (2.2)
such that

  * (i)

$\|u_{k}\|_{L^{\alpha_{k}}(B_{R})}\leq C$, for all $k\geq 0$, where
$\alpha_{k}=\max\big{(}2^{*},\frac{N}{2}p_{k}-N\big{)}$,

  * (ii)

$\lim_{k\to+\infty}\|u_{k}\|_{L^{\infty}(B_{R})}=+\infty.$

Assume moreover that $u_{k}\in C^{2}(\overline{B_{R}})$ for all $k\geq 0$ if
$p_{k}\geq 2^{*}$ and denote by $u_{0}$ its weak limit.

$1)$ Assume that $p_{k}\leq 2^{*}$ for all $k\geq 0$ and
$\lim_{k\to+\infty}p_{k}=2^{*}$. Then

  * •

$u_{0}\equiv 0$ for $3\leq N\leq 5$ and $u_{0}(0)\leq\frac{1}{2}$ when $N=6$.

  * •

If $N=3$ we have in addition $H_{1}\leq 0$ where $H_{1}$ is the mass of
$G_{1}$ at $0$ as in (3.2).

  * •

When $3\leq N\leq 6$ blow-up occurs with a single positive bubble at $0$.

$2)$ Assume that $p_{k}=2^{*}$ for all $k\geq 0$. Then

  * •

either $N=3$ and then $u_{0}\equiv 0$, $H_{1}=0$, where $H_{1}$ is the mass of
$G_{1}$ at $0$ as in (3.2), and the blow-up occurs with a single positive
bubble at $0$.

  * •

or $N=6$, $u_{0}(0)=\frac{1}{2}$ and the blow-up occurs with a single positive
bubble at $0$.

$3)$ Assume that $p_{k}\geq 2^{*}$ for all $k\geq 0$ and
$\lim_{k\to+\infty}p_{k}=2^{*}$. Then $3\leq N\leq 6$.

  * •

If $4\leq N\leq 6$ we have $u_{0}>0$ and $u_{0}(0)\geq\frac{1}{2}$ when $N=6$.

  * •

If $N=3$ we either have $u_{0}>0$ or $(u_{0}\equiv 0$ and $H_{1}\geq 0)$,
where $H_{1}$ is the mass of $G_{1}$ at $0$ as in (3.2).

The conditions in Theorem 3.1, in particular the mass conditions when $N=3$,
are well-known and have been highlighted for instance in [LZ99, Dru02, Dru03].
Simplicity of the blow-up (that is the nonexistence of towers of bubbles) is
here a consequence of the radiality of the solutions. As is easily seen,
Theorem 3.1 immediately implies Theorems 1.4 and 1.5. The consequence of the
sign of the mass on the admissible values of the radius $R$ in dimension $N=3$
is discussed in Remark 3.3.

Before proving Theorem 3.1, we show that Theorem 1.2 is a consequence of it :

###### Proof of Theorem 1.2 assuming Theorem 3.1..

We first assume that $N=6$. We let $(u_{k})_{k\geq 0}$ be a sequence of
positive radial solutions to (2.2) with $p_{k}=3$ satisfying the boundedness
assumption (i) and assume by contradiction that $(u_{k})_{k\geq 0}$ blows up.
We let $u_{0}$ be the weak limit of $(u_{k})_{k\geq 0}$. Theorem 3.1 shows
that $u_{k}$ blows-up with a single bubble at the origin and that
$u_{0}(0)=\frac{1}{2}$. Since $u_{0}$ is radial this completely determines it:
in particular by local uniqueness $u_{0}$ coincides with the restriction to
$[0,R]$ of the unique radial positive solution of

(3.5) $-u^{\prime\prime}-\frac{5}{r}u^{\prime}+u-u^{2}=0\text{ in
}[0,+\infty),\quad u^{\prime}(0)=0,u(0)=\frac{1}{2}$

constructed in Theorem $1.6$ of [Ni83]. It is proven in [Ni83] that this
solution tends to $1$ as $r\to+\infty$ and oscillates an infinite number of
times around the constant solution $1$. We denote by $(R_{\ell})_{\ell\geq 1}$
the sequence of the local extrema of $u_{0}$ in $(0,+\infty)$, which is a
discrete set by local uniqueness. Since $u_{0}$ solves (2.2) with $p_{k}=3$ we
necessarily have $R=R_{\ell}$ for some $k\geq 0$, otherwise the Neumann
boundary condition is not satisfied. That
$\lim_{\ell\to+\infty}R_{\ell}=+\infty$ and
$\liminf_{\ell\to+\infty}(R_{\ell}-R_{\ell-1})>0$ also follows from [Ni83].
This concludes the proof of Theorem 1.2 when $N=6$.

We now consider the case $N=3$. Let $(u_{k})_{k\geq 0}$ be a sequence of
positive radial solutions to (2.2) with $p_{k}=6$ satisfying the boundedness
assumption (i) and assume by contradiction that $(u_{k})_{k\geq 0}$ blows up.
Theorem 3.1 then shows that $u_{k}$ blows up with a zero weak limit and one
bubble at the origin and that the mass of $G$, the Green’s function of
$-\triangle+1$, defined as in (3.2), vanishes at $0$. For the ball $B(0,R)$
the mass of $G$ can be explicitly computed. A simple scaling argument similar
to that of Remark 4.1 shows that it vanishes if and only if the mass of the
Green’s function of $-\triangle+R^{2}$ in the ball
$B(0,1)\subset\mathbb{R}^{3}$ vanishes. The mass of $-\triangle+R^{2}$ in
$B(0,1)$ has been explicitly computed in [dPMRW19], Lemma $2.3$, where it was
shown that there exists a unique value of $R$, that we call $R^{*}$, for which
it vanishes. Thus $R=R^{*}$ which proves Theorem 1.2. ∎

######  Remark 3.2 (Critical case : $N=3$ and $R=R^{*}$).

In the exactly critical case $p_{k}=6$ for all $k\geq 0$ it seems very likely
to us that blow-up for finite-energy positive radial solutions of (2.2) will
not occur even when $R=R^{*}$. We believe a rigorous proof of this fact should
follow from an adaptation of the techniques developed in [KL22]. It would
consist to go to the next order in the analysis. Since this would be quite
lengthly, we leave that point open. Note also that our setting here is
different from the one in [dPMRW19] where perturbations of $R$ were allowed.

######  Remark 3.3 (Critical case : mass condition in dimension $N=3$).

In dimension $N=3$, as previously mentioned, the mass of $G_{1}$ at $0$
vanishes for one and only one radius $R^{*}$. In the assertion 1) of Theorem
3.1, the conclusion $H_{1}\leq 0$ implies $R\geq R^{*}$ whereas $H_{1}\geq 0$
implies $R\leq R^{*}$ in 3). This easily follows from (3.4) and the analysis
in [RW04] (see also [dPMRW19]).

###### Proof of Theorem 3.1.

Throughout this proof we let $(u_{k})_{k\geq 0}$ be a sequence of _positive_
functions satisfying (2.2), and the assumptions (i) and (ii). We let $L\geq 1$
be as in Proposition 2.1.

Claim 1:  if $p_{k}\leq 2^{*}$ (up to a subsequence), then $u_{0}\equiv 0$
when $3\leq N\leq 5$ and $u_{0}(0)\leq\frac{1}{2}$ when $N=6$ with equality if
$p_{k}=2^{*}$ for all $k\geq 0$.

We choose $\delta_{k}=\sqrt{\mu_{L,k}}$. Straightforward computations using
Proposition 2.1 and (2.34) show that

$\displaystyle\omega_{N-1}\delta_{k}^{N-1}\Bigg{(}-\frac{\delta_{k}}{2}(u_{k}^{\prime}(\delta_{k}))^{2}-\frac{N-2}{2}u_{k}(\delta_{k})u_{k}^{\prime}(\delta_{k})+\frac{\delta_{k}}{2}(u_{k}(\delta_{k}))^{2}-\frac{\delta_{k}}{p_{k}}|u_{k}(\delta_{k})|^{p_{k}}\Bigg{)}$
$\displaystyle=O\Big{(}\mu_{L,k}^{\frac{N-2}{2}+N-2-\frac{4}{p_{k}-2}}\Big{)},$

and that

$\displaystyle\int_{B_{\delta_{k}}}u_{k}^{2}dx=\left\\{\begin{aligned}
&O\big{(}\mu_{L,k}^{2-\frac{4}{p_{k}-2}}\big{)}&\textrm{ if }N=3\\\
&O\big{(}\mu_{L,k}^{\frac{7}{2}-\frac{4}{p_{k}-2}}\big{)}&\textrm{ if }N=4\\\
&\big{(}C_{1}(N)+o(1)\big{)}\mu_{L,k}^{N-\frac{4}{p_{k}-2}}&\textrm{ if }N\geq
5,\\\ \end{aligned}\right.$

where $C_{1}(N)=\int_{\mathbb{R}^{N}}B_{0}^{2}dx$ when $N\geq 5$.
Independently, Proposition 2.1 with Fatou’s lemma shows that

$\int_{B_{\delta_{k}}}u_{k}^{p_{k}}dx\geq\mu_{L,k}^{N-2-\frac{4}{p_{k}-2}}\big{(}K_{N}^{-N}+o(1)\big{)}$

as $k\to+\infty$. Since $p_{k}\leq 2^{*}$ for all $k\geq 0$ plugging the
latter computations in (3.1) yields

$2^{*}-p_{k}=O\big{(}\mu_{L,k}^{\frac{N-2}{2}}\big{)}+\left\\{\begin{aligned}
&O\big{(}\mu_{L,k}\big{)}&\textrm{ if }N=3\\\
&O\big{(}\mu_{L,k}^{\frac{3}{2}}\big{)}&\textrm{ if }N=4\\\
&O\big{(}\mu_{L,k}^{2}\big{)}&\textrm{ if }N\geq 5\\\ \end{aligned}\right..$

As a consequence, and since $N-2-\frac{4}{p_{k}-2}\to 0$ as $k\to+\infty$, we
have that

(3.6) $\mu_{L,k}^{N-2-\frac{4}{p_{k}-2}}=1+o(1)$

as $k\to+\infty$. Let, for $0\leq r\leq\frac{R}{\mu_{L,k}}$,
$\hat{u}_{k}(r)=\mu_{L,k}^{\frac{2}{p_{k}-2}-\frac{N-2}{2}}u_{k}(\sqrt{\mu_{L,k}}r)$.
With (1.3), Proposition 2.1, (3.6) and standard elliptic theory it is easily
seen that

$\hat{u}_{k}(r)\to\hat{u}_{\infty}(r)=\frac{(N(N-2))^{\frac{N-2}{4}}}{r^{N-2}}+u_{0}(0)$

in $C^{2}_{loc}(]0,1])$ as $k\to+\infty$. Using the latter convergence we can
compute again the boundary integral in (3.1). Scaling by a factor
$\sqrt{\mu_{L,k}}$ and using the exact expression of $\hat{u}_{\infty}$ and
(3.6) we have

$\displaystyle\omega_{N-1}\delta_{k}^{N-1}\Bigg{(}-\frac{\delta_{k}}{2}(u_{k}^{\prime}(\delta_{k}))^{2}-\frac{N-2}{2}u_{k}(\delta_{k})u_{k}^{\prime}(\delta_{k})+\frac{\delta_{k}}{2}(u_{k}(\delta_{k}))^{2}-\frac{\delta_{k}}{p_{k}}|u_{k}(\delta_{k})|^{p_{k}}\Bigg{)}$
$\displaystyle=\Big{(}\frac{1}{2}(N-2)^{2}(N(N-2))^{\frac{N-2}{4}}\omega_{N-1}u_{0}(0)+o(1)\Big{)}\mu_{L,k}^{\frac{N-2}{2}}.$

It remains to plug this new expression in (3.1). Using (3.6) and combining it
with the previous integral estimates, and since $p_{k}\leq 2^{*}$, we obtain
that

$u_{0}(0)+o(1)=O\big{(}\mu_{L,k}^{\frac{1}{2}}\big{)}\quad\text{ if }3\leq
N\leq 5.$

Hence $u_{0}(0)=0$ (and thus $u_{0}\equiv 0$) if $3\leq N\leq 5$. If $N=6$ it
is known that $\omega_{6}=\frac{16}{15}\pi^{3}$, so that straightforward
computations show that $C_{1}(6)=106\pi^{3}$. Since $\omega_{5}=\pi^{3}$, the
identity (3.1) now gives, with the two previous estimates and after passing to
the limit:

(3.7) $\displaystyle u_{0}(0)\leq\frac{1}{2}\quad$ $\displaystyle\textrm{ if
}p_{k}\leq 2^{*}\text{ for all }k\geq 0$ $\displaystyle
u_{0}(0)=\frac{1}{2}\quad$ $\displaystyle\textrm{ if }p_{k}=2^{*}\text{ for
all }k\geq 0.$

Claim 2: if $p_{k}\leq 2^{*}$ (up to a subsequence), then, in dimensions
$3\leq N\leq 6$, a blow-up can only occur with a single positive bubble at the
origin (regardless of the nature of $u_{0}$).

Assume by contradiction that $L\geq 2$ and let

(3.8) $\delta_{k}=\sqrt{\mu_{1,k}\mu_{2,k}}.$

Using Proposition 2.1 and (2.34) we have that

$\displaystyle\omega_{N-1}\delta_{k}^{N-1}\Bigg{(}-\frac{\delta_{k}}{2}(u_{k}^{\prime}(\delta_{k}))^{2}-\frac{N-2}{2}u_{k}(\delta_{k})u_{k}^{\prime}(\delta_{k})+\frac{\delta_{k}}{2}(u_{k}(\delta_{k}))^{2}-\frac{\delta_{k}}{p_{k}}|u_{k}(\delta_{k})|^{p_{k}}\Bigg{)}$
$\displaystyle=O\Big{(}\mu_{1,k}^{N-2-\frac{4}{p_{k}-2}}\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}}\Big{)},$

that

(3.9) $\displaystyle\int_{B_{\delta_{k}}}u_{k}^{2}dx=\left\\{\begin{aligned}
&O\big{(}\mu_{1,k}^{2-\frac{4}{p_{k}-2}}\big{)}&\textrm{ if }N=3\\\
&O\big{(}\mu_{1,k}^{4-\frac{4}{p_{k}-2}}\ln\frac{1}{\mu_{1,k}}\big{)}&\textrm{
if }N=4\\\
&\big{(}C_{1}(N)+o(1)\big{)}\mu_{1,k}^{N-\frac{4}{p_{k}-2}}&\textrm{ if }N\geq
5,\\\ \end{aligned}\right.$

and that

(3.10)
$\int_{B_{\delta_{k}}}u_{k}^{p_{k}}dx=\mu_{1,k}^{N-2-\frac{4}{p_{k}-2}}\big{(}K_{N}^{-N}+o(1)\big{)}$

as $k\to+\infty$. Estimate (3.9) implies in particular that

$\int_{B_{\delta_{k}}}u_{k}^{2}dx=O\Big{(}\mu_{1,k}^{N-2-\frac{4}{p_{k}-2}}\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}}\Big{)}$

for any $3\leq N\leq 6$. Plugging the latter estimates in (3.1) we thus obtain
that

(3.11)
$p_{k}-2^{*}=O\Bigg{(}\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}}\Bigg{)}$

(remark that (3.11) does not require the assumption $p_{k}\leq 2^{*}$). For
$0\leq r\leq\frac{R}{\delta_{k}}$ we now let

$\hat{u}_{k}(r)=\mu_{1,k}^{\frac{2}{p_{k}-2}-(N-2)}\delta_{k}^{N-2}u_{k}\big{(}\delta_{k}r\big{)}$

where $\delta_{k}$ is still given by (3.8). With (3.11) we now have

$\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}-\frac{2}{p_{k}-2}}=1+o(1)$

as $k\to+\infty$, so that by Proposition 2.1 we have

$\hat{u}_{k}(r)\to\frac{(N(N-2))^{\frac{N-2}{4}}}{r^{N-2}}+\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}\quad\text{
in }C^{2}_{loc}(]0,1])$

as $k\to+\infty$. The boundary term can thus be computed again as

(3.12)
$\displaystyle\omega_{N-1}\delta_{k}^{N-1}\Bigg{(}-\frac{\delta_{k}}{2}(u_{k}^{\prime}(\delta_{k}))^{2}-\frac{N-2}{2}u_{k}(\delta_{k})u_{k}^{\prime}(\delta_{k})+\frac{\delta_{k}}{2}(u_{k}(\delta_{k}))^{2}-\frac{\delta_{k}}{p_{k}}|u_{k}(\delta_{k})|^{p_{k}}\Bigg{)}$
$\displaystyle=\Big{(}\frac{1}{2}(N-2)^{2}\big{(}N(N-2)\big{)}^{\frac{N-2}{4}}\omega_{N-1}+o(1)\Big{)}\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}}\mu_{1,k}^{N-2-\frac{4}{p_{k}-2}}.$

The latter estimate in (3.1), together with (3.9), now shows that

$\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}}=\left\\{\begin{aligned}
&O\big{(}\mu_{1,k}\big{)}&\textrm{ if }N=3\\\
&O\big{(}\mu_{1,k}^{\frac{3}{2}}\big{)}&\textrm{ if }N=4\\\
&\big{(}C_{1}(N)+o(1)\big{)}\mu_{1,k}^{2}&\textrm{ if }N\geq 5.\\\
\end{aligned}\right.$

Since $\mu_{2,k}\to 0$ as $k\to+\infty$ this is a contradiction when $N\leq 6$
and the claim is proved.

Claim 3: if $p_{k}=2^{*}$ for all $k\geq 0$, then a blow-up is not possible
when $N\not\in\\{3,6\\}$.

_Assume first that $N=4,5$._ We have already proven in Claim 1 that
$u_{0}\equiv 0$ and in Claim 2 that $L=1$. We choose $\delta_{k}=\delta$ for
some $0<\delta<R$ fixed. For this choice of $\delta_{k}$ it is easily seen,
using again Proposition 2.1 and (2.34), that we have

$\displaystyle\omega_{N-1}\delta^{N-1}\Bigg{(}-\frac{\delta}{2}(u_{k}^{\prime}(\delta))^{2}-\frac{N-2}{2}u_{k}(\delta)u_{k}^{\prime}(\delta)+\frac{\delta}{2}(u_{k}(\delta))^{2}-\frac{\delta}{p_{k}}|u_{k}(\delta)|^{p_{k}}\Bigg{)}=O\big{(}\mu_{1,k}^{N-2}\big{)}$

and

$\displaystyle\int_{B_{\delta_{k}}}u_{k}^{2}dx=\left\\{\begin{aligned}
&64\omega_{3}\mu_{1,k}^{2}\ln\frac{1}{\mu_{1,k}}+o\big{(}\mu_{1,k}^{2}\ln\frac{1}{\mu_{1,k}}\big{)}&\textrm{
if }N=4\\\ &\big{(}C_{1}(5)+o(1)\big{)}\mu_{1,k}^{2}&\textrm{ if }N=5.\\\
\end{aligned}\right.$

Plugging the latter estimates in (3.1) shows that

$\displaystyle
64\omega_{3}\mu_{1,k}^{2}\ln\frac{1}{\mu_{1,k}}+o\big{(}\mu_{1,k}^{2}\ln\frac{1}{\mu_{1,k}}\big{)}=O\big{(}\mu_{1,k}^{2}\big{)}\quad\textrm{
if }N=4$
$\displaystyle\big{(}C_{1}(5)+o(1)\big{)}\mu_{1,k}^{2}=O(\mu_{1,k}^{3})\quad\textrm{
if }N=5,$

a contradiction in both cases.

_Consider now the dimensions $N\geq 7$._ In this setting, it is possible that
$u_{0}\not\equiv 0$ and $L>1$. We choose $\delta_{k}=\sqrt{\mu_{L,k}}$. With
Proposition 2.1 and (2.34) we now have

$\displaystyle\omega_{N-1}\delta_{k}^{N-1}\Bigg{(}-\frac{\delta_{k}}{2}(u_{k}^{\prime}(\delta_{k}))^{2}-\frac{N-2}{2}u_{k}(\delta_{k})u_{k}^{\prime}(\delta_{k})+\frac{\delta_{k}}{2}(u_{k}(\delta_{k}))^{2}-\frac{\delta_{k}}{p_{k}}|u_{k}(\delta_{k})|^{p_{k}}\Bigg{)}=O\big{(}\mu_{L,k}^{\frac{N-2}{2}}\big{)}.$

Straightforward computations using Proposition 2.1 again show that

$\int_{B_{\delta_{k}}}u_{k}^{2}dx=\big{(}C_{1}(N)+o(1)\big{)}\mu_{L,k}^{2}\textrm{
if }N\geq 7.$

Plugging the latter two identities in (3.1) then shows that

$\big{(}C_{1}(N)+o(1)\big{)}\mu_{L,k}^{2}=O(\mu_{L,k}^{\frac{N-2}{2}}),$

a contradiction since $N\geq 7$.

Claim 4: if $N=3$ and $u_{0}=0$, then $H_{1}=0$ if $p_{k}=2^{*}$ for all
$k\geq 0$ whereas $H_{1}\stackrel{{\scriptstyle\leq}}{{\geq}}0$ if
$p_{k}\stackrel{{\scriptstyle\leq}}{{\geq}}2^{*}$ for all $k\geq 0$ .

If $p_{k}\leq 2^{*}$ we have in fact proven that $u_{0}\equiv 0$ and $L=1$. If
$p_{k}\geq 2^{*}$ for all $k\geq 0$, one can have $u_{0}>0$ and it is an
assumption here that $u_{0}\equiv 0$. Since $u_{0}\equiv 0$, by (1.3) and
Proposition 2.1 we have

$\mu_{L,k}^{\frac{2}{p_{k}-2}-1}u_{k}(r)\to G_{1}(r)\quad\text{ in
}C^{1}_{loc}(]0,R]),$

where $G_{1}$ is as in (3.2) and where we already know that $L=1$ if
$p_{k}\leq 2^{*}$ for all $k\geq 0$. Choose again $\delta_{k}=\delta$ for some
$0<\delta<R$ fixed. Straightforward computations with (3.2) show that

$\displaystyle\omega_{N-1}\delta^{N-1}\Bigg{(}-\frac{\delta}{2}(u_{k}^{\prime}(\delta))^{2}-\frac{N-2}{2}u_{k}(\delta)u_{k}^{\prime}(\delta)+\frac{\delta}{2}(u_{k}(\delta))^{2}-\frac{\delta}{p_{k}}|u_{k}(\delta)|^{p_{k}}\Bigg{)}$
$\displaystyle=\Big{(}\frac{1}{2}3^{\frac{1}{4}}\omega_{2}H_{1}+O(\delta)+o(1)\Big{)}\mu_{L,k}^{1-\frac{4}{p_{k}-2}},$

where $H_{1}$ is as in (3.2). Independently, we have by Proposition 2.1 that

$\int_{B_{\delta}}u_{k}^{2}dx=O\big{(}\delta\mu_{L,k}^{2-\frac{4}{p_{k}-2}}\big{)}+o\big{(}\mu_{L,k}^{2-\frac{4}{p_{k}-2}}\big{)}$

and that, up to passing to a subsequence, that

(3.13)
$\int_{B_{\delta}}u_{k}^{p_{k}}dx=\big{(}C_{2}+o(1)\big{)}\mu_{L,k}^{1-\frac{4}{p_{k}-2}}$

for some $C_{2}>0$. The latter equality is straightforward when $p_{k}\leq
2^{*}$ since $L=1$. If $p_{k}>2^{*}$ it is obtained, up to passing to a
subsequence, as a double inequality. On the one side a bound from below
follows from Fatou’s theorem, and on the other side a bound from above is
obtained from Proposition 2.1 and the estimate

$\left(\frac{\mu_{i,k}}{\mu_{L,k}}\right)^{N-\frac{2p_{k}}{p_{k}-2}}=O(1)$

for all $1\leq i\leq L-1$, that follows since $N-\frac{2p_{k}}{p_{k}-2}\geq 0$
when $p_{k}\geq 2^{*}$. Note that estimate (3.13) is different than (3.10): in
(3.13) we are integrating in $B_{\delta}$ – and not on
$B_{\sqrt{\mu_{1,k}\mu_{2,k}}}$ as before – hence possible contributions from
all the bubbles have to be taken into account. Plugging the latter estimates
in (3.1) finally gives

$\big{(}H_{1}+O(\delta)+o(1)\big{)}\mu_{L,k}=\big{(}C+o(1)\big{)}(p_{k}-2^{*})$

for some positive constant $C$. If $p_{k}=2^{*}$ for all $k$ this shows, after
passing to the limit $\delta\to 0$, that $H_{1}=0$. If $p_{k}\leq 2^{*}$ for
all $k\geq 0$ we similarly get that $H_{1}\leq 0$ and, if $p_{k}\geq 2^{*}$
for all $k\geq 0$, we obtain that $H_{1}\geq 0$.

With Claim 1 to Claim 4, we are done with the assertions concerning $p_{k}\leq
2^{*}$ and $p_{k}=2^{*}$ for all $k\geq 0$. We have also completed the proof
of the case $p_{k}\geq 2^{*}$ when $N=3$.

Claim 5: if $p_{k}\geq 2^{*}$ for all $k\geq 0$ and $4\leq N\leq 6$, then
$u_{0}>0$ and $u_{0}(0)\geq\frac{1}{2}$ when $N=6$.

We proceed by contradiction and assume that $u_{0}\equiv 0$. Choose again
$\delta_{k}=\delta$ for some $0<\delta<R$. Proposition 2.1 and (2.34) show
that

$\displaystyle\omega_{N-1}\delta^{N-1}\Bigg{(}-\frac{\delta}{2}(u_{k}^{\prime}(\delta))^{2}-\frac{N-2}{2}u_{k}(\delta)u_{k}^{\prime}(\delta)+\frac{\delta}{2}(u_{k}(\delta))^{2}-\frac{\delta}{p_{k}}|u_{k}(\delta)|^{p_{k}}\Bigg{)}=O\big{(}\mu_{L,k}^{2(N-2)-\frac{4}{p_{k}-2}}\big{)}$

and that

$\displaystyle\int_{B_{\delta_{k}}}u_{k}^{2}dx\geq\left\\{\begin{aligned}
&64\omega_{3}\mu_{L,k}^{4-\frac{4}{p_{k}-2}}\ln\frac{1}{\mu_{L,k}}+o\big{(}\mu_{L,k}^{4-\frac{4}{p_{k}-2}}\ln\frac{1}{\mu_{L,k}}\big{)}&\textrm{
if }N=4\\\
&\big{(}C_{1}(N)+o(1)\big{)}\mu_{L,k}^{N-\frac{4}{p_{k}-2}}&\textrm{ if
}N=5,6.\\\ \end{aligned}\right.$

Since $p_{k}\geq 2^{*}$ for all $k\geq 0$, we infer from the Pohozaev identity
(3.1) that

$\displaystyle\ln\frac{1}{\mu_{1,k}}+o\big{(}\ln\frac{1}{\mu_{1,k}}\big{)}=O\big{(}1\big{)}\quad\textrm{
if }N=4$
$\displaystyle\big{(}C_{1}(N)+o(1)\big{)}\mu_{L,k}^{2}=O(\mu_{1,k}^{N-2})\quad\textrm{
if }N=5,6,$

and this yields a contradiction. Whence $u_{0}>0$. Finally, the same arguments
that led to (3.7) similarly show that $u_{0}(0)\geq\frac{1}{2}$ when
$p_{k}\geq 2^{*}$.

This concludes the proof of Theorem 3.1. ∎

In the next section, we construct examples of blowing up sequences of positive
solutions in all the cases mentioned in Theorem 3.1 when $N\geq 4$. Unless
specified otherwise, towers of positive bubbles at the origin (corresponding
to $L\geq 2$ in Proposition 2.1) are likely to occur. In this case the
identity (3.1) also allows us to obtain necessary conditions on the blow-up.
Assume that $(u_{k})_{k\geq 0}$ is a sequence of radial positive solutions of
(1.3) satisfying (1.4) and (2.1). We claim for instance that the following
holds:

######  Lemma 3.4.

Assume that $p_{k}\geq 2^{*}$ for all $k\geq 0$, $3\leq N\leq 6$ and that
$(u_{k})_{k\geq 0}$ blows up with at least two bubbles, that is $L\geq 2$.
Then, for any $1\leq i\leq L-1$, there exists a constant $C_{i}>0$ such that

$\left(\frac{\mu_{i,k}}{\mu_{i+1,k}}\right)^{\frac{N-2}{2}}=\big{(}C_{i}+o(1)\big{)}(p_{k}-2^{*}).$

###### Proof.

We first prove that there exists a constant $C_{1}>0$ such that

(3.14)
$p_{k}-2^{*}=\big{(}C_{1}+o(1)\big{)}\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}}.$

Assume by contradiction that
$p_{k}-2^{*}=o\big{(}(\mu_{1,k}\mu_{2,k}^{-1})^{\frac{N-2}{2}}\big{)}$. This
implies in particular that

$\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}-\frac{2}{p_{k}-2}}=1+o(1)$

as $k\to+\infty$. With the latter estimates (3.9), (3.11) and (3.12) obtained
in the proof of Theorem 3.1 are still true (since they did not use the
assumption $p_{k}\leq 2^{*}$). Then plugging (3.12) and (3.9) in (3.1) gives

$\big{(}1+o(1)\big{)}\left(\frac{\mu_{1,k}}{\mu_{2,k}}\right)^{\frac{N-2}{2}}=\left\\{\begin{aligned}
&O\big{(}\mu_{1,k}\big{)}&\textrm{ if }N=3\\\
&O\big{(}\mu_{1,k}^{2}\ln\frac{1}{\mu_{1,k}}\big{)}&\textrm{ if }N=4\\\
&\big{(}C_{1}(N)+o(1)\big{)}\mu_{1,k}^{2}&\textrm{ if }N=5,6.\\\
\end{aligned}\right.$

Since $\mu_{2,k}\to 0$ as $k\to+\infty$ this is a contradiction. This proves
that
$p_{k}-2^{*}\gtrsim\big{(}(\mu_{1,k}\mu_{2,k}^{-1})^{\frac{N-2}{2}}\big{)}$.
The other inequality is given by (3.11), and this proves (3.14) up to passing
to a subsequence. If now $i\in\\{1,\dots,L-1\\}$, we apply a Pohozaev identity
on $B_{r_{i,k}}$, where $r_{i,k}=\sqrt{\mu_{i,k}\mu_{i+1,k}}$. Estimates
(3.9), (3.11) and (3.12) remain true with $\mu_{1,k}$ and $\mu_{2,k}$ replaced
by $\mu_{i,k}$ and $\mu_{i+1,k}$ respectively. The proof then follows the same
lines as the proof of (3.14). ∎

###  3.2. Proof of Theorem 1.1

###### Proof of Theorem 1.1.

We assume that $N\geq 7$, $p_{k}\geq 2^{*}$ for all $k\geq 0$ and
$\lim_{k\to+\infty}p_{k}=2^{*}$, and we let $(u_{k})_{k\geq 0}$ be a sequence
of solutions of (2.2) satisfying (1.4). As mentioned before, by the result of
[Ni83], radial solutions of (2.2) when $p_{k}\geq 2^{*}$ have fixed sign. We
will however not use this observation in this proof but instead rely on
Proposition 2.1 and (3.1), that are independent of the sign of the solutions.
We proceed again by contradiction and assume that (2.1) holds. We will again
conclude by using the Pohozaev identity (3.1). Let $L\geq 1$ be the integer
given by Proposition 2.1 and let $\delta_{k}=\sqrt{\mu_{L,k}}$.
Straightforward computations using Proposition 2.1 and (2.34) show that

$\displaystyle\int_{B_{\delta_{k}}}u_{k}^{2}dx\geq\big{(}C_{1}(N)+o(1)\big{)}\mu_{L,k}^{N-\frac{4}{p_{k}-2}}$

for any $N\geq 7$, for some positive constant $C_{1}(N)$ only depending on $N$
and

$\displaystyle\omega_{N-1}\delta_{k}^{N-1}\Bigg{(}-\frac{\delta_{k}}{2}(u_{k}^{\prime}(\delta_{k}))^{2}-\frac{N-2}{2}u_{k}(\delta_{k})u_{k}^{\prime}(\delta_{k})+\frac{\delta_{k}}{2}(u_{k}(\delta_{k}))^{2}-\frac{\delta_{k}}{p_{k}}|u_{k}(\delta_{k})|^{p_{k}}\Bigg{)}$
$\displaystyle=O\big{(}\mu_{L,k}^{\frac{N-2}{2}+N-2-\frac{4}{p_{k}-2}}\big{)}.$

Since $p_{k}\geq 2^{*}$, plugging these estimates in (3.1) gives,

$C_{1}(N)\mu_{L,k}^{2}\leq O\big{(}\mu_{L,k}^{\frac{N-2}{2}}\big{)}.$

Since $N\geq 7$, this is a contradiction so that (2.1) cannot occur. The
sequence $(u_{k})_{k\geq 0}$ is thus uniformly bounded in $L^{\infty}(B_{R})$
and converges, up to a subsequence, in $C^{2}(\overline{B_{R}})$ by standard
elliptic theory. This proves Theorem 1.1. ∎

We conclude this subsection by showing that if an additional assumption is
added we can improve the conclusion of Theorem 1.2 when $N=6$.

######  Lemma 3.5.

Assume that $N=6$ and $p_{k}=3$ for all $k\geq 0$. Let $(u_{k})_{k\geq 0}$ be
a sequence of radial solutions of (2.2) that satisfies (1.4) and let $u_{0}$
be its weak limit in $H^{1}(B_{R})$. If $u_{0}$ is non-degenerate in the sense
of (1.7) then $(u_{k})_{k\geq 0}$ is bounded in $L^{\infty}(B_{R})$ (and hence
precompact in $C^{2}(B_{R})$).

###### Proof.

By Theorem 3.1 we know that $u_{0}$ is positive, satisfies
$u_{0}(0)=\frac{1}{2}$ and the blow-up arises with a single bubble at the
origin. For any $k\geq 0$, we let $v_{k}=u_{k}-u_{0}$. By (1.3) $v_{k}$
satisfies

(3.15) $-\triangle v_{k}+h_{0}v_{k}=v_{k}^{2}\quad\text{ in }B_{R},$

where we have let $h_{0}=1-2u_{0}$. As a consequence of the non-degeneracy of
$u_{0}$ (see (1.7)) it is easily seen that $-\triangle+h_{0}$ has no kernel in
the sense of (2.35). Hence $G_{h_{0}}$, the Green’s function of
$-\triangle+h_{0}$ with Neumann boundary condition, is uniquely defined. Since
$h_{0}(0)=0$ and $h_{0}^{\prime}(0)=0$, $G_{h_{0}}$ satisfies (3.3).

We will need a refinement of the Pohozaev identity for (3.15). Let
$0<\delta<R$ be fixed. Integrating (3.15) against $rv_{k}^{\prime}(r)$ gives,
after a few integration by parts (see again [Heb14], Proposition $6.2$), that
for any $k\geq 0$

(3.16)
$\displaystyle\int_{B_{\delta}}\big{(}h_{0}+\frac{1}{2}rh_{0}^{\prime}(r)\big{)}v_{k}^{2}dx$
$\displaystyle=\omega_{5}\delta^{5}\Bigg{(}-\frac{\delta}{2}(v_{k}^{\prime}(\delta))^{2}-2v_{k}(\delta)v_{k}^{\prime}(\delta)+\delta\frac{h_{0}(\delta)}{2}(v_{k}(\delta))^{2}-\frac{\delta}{3}v_{k}(\delta)^{3}\Bigg{)}$

holds. By Proposition 2.1 applied to $u_{k}$ and by definition of $v_{k}$ we
have

(3.17)
$v_{k}(r)=\big{(}1+o(1)\big{)}B_{1,k}(r)+O(\mu_{1,k}^{2})+o(\|u_{0}\|_{L^{\infty}(B_{R})})$

as $k\to+\infty$, for any $r\in[0,R]$, where $B_{1,k}$ is as in (2.4). We
claim that we actually have

(3.18) $v_{k}(r)=\big{(}1+o(1)\big{)}B_{1,k}(r)+O(\mu_{1,k}^{2})$

for any $r\in[0,R]$. This follows from an adaptation of the arguments in the
proof of Proposition 2.1. More precisely, it is easily seen with (3.17) that
(2.15) still holds for $v_{k}$ with $L=1$ and $\eta_{k}(\delta_{\varepsilon})$
now defined as
$\eta_{k}(\delta_{\varepsilon})=\max_{[\delta_{\varepsilon},R]}|v_{k}|$. As a
consequence, (2.19) remains true and is proven in the same way. Since
$-\triangle+h_{0}$ has no kernel the same arguments as in the proof of (2.24)
then show that $\eta_{k}(\delta_{\varepsilon})=O(\mu_{1,k}^{2})$, from which
(3.18) follows.

With (3.15) and (3.18) we then get that $\mu_{1,k}^{-2}v_{k}(r)\to
G_{h_{0}}(r)$ in $C^{2}_{loc}(]0,\delta])$ as $k\to+\infty$. Using (3.3) we
then have

$\displaystyle\mu_{1,k}^{-4}\omega_{5}\delta^{5}\Bigg{(}-\frac{\delta}{2}(v_{k}^{\prime}(\delta))^{2}-2v_{k}(\delta)v_{k}^{\prime}(\delta)+\delta\frac{h_{0}}{2}(v_{k}(\delta))^{2}-\frac{\delta}{3}v_{k}(\delta)^{3}\Bigg{)}\to
2C_{0}\ln\delta+O(1)\quad\text{ as }k\to+\infty,$

where $O(1)$ denotes a quantity that remains bounded as $\delta\to 0$. (Note
however that $\delta$ is fixed in this proof). Independently, and by
Proposition 2.1, we have

$\int_{B_{\delta}}\big{(}h_{0}+\frac{1}{2}rh_{0}^{\prime}(r)\big{)}v_{k}^{2}dx=(24)^{5}\omega_{5}h_{0}^{\prime\prime}(0)\mu_{1,k}^{4}\ln\frac{1}{\mu_{1,k}}+o\big{(}\mu_{1,k}^{4}\ln\frac{1}{\mu_{1,k}}\big{)}$

as $k\to+\infty$. Plugging the latter two estimates in (3.16) shows that

$h_{0}^{\prime\prime}(0)=O\Big{(}\big{(}\ln\frac{1}{\mu_{1,k}}\big{)}^{-1}\Big{)}$

as $k\to+\infty$, hence $h_{0}^{\prime\prime}(0)=0$. Since $h_{0}=1-2u_{0}$ we
thus have $u_{0}^{\prime}(0)=u_{0}^{\prime\prime}(0)=0$. However, applying
(2.2) at $r=0$ with $u_{0}(0)=\frac{1}{2}$ and $u_{0}^{\prime}(0)=0$ yields
$u_{0}^{\prime\prime}(0)=\frac{1}{24}$, a contradiction. ∎

Lemma 3.5 provides an additional precompactness results in case the non-
degeneracy assumption is satisfied. The peculiarity of the six-dimensional
case for critical equations like (1.1) has been known since [Dru03] (see also
[Heb14]). Theorem 1.2 and Lemma 3.5 are among the few precompactness results
available in dimension $6$ for equations like (1.3), and they entirely rely on
the assumption that the solutions considered are radial. Note that the non-
degeneracy of $u_{0}$ is a generic assumption in the choice of $R$, see
Proposition 4.2 below. This genericity is however not enough to close Theorem
1.2.

######  Remark 3.6 (Critical case, $N=6$).

Theorem 1.2 states that precompactness for finite-energy solutions of (1.3)
when $N=6$ and $p_{k}=3$ for all $k\geq 0$ holds at least if
$R\not\in\\{R_{\ell}\\}_{\ell\geq 1}$ for a given discrete sequence
$(R_{\ell})_{\ell\geq 1}$. If $R=R_{\ell}$ for some $k\geq 0$, as shown in the
proof of Theorem 1.2, every finite-energy sequence $(u_{k})_{k\geq 0}$ of
(1.3) has the same weak limit which is given by (3.5). Lemma 3.5 then allows
to completely prove Conjecture 1.3 in dimension $N=6$ if one can show $u_{0}$
is non-degenerate in the sense of (1.7) when $R=R_{\ell}$. The solution
$u_{0}$ is non-degenerate (for $R=R_{\ell}$) if the unique solution $v_{0}$ of

(3.19) $-v^{\prime\prime}-\frac{5}{r}v^{\prime}+v-2u_{0}v=0\text{ in
}[0,+\infty),\quad v^{\prime}(0)=0,v(0)=1$

is such that $v^{\prime}(R_{\ell})\neq 0$. Indeed this linear equation has a
two-dimensional basis of solutions and $v_{0}$ is the only one which can span
an element of the kernel of the linearization at $u_{0}$. Numerics provide an
evidence that $u_{0}$ is non-degenerate, see Figure 3. Proving this fact
theoretically seems quite tricky and very sensitive on the parameters
involved. For instance, degenerate radial solutions have been highlighted in
[BGT16] for other values of the parameters (dimension and power).

Figure 3. The derivative of (the numerically computed) $u_{0}$ is plotted in
blue while the derivative of the (numerical) solution $v_{0}$ of (3.19) is in
orange. Both curves have been normalized to fit in the same plot (since we are
only interested by their roots). It seems clear, at least for the first zeros,
that the derivative of $u_{0}$ and $v_{0}$ do not vanish at the same points.

##  4\. Lyapunov-Schmidt construction

The goal of this section is to prove Theorems 1.6 to 1.8. For clarity, we will
change the notations with respect to the previous sections. For
$\varepsilon\neq 0,|\varepsilon|<1$, we consider solutions of the Neumann
problem

(4.1) $\left\\{\begin{aligned} -\Delta u+u&=u^{\frac{N+2}{N-2}-\varepsilon}\
\text{in}\ B_{R}\subset\mathbb{R}^{N},\\\ u&>0,\\\ \partial_{\nu}u&=0\
\text{on}\ \partial B_{R}.\end{aligned}\right.$

If $(\varepsilon_{k})_{k\geq 0}$ is a sequence of real numbers going to $0$ as
$k\to+\infty$, a solution of (4.1) with $\varepsilon=\varepsilon_{k}$ is a
positive solution of (1.3) with $p_{k}=2^{*}-\varepsilon_{k}$. We let
$u_{0}\in C^{2,\theta}(\overline{B_{R}})$, $\theta\in(0,1)$, be a
nondegenerate radial solution of (4.1) with $\varepsilon=0$. Solutions of type
$u_{0}+B$ or $B$ are defined as in (1.5).

We will construct blowing up famillies $(u_{\varepsilon})_{\varepsilon>0}$ of
radial solutions of (4.1) in the following cases:

  1. (a)

(subcritical blow-up) for $\varepsilon\to 0$ with $\varepsilon>0$,

     * •

solutions of type $B$ for any $N\geq 4$ as announced in Theorem 1.6.

     * •

solutions of type $u_{0}+B$ as announced in Theorem 1.7 under the following
assumptions:

$\diamond$ $N\geq 7$ or

$\diamond$ $N=6$ and $u_{0}(0)<\frac{1}{2}$.

  2. (b)

(supercritical blow-up) for $\varepsilon\to 0$ with $\varepsilon<0$, solutions
of type $u_{0}+B$ as announced in Theorem 1.8 under the following assumptions:

$\diamond$ $3\leq N\leq 5$ or

$\diamond$ $N=6$ and $u_{0}(0)>\frac{1}{2}$.

In view of Theorem 1.2 to 1.5, this covers all the cases where radial blow-up
can occur, except the case $N=3$ for which we refer to Remark 4.6 below. Also,
we do not address the existence of towers of bubbles. As proven in Theorem
3.1, the assumption $u_{0}>0$ is necessary when $\varepsilon<0$.

###  4.1. Preliminaries and notation

In all the following, we will denote by $\left\langle.,.\right\rangle$ the
scalar product given, for $u,v\in H^{1}(B_{R})$, by

$\left\langle u,v\right\rangle=\int_{B_{R}}(\nabla u\cdot\nabla v+uv)dx,$

where $\cdot$ denotes the scalar product in $\mathbb{R}^{n}$, and by
$\|\cdot\|$ the associated norm in $H^{1}(B_{R})$. We denote by
$i^{\ast}:L^{\frac{2N}{N+2}}(B_{R})\rightarrow H^{1}(B_{R})$ the adjoint
operator of the embedding $i:H^{1}(B_{R})\rightarrow
L^{\frac{2N}{N-2}}(B_{R})$, i.e. for all $\varphi\in
L^{\frac{2N}{N+2}}(B_{R})$, the function $u=i^{\ast}(\varphi)\in H^{1}(B_{R})$
is the unique solution of

$\left\\{\begin{aligned} -\Delta u+u&=\varphi\ \text{in}\
B_{R}\subset\mathbb{R}^{N},\\\ \partial_{\nu}u&=0,\ \text{on}\ \partial
B_{R}.\end{aligned}\right.$

In particular, $\|u\|\leq C\|\varphi\|_{L^{\frac{2N}{N+2}}(B_{R})}$ for some
$C=C(N,R)>0$. In view of equation (4.1), we consider the following problem:
find a positive $u\in H^{1}(B_{R})$ such that

$u=i^{\ast}(f_{\varepsilon}(u)),$

where we have let, for $u\in\mathbb{R}$,

$f_{\varepsilon}(u)=(u)_{+}^{2^{\ast}-1-\varepsilon}$

and $u_{+}=\max(u,0)$. For $s>\frac{2N}{N-2}$ and by definition of $i$ we
have, by standard elliptic regularity results,

(4.2) $\|i^{\ast}(w)\|+\|i^{\ast}(w)\|_{L^{s}(B_{R})}\leq
C\|w\|_{L^{\frac{Ns}{N+2s}}(B_{R})}$

for some $C=C(N,R,s)>0$. For $\varepsilon$ small enough, we define

$\alpha_{\varepsilon}=\begin{cases}2^{\ast}-\frac{N}{2}\varepsilon,&\text{if}\
\varepsilon<0,\\\ 2^{\ast},&\text{if}\ \varepsilon>0.\end{cases}$

This exponent coincides with the exponent $\alpha_{k}$ defined by (1.4) since
$p_{k}=2^{*}-\varepsilon$ in this section. We let
$\mathcal{H}_{\varepsilon}=H^{1}(B_{R})\cap L^{\alpha_{\varepsilon}}(B_{R})$
endowed with the norm

$\|w\|_{\alpha_{\varepsilon}}=\|w\|+\|w\|_{L^{\alpha_{\varepsilon}}(B_{R})},$

which is a Banach space. It is known (see [CGS89], [Oba72]) that all the
positive solutions $u$ of the equation

$-\Delta u=u^{2^{\ast}-1}\text{ in }\mathbb{R}^{N}$

are given by

$B_{\delta,y}(x)=\delta^{\frac{2-N}{2}}B_{0}\Big{(}\frac{x-y}{\delta}\Big{)},\
\delta>0,\ y\in\mathbb{R}^{n}$

where $B_{0}$ is as in (2.5). It is also well known (see [BE91]) that all
$v\in\dot{H}^{1}(\mathbb{R}^{N})$ solving

$-\Delta v=(2^{\ast}-1)B_{0}^{2^{\ast}-2}v$

are linear combinations of

$V_{0}(x)=\dfrac{|x|^{2}-1}{(1+|x|^{2})^{\frac{N}{2}}}$

and

$V_{i}(x)=\dfrac{x_{i}}{(1+|x|^{2})^{\frac{N}{2}}},\ i=1,\ldots,N.$

If $v$ is in addition assumed to be radial it is then a multiple of $V_{0}$.

From now on we will only consider _radial_ solutions of (4.1). Let
$\chi:\mathbb{R}\rightarrow\mathbb{R}$ be a smooth cutoff function such that
$0\leq\chi\leq 1$, $\chi(x)=1$ if $|x|\leq\frac{R}{4}$ and $\chi(x)=0$ if
$|x|\geq\frac{R}{2}$. We define, for any $\lambda>0$,

$B_{\lambda}(x)=\chi(|x|)\lambda^{\frac{2-N}{2}}B_{0}(\lambda^{-1}|x|)$

and

$Z_{\lambda,0}(x)=\chi(|x|)\lambda^{\frac{2-N}{2}}V_{0}(\lambda^{-1}|x|).$

These are radial functions in $B_{R}$ and are compactly supported in $B_{R}$,
thus have zero Neumann derivative. We denote by $\Pi_{\lambda}$, respectively
$\Pi_{\lambda}^{\bot}$, the orthogonal projection of $H^{1}(B_{R})$ onto

$K_{\lambda}=\mathrm{span}\left\\{Z_{\lambda,0}\right\\},$

respectively onto

(4.3) $K_{\lambda}^{\bot}=\left\\{\phi\in
H^{1}(B_{R})/\left\langle\phi,Z_{\lambda,0}\right\rangle=0\right\\}.$

We recall that a solution $u_{0}$ of (4.1) with $\varepsilon=0$ is said to be
nondegenerate if (1.7) is satisfied with $h_{0}\equiv 1$.

######  Remark 4.1 (Encoding the radius of the ball with a parameter).

If $\mu>0$ is fixed it is easily seen that $u$ solves $-\Delta u+\mu
u=u^{2^{*}-1}$ in $B_{R}$, with $\partial_{\nu}u=0$ on $\partial B_{R}$, if
and only if $v(x):=\mu^{-\frac{N-2}{4}}u(\frac{x}{\sqrt{\mu}})$ solves
$-\Delta v+v=v^{2^{*}-1}$ in $B_{R\sqrt{\mu}}$ with $\partial_{\nu}v=0$ on
$\partial B_{R\sqrt{\mu}}$ (no radiality is assumed here). Clearly, $v$ is
non-degenerate if and only if $u$ is non-degenerate.

With Remark 4.1 in mind, the following result shows that the existence of non-
degenerate solutions of (4.1) when $\varepsilon=0$ is a generic property in
the choice of the radius $R$:

######  Proposition 4.2.

Let $u_{0}\in H^{1}(B_{R})$ be a positive solution to

$\left\\{\begin{aligned} -\Delta u_{0}+\mu
u_{0}&=u_{0}^{2^{\ast}-1},&\text{in}\ B_{R},\\\
\partial_{\nu}u_{0}&=0,&\text{on}\ \partial B_{R}\end{aligned}\right.$

for some $\mu>0$. Fix $\nu>0$. Then, there exist $\mu_{\nu}>0$ and
$u_{0,\nu}>0$ such that
$|\mu_{\nu}-\mu|+\|u_{0}-u_{0,\nu}\|_{C^{2,\theta}(\overline{B_{R}})}<\nu$,
$\theta\in(0,1)$, and $u_{0,\nu}>0$ is a non-degenerate solution of

$\left\\{\begin{aligned} -\Delta
u_{0,\nu}+\mu_{\nu}u_{0,\nu}&=u_{0,\nu}^{2^{\ast}-1},\text{ in }\ B_{R},\\\
\partial_{\nu}u_{0,\nu}&=0\text{ on }\ \partial B_{R}.\end{aligned}\right.$

###### Proof.

We proceed as in [RV13]. We set, for $\eta\geq 0$,

$A_{\eta}=\inf_{u\in
H^{1}(B_{R})\backslash\\{0\\}}\dfrac{\int_{B_{R}}\Big{(}|\nabla
u|^{2}+(\mu-(2^{\ast}-1)u_{0}^{2^{\ast}-2}-\eta)u^{2}\Big{)}dx}{(\int_{B_{R}}|u|^{2^{\ast}}dx)^{\frac{2}{2^{\ast}}}}.$

It is easy to see that $A_{\eta}<0$ and $\lim_{\eta\rightarrow
0}A_{\eta}=A_{0}<0$. Direct minimisation yields, for all $\eta\geq 0$, a
minimizer $w_{\eta}\in C^{2,\theta}(\overline{B_{R}})$, $w_{\eta}>0$, such
that

(4.4) $\left\\{\begin{aligned} -\Delta
w_{\eta}+(\mu-(2^{\ast}-1)u_{0}^{2^{\ast}-2}-\eta)w_{\eta}&=A_{\eta}w_{\eta}^{2^{\ast}-1}\text{
in }B_{R}\\\ \partial_{\nu}w_{\eta}&=0\text{ on }\partial
B_{R}\end{aligned}\right..$

With (4.4), and since $A_{\eta}<0$ for all $\eta\geq 0$, it is easily seen
that the family $(w_{\eta})_{\eta\geq 0}$ is relatively compact in
$C^{2}(\overline{B_{R}})$.

We now claim that, when $\eta=0$, $u_{0}$ is the only solution of (4.4) that
is positive in $\overline{B_{R}}$. We assume that $u$ and $v$ are two such
solutions and we let $x_{0}\in\overline{B_{R}}$ be the maximum point of
$\frac{u}{v}$ in $\overline{B_{R}}$. Straightforward computations show that
$w:=\frac{u}{v}$ satisfies

$-\triangle
w-A_{0}w^{2^{*}-1}\Big{(}1-\frac{1}{w^{2^{*}-2}}\Big{)}=2\langle\nabla
w,\nabla\ln v\rangle\quad\text{ in }\overline{B_{R}}.$

By definition of $x_{0}$, and since $\partial_{\nu}w=0$ in $\partial B_{R}$,
we always have $\nabla w(x_{0})=0$, regardless of whether $x_{0}$ is an
interior or boundary point. If we write $w(x)=w(r\omega)$ in spherical
coordinates for $x=r\omega\in\overline{B_{R}}$, the condition that $x_{0}$ is
the global maximum of $w$ in $\overline{B_{R}}$ implies both
$\triangle_{\mathbb{S}^{n-1}}w(R\cdot)\leq 0$ and, since
$\partial_{\nu}w(x_{0})=0$, $\partial_{\nu}^{2}w(x_{0})\leq 0$. Therefore
$-\triangle w(x_{0})\geq 0$, so that applying the latter equation at $x_{0}$
and since $A_{0}<0$ yields $w(x_{0})\leq 1$. This implies that $u\leq v$ in
$\overline{B_{R}}$. The same argument with the roles of $u$ and $v$ reversed
also shows that $v\leq u$ in $\overline{B_{R}}$, and thus $u\equiv v$ in
$\overline{B_{R}}$.

A first consequence of this uniqueness result is that $A_{0}=-(2^{*}-2)$,
which is easily seen integrating (4.4) by parts against $u_{0}$ when $\eta=0$.
As a second consequence, we get that
$\|w_{\eta}-u_{0}\|_{C^{2,\theta}(\overline{B_{R}})}\rightarrow 0$ as
$\eta\rightarrow 0$. Let now, for $\eta>0$,
$\mu_{\eta}=\mu-(2^{\ast}-1)(u_{0}^{2^{\ast}-2}-w_{\eta}^{2^{\ast}-2})-\eta$.
Clearly $\lim_{\eta\to 0}\mu_{\eta}=\mu>0$. By (4.4) $w_{\eta}$ satisfies

$-\triangle
w_{\eta}+\mu_{\eta}w_{\eta}=\big{(}2^{*}-1+A_{\eta}\big{)}w_{\eta}^{2^{*}-1}\quad\text{
in }\overline{B_{R}}.$

We let $u_{0,\eta}=\big{(}2^{*}-1+A_{\eta}\big{)}^{\frac{N-2}{4}}w_{\eta}$.
Since $2^{*}-1+A_{\eta}\to 1$ as $\eta\to 0$ we have $u_{0,\eta}\to u_{0}$ in
$C^{2}(\overline{B_{R}})$ as $\eta\to 0$. Let $\nu>0$ be fixed. It remains to
choose $\eta>0$ small enough so that
$-\triangle+\big{(}1-(2^{*}-1)u_{0}^{2^{*}-2}-\eta\big{)}$ has no kernel in
the sense of (2.35) and
$|\mu_{\eta}-\mu|+\|u_{0}-u_{0,\eta}\|_{C^{2,\theta}(\overline{B_{R}})}<\nu$.
This choice of $\eta$ implies that $u_{0,\eta}$ is still non-degenerate and
concludes the proof. ∎

###  4.2. Finite dimensional reduction

To prove Theorem 1.8 we will look, for $\varepsilon\neq 0$, for a radial
solution $u_{\varepsilon}$ to (4.1) of the form

$u_{\varepsilon}=u_{0}+B_{\lambda_{\varepsilon}(t)}+\phi_{\varepsilon}(t),$

where $u_{0}$ is a nondegenerate nonnegative radial solution of (4.1),
$\phi_{\varepsilon}(t)\in K_{\lambda_{\varepsilon}(t)}^{\bot}$ is a radial
function and, for any $t>0$ and $\varepsilon\neq 0$,

(4.5) $\lambda_{\varepsilon}(t)=\left\\{\begin{aligned}
&(|\varepsilon|t)^{2}&\text{ if }N=3,\\\ &|\varepsilon|t&\text{ if }N=4\text{
and }u_{0}>0,\\\
&\Big{(}\frac{|\varepsilon|}{\ln\frac{1}{|\varepsilon|}}\Big{)}^{\frac{1}{2}}t&\text{
if }\ N=4\text{ and }u_{0}=0,\\\ &(|\varepsilon|t)^{\frac{2}{3}}&\text{ if }\
N=5\text{ and }u_{0}>0,\\\ &\sqrt{|\varepsilon|t}&\text{ if }N=5\text{ and
}u_{0}=0,\\\ &\sqrt{|\varepsilon|t}&\text{ if }N\geq 6.\ \end{aligned}\right.$

In the following, for the sake of simplicity, we will often write $\lambda(t)$
and $\phi(t)$ instead of $\lambda_{\varepsilon}(t)$ and
$\phi_{\varepsilon}(t)$. It is well-known that equation (4.1) is equivalent to
the following system

(4.6)
$\Pi_{\lambda(t)}(u_{0}+B_{\lambda(t)}+\phi(t)-i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)}+\phi(t))))=0,$

and

(4.7)
$\Pi_{\lambda(t)}^{\bot}(u_{0}+B_{\lambda(t)}+\phi(t)-i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)}+\phi(t))))=0.$

Since $u_{0}$ and $B_{\lambda(t)}$ satisfy the Neumann boundary condition, and
by definition of $i^{\ast}$, $\phi(t)$ also satisfies the Neumann boundary
condition in the two previous equations. We begin by solving (4.7).
Classically this requires a quantitative estimate on the error of the
approximate solution $u_{0}+B_{\lambda(t)}$

From now on, if $a_{\varepsilon}$ and $b_{\varepsilon}$ denote two positive
families of real numbers, we will write $a_{\varepsilon}\lesssim
b_{\varepsilon}$ if there exists a constant $C$ independent of $\varepsilon$
and $t$ such that

$a_{\varepsilon}\leq Cb_{\varepsilon}\text{ for all }\varepsilon.$

######  Lemma 4.3.

Let $0<a<b$ be real numbers. There exists a positive constant $C_{a,b}$ such
that for $|\varepsilon|$ small we have

$\displaystyle\left\|i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)}))-u_{0}-B_{\lambda(t)}\right\|_{\alpha_{\varepsilon}}$
$\displaystyle\leq C_{a,b}\left\\{\begin{aligned}
&\Big{(}\frac{|\varepsilon|}{\ln\frac{1}{|\varepsilon|}}\Big{)}^{\frac{1}{2}}\text{
if }N=4\ \text{ and }u_{0}=0,\\\ &|\varepsilon|^{\frac{3}{4}}\text{ if
}N=5\text{ and }u_{0}=0,\\\ &|\varepsilon|\ln\frac{1}{|\varepsilon|}\text{
otherwise }.\end{aligned}\right.$

uniformly for $t\in[a,b]$.

###### Proof.

Let $0<a<b$ be real numbers. By (4.5), unless $N=4$ and $u_{0}\equiv 0$,
$\lambda_{\varepsilon}(t)$ converges uniformly to zero for $t\in[a,b]$ as
$\varepsilon\to 0$. If $N=4$ and $u_{0}\equiv 0$, again by (4.5),
$\lambda_{\varepsilon}(t)$ converges uniformly to zero for $t\in[a,+\infty)$
as $\varepsilon\to 0$. For simplicity we set $\alpha_{\varepsilon}=\alpha$ in
this proof. We have, by (4.2)

$\displaystyle\left\|i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)}))-u_{0}-B_{\lambda(t)}\right\|_{\alpha}$
$\displaystyle\lesssim\left\|(f_{\varepsilon}(u_{0}+B_{\lambda(t)}))-(-\Delta+Id)(u_{0}+B_{\lambda(t)})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R}).}$

The triangle inequality yields

$\displaystyle\left\|i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)}))-u_{0}-B_{\lambda(t)}\right\|_{\alpha}$
$\displaystyle\lesssim\left\|f_{\varepsilon}(u_{0}+B_{\lambda(t)})-f_{\varepsilon}(u_{0})-f_{\varepsilon}(B_{\lambda(t)})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\quad+\left\|f_{\varepsilon}(u_{0})+\Delta
u_{0}-u_{0}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\quad+\left\|f_{\varepsilon}(B_{\lambda(t)})+(\Delta-
Id)(B_{\lambda(t)})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\lesssim(I_{1}+I_{2}+I_{3}).$

We begin by estimating $I_{1}$. Using once more the triangle inequality, we
get

$\displaystyle I_{1}$
$\displaystyle\lesssim\left\|(f_{\varepsilon}(u_{0}+B_{\lambda(t)})-f_{\varepsilon}(B_{\lambda(t)}))1_{B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\quad+\left\|(f_{\varepsilon}(u_{0}+B_{\lambda(t)})-f_{\varepsilon}(u_{0}))1_{B_{R}\backslash
B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\quad+\left\|f_{\varepsilon}(B_{\lambda(t)})1_{B_{R}\backslash
B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}}+\left\|f_{\varepsilon}(u_{0})1_{B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\lesssim\left\|u_{0}B_{\lambda(t)}^{2^{\ast}-2-\varepsilon}1_{B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}+\left\|u_{0}^{2^{\ast}-2-\varepsilon}B_{\lambda(t)}1_{B_{R}\backslash
B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$ (4.8)
$\displaystyle\quad+\left\|f_{\varepsilon}(B_{\lambda(t)})1_{B_{R}\backslash
B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}+\left\|f_{\varepsilon}(u_{0})1_{B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}.$

A first remark is that, by definition of $\lambda(t)=\lambda_{\varepsilon}(t)$
in (4.5), we have

(4.9) $\lambda_{\varepsilon}(t)^{\varepsilon}=1+o(1)\quad\text{ uniformly for
}t\in[a,b]$

and

(4.10)
$\varepsilon|\ln\lambda(t)|\lesssim|\varepsilon|\ln\frac{1}{|\varepsilon|}\quad\text{
uniformly for }t\in[a,b].$

By definition of $\alpha=\alpha_{\varepsilon}$, we have
$\frac{N\alpha}{N+2\alpha}=\frac{2N}{N+2}+O(|\varepsilon|)$ as $\varepsilon\to
0$. Since $B_{\lambda_{\varepsilon}(t)}\lesssim\lambda(t)^{-\frac{N-2}{2}}$ in
$\overline{B_{R}}$ we can compute, using (4.9) and (4.10):

$\displaystyle\left\|u_{0}B_{\lambda(t)}^{2^{\ast}-2-\varepsilon}1_{B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\lesssim\left\|u_{0}B_{\lambda(t)}^{2^{\ast}-2}1_{B_{\sqrt{\lambda(t)}}}\right\|_{L^{\frac{2N}{N+2}}(B_{R})}$
$\displaystyle\lesssim\left\\{\begin{aligned}
&\lambda(t)^{\frac{N-2}{2}}&\text{ if }N=3,4,5,\\\
&\lambda(t)^{2}|\ln\lambda(t)|^{\frac{2}{3}}&\text{ if }N=6,\\\
&\lambda(t)^{2}&\text{ if }N\geq 7.\end{aligned}\right.$

Similar computations using (4.9) and (4.10) show that

$\displaystyle I_{1}\lesssim\lambda(t)^{\frac{N+2}{4}}+\left\\{\begin{aligned}
&\lambda(t)^{\frac{N-2}{2}}&\text{ if }N=3,4,5,\\\
&\lambda(t)^{2}|\ln\lambda(t)|^{\frac{2}{3}}&\text{ if }N=6,\\\
&\lambda(t)^{2}&\text{ if }N\geq 7.\end{aligned}\right.$

Let us turn to $I_{2}$. From Taylor’s expansion, we see that

$I_{2}=\left\|f_{\varepsilon}(u_{0})-f_{0}(u_{0})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}\lesssim|\varepsilon|.$

Next we estimate $I_{3}$. We have that

$\displaystyle I_{3}$
$\displaystyle\lesssim\left\|\chi^{2^{\ast}-1-\varepsilon}(B_{\lambda(t)}^{2^{\ast}-1-\varepsilon}-B_{\lambda(t)}^{2^{\ast}-1})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\quad+\left\|(\chi^{2^{\ast}-1-\varepsilon}-\chi^{2^{\ast}-1})B^{2^{\ast}-1}_{\lambda(t)}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\quad+\left\|f_{0}(B_{\lambda(t)})-(-\Delta+Id)(B_{\lambda(t)})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}.$

Using Taylor’s expansions, (4.9) and (4.10) we find

$\left\|\chi^{2^{\ast}-1-\varepsilon}(B_{\lambda(t)}^{2^{\ast}-1-\varepsilon}-B_{\lambda(t)}^{2^{\ast}-1})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}\lesssim|\varepsilon|\ln\frac{1}{|\varepsilon|},$

and

$\left\|(\chi^{2^{\ast}-1-\varepsilon}-\chi^{2^{\ast}-1})B_{\lambda(t)}^{2^{\ast}-1}\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}\lesssim\lambda(t)^{\frac{N+2}{2}},$

where it is intended that all these expansions are uniform for $t\in[a,b]$.
Finally, for the last term in the estimation of $I_{3}$ we have, using again
(4.9) and (4.10):

$\displaystyle\left\|f_{0}(B_{\lambda(t)})-(-\Delta+Id)(B_{\lambda(t)})\right\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\lesssim\|B_{\lambda(t)}\Delta\chi\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}+\|\nabla\chi\nabla
B_{\lambda(t)}\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}+\|\chi
B_{\lambda(t)}\|_{L^{\frac{N\alpha}{N+2\alpha}}(B_{R})}$
$\displaystyle\lesssim\lambda(t)^{\frac{N-2}{2}}+\left\\{\begin{aligned}
&\lambda(t)^{\frac{N-2}{2}}&\text{ if }N=3,4,5,\\\
&\lambda(t)^{2}|\ln\lambda(t)|^{\frac{2}{3}}&\text{ if }N=6,\\\
&\lambda(t)^{2}&\text{ if }N\geq 7.\end{aligned}\right.$

Combining the previous estimates shows that

(4.11)
$\displaystyle\|i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)}))-u_{0}-B_{\lambda(t)}\|_{\alpha}$
$\displaystyle\lesssim\lambda(t)^{\frac{N+2}{4}}+|\varepsilon|\ln\frac{1}{|\varepsilon|}$
$\displaystyle\quad+\lambda(t)^{\frac{N-2}{2}}+\left\\{\begin{aligned}
&\lambda(t)^{\frac{N-2}{2}}&\text{ if }N=3,4,5\\\
&\lambda(t)^{2}|\ln\lambda(t)|^{\frac{2}{3}}&\text{ if }N=6\\\
&\lambda(t)^{2}&\text{ if }N\geq 7\end{aligned}\right.$

It is now easily seen with (4.5) that (4.11) concludes the proof of Lemma 4.3.
∎

The general resolution of (4.7) up to kernel elements is given by the
following Proposition:

######  Proposition 4.4.

Let $u_{0}\in C^{2,\theta}(B_{R})$ be a nondegenerate radial solution of (4.1)
with $\varepsilon=0$. Let $a<b$ be two real numbers. There exists a positive
constant $C_{a,b}$ depending only on $a$ and $b$ such that for $|\varepsilon|$
small enough and for all $t\in[a,b]$ there exists a unique radial function
$\phi(t)=\phi_{\varepsilon}(t)\in K_{\lambda(t)}^{\bot}$ solving equation
(4.7) and satisfying

(4.12) $\displaystyle\left\|\phi(t)\right\|_{\alpha_{\varepsilon}}\leq
C_{a,b}\left\|i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)}))-u_{0}-B_{\lambda(t)}\right\|_{\alpha_{\varepsilon}}$

Moreover, $t\in[a,b]\mapsto\phi(t)\in H^{1}(B_{R})$ is continuously
differentiable.

The proof is very classical and we omit it. We refer for instance to [RV13,
RV14] for the same kind of result (see also [RW04]).

###  4.3. The reduced problem.

For $\varepsilon$ small enough, we define the energy in $H^{1}(B_{R})$
associated to (4.1) by

$J_{\varepsilon}(u)=\dfrac{1}{2}\int_{B_{R}}(|\nabla
u|^{2}+u^{2})dx-\int_{B_{R}}F_{\varepsilon}(u)dx,$

where
$F_{\varepsilon}(s)=\frac{1}{2^{*}-\varepsilon}s_{+}^{2^{*}-\varepsilon}$. We
set

(4.13)
$I_{\varepsilon}(t)=J_{\varepsilon}\big{(}u_{0}+B_{\lambda(t)}+\phi(t)\big{)},\quad
t\in[a,b],$

where $\phi(t)\in K_{\lambda(t)}^{\bot}$ is the function defined in
Proposition 4.4. In the next proposition, we give an expansion of
$I_{\varepsilon}$ as $\varepsilon\to 0$.

######  Proposition 4.5.

Let $u_{0}\in C^{2,\theta}(B_{R})$, $\theta\in(0,1)$ be a nondegenerate radial
solution of (4.1) with $\varepsilon=0$. For $N\geq 3$ we let

(4.14)
$c_{5}(N,u_{0})=\dfrac{1}{N}K_{N}^{-N}\left(\dfrac{2(N-1)}{(N-2)(N-4)}1_{N\geq
6}-\dfrac{2^{N}u_{0}(0)\omega_{N-1}}{(N(N-2))^{\frac{N-2}{4}}\omega_{N}}1_{N\leq
6}\right),$

where $\omega_{N}$ stands for the volume of $\mathbb{S}^{N}$ and $K_{N}$ is as
in (2.6).We consider $I_{\varepsilon}$ defined in (4.13).

  * •

Assume that $N=4$ and $u_{0}\equiv 0$. There exist numerical constants
$\hat{c}_{i},1\leq i\leq 5$, with $\hat{c}_{5}>0$, such that, as
$\varepsilon\to 0$,

(4.15)
$I_{\varepsilon}(t)=\hat{c}_{1}+\hat{c}_{2}\varepsilon+\hat{c}_{3}\varepsilon\ln\frac{1}{|\varepsilon|}+\hat{c}_{4}\varepsilon\ln\ln\frac{1}{|\varepsilon|}+\hat{c}_{5}\big{(}\varepsilon\ln\frac{1}{t}+|\varepsilon|\frac{3}{2}t^{2}\big{)}+o(\varepsilon).$

  * •

In all the other cases: there exist constants $c_{i}(N,u_{0})$, $i=1,2,5$
depending on $N$ and $u_{0}$ and $c_{i}(N)$, $i=3,4$, depending only $N$ such
that, as $\varepsilon\to 0$:

(4.16) $\displaystyle I_{\varepsilon}(t)$
$\displaystyle=c_{1}(N,u_{0})+c_{2}(N,u_{0})\varepsilon+c_{3}(N)\varepsilon\ln\frac{1}{|\varepsilon|}+c_{4}(N)\varepsilon\ln\frac{1}{t}+c_{5}(N,u_{0})|\varepsilon|t+o(\varepsilon),$

and the constant $c_{4}(N)$ is positive in these cases.

All these expansions are uniform with respect to $t\in[a,b]$.

###### Proof.

As before we write $\lambda(t)$ instead of $\lambda_{\varepsilon}(t)$ in this
proof, where $\lambda(t)$ is given by (4.5). Using Lemma 4.3 and Proposition
4.4, we have

$\displaystyle I_{\varepsilon}(t)-J_{\varepsilon}(u_{0}+B_{\lambda(t)})$
$\displaystyle=\left\langle
u_{0}+B_{\lambda(t)}-i^{\ast}(f_{\varepsilon}(u_{0}+B_{\lambda(t)})),\phi(t)\right\rangle+O(\left\|\phi(t)\right\|^{2})$
$\displaystyle=\begin{cases}O\Big{(}\frac{|\varepsilon|}{\ln\frac{1}{|\varepsilon|}}\Big{)},&\text{if}\
N=4\ \text{and}\ u_{0}=0,\\\
O\big{(}|\varepsilon|^{\frac{3}{2}}\big{)},&\text{if}\ N=5\ \text{and}\
u_{0}=0,\\\
O\big{(}\varepsilon^{2}\ln\frac{1}{|\varepsilon|}^{2}\big{)},&\text{otherwise},\end{cases}$
$\displaystyle=o(\varepsilon)$

when $\varepsilon\rightarrow 0$, uniformly for $t\in[a,b]$. Now, the proof of
the proposition is reduced to estimate
$J_{\varepsilon}(u_{0}+B_{\lambda(t)})$. Since $u_{0}$ is a solution of (4.1),
it follows that

$\displaystyle J_{\varepsilon}(u_{0}+B_{\lambda(t)})$
$\displaystyle=\dfrac{1}{N}\int_{B_{R}}u_{0}^{2^{\ast}}dx+\dfrac{\varepsilon}{2^{\ast}}\int_{B_{R}}u_{0}^{2^{\ast}}(\ln
u_{0}-\dfrac{1}{2^{\ast}})dx+I_{1,\varepsilon,t}+I_{2,\varepsilon,t}+I_{3,\varepsilon,t}+O(\varepsilon^{2}),$

where

$I_{1,\varepsilon,t}=\dfrac{1}{2}\int_{B_{R}}\big{(}|\nabla
B_{\lambda(t)}|^{2}+B_{\lambda(t)}^{2}\big{)}dx-\int_{B_{R}}F_{\varepsilon}(B_{\lambda(t)})dx,\quad
I_{2,\varepsilon,t}=-\int_{B_{R}}f_{\varepsilon}(B_{\lambda(t)})u_{0}dx,$

and

$I_{3,\varepsilon,t}=-\int_{B_{R}}(F_{\varepsilon}(u_{0}+B_{\lambda(t)})-F_{\varepsilon}(u_{0})-F_{\varepsilon}(B_{\lambda(t)})-f_{\varepsilon}(u_{0})B_{\lambda(t)}-f_{\varepsilon}(B_{\lambda(t)})u_{0})dx.$

We begin by the estimate of $I_{3,\varepsilon,t}$. Remark that this term
vanishes when $u_{0}\equiv 0$. Using Taylor expansion and rough estimates, we
get

$\displaystyle|I_{3,\varepsilon,t}|\lesssim{}$
$\displaystyle\left\|(F_{\varepsilon}(u_{0}+B_{\lambda(t)})-F_{\varepsilon}(B_{\lambda(t)})-f_{\varepsilon}(B_{\lambda(t)})u_{0})\right\|_{L^{1}(B_{\sqrt{\lambda(t))}})}$
$\displaystyle+\left\|(F_{\varepsilon}(u_{0}+B_{\lambda(t)})-F_{\varepsilon}(u_{0})-f_{\varepsilon}(u_{0})B_{\lambda(t)})\right\|_{L^{1}(B_{R}\backslash
B_{\sqrt{\lambda(t)}})}$
$\displaystyle+\left\|F_{\varepsilon}(u_{0})\right\|_{L^{1}(B_{\sqrt{\lambda(t)}})}+\left\|f_{\varepsilon}(u_{0})B_{\lambda(t)}\right\|_{L^{1}(B_{\sqrt{\lambda(t)}})}$
$\displaystyle+\left\|F_{\varepsilon}(B_{\lambda(t)})\right\|_{L^{1}(B_{R}\backslash
B_{\sqrt{\lambda(t)}})}+\left\|u_{0}f_{\varepsilon}(B_{\lambda(t)})\right\|_{L^{1}(B_{R}\backslash
B_{\sqrt{\lambda(t)}})}$ $\displaystyle\lesssim{}$
$\displaystyle\left\|u_{0}^{2}B_{\lambda(t)}^{2^{\ast}-2-\varepsilon}\right\|_{L^{1}(B_{\sqrt{\lambda(t)}})}+\left\|u_{0}^{2^{\ast}-2-\varepsilon}B_{\lambda(t)}^{2}\right\|_{L^{1}(B_{R}\backslash
B_{\sqrt{\lambda(t)}})}$
$\displaystyle+\left\|F_{\varepsilon}(B_{\lambda(t)})\right\|_{L^{1}(B_{R}\backslash
B_{\sqrt{\lambda(t)}})}+\left\|u_{0}f_{\varepsilon}(B_{\lambda(t)})\right\|_{L^{1}(B_{R}\backslash
B_{\sqrt{\lambda(t)}})}$
$\displaystyle+\left\|F_{\varepsilon}(u_{0})\right\|_{L^{1}(B_{\sqrt{\lambda(t)}})}+\left\|f_{\varepsilon}(u_{0})B_{\lambda(t)}\right\|_{L^{1}(B_{\sqrt{\lambda(t)}})}$
$\displaystyle\lesssim{}$
$\displaystyle\left\|u_{0}^{2}B_{\lambda(t)}^{2^{\ast}-2-\varepsilon}\right\|_{L^{1}(B_{\sqrt{\lambda(t)}})}+\left\|u_{0}^{2^{\ast}-2-\varepsilon}B_{\lambda(t)}^{2}\right\|_{L^{1}(B_{R}\backslash
B_{\sqrt{\lambda(t)}})}$ $\displaystyle+\lambda(t)^{\frac{N}{2}}.$

Therefore estimating the last two terms and using the definition of
$\lambda(t)$ given by (4.5), we obtain

(4.17) $\displaystyle|I_{3,\varepsilon,t}|$
$\displaystyle\lesssim\left\\{\begin{array}[]{lll}\lambda(t)\ \mathrm{if}\
N=3\\\ \lambda(t)^{2}|\ln\lambda(t)|\ \mathrm{if}\ N=4\ \mathrm{and}\
u_{0}>0\\\ \lambda(t)^{\frac{N}{2}}\ \mathrm{if}\ N\geq 5\\\
\end{array}\right.$ $\displaystyle=o(\varepsilon)$

uniformly for $t\in[a,b]$. Now, let us estimate $I_{2,\varepsilon,t}$. For any
$N\geq 3$ we have, with (4.5) (4.9) and (4.10),

$\displaystyle I_{2,\varepsilon,t}={}$ $\displaystyle-
u_{0}(0)\omega_{N-1}(N(N-2))^{\frac{N-2}{4}(2^{\ast}-1-\varepsilon)}\lambda(t)^{\frac{N-2}{2}(1+\varepsilon)}$
$\displaystyle\times\int_{0}^{\frac{R}{4\lambda(t)}}\dfrac{r^{N-1}}{(1+r^{2})^{\frac{N+2}{2}-\varepsilon\frac{N-2}{2}}}dr+O(\lambda(t)^{\frac{N+2}{2}-\frac{N-2}{2}\varepsilon})$
(4.18) $\displaystyle={}$
$\displaystyle-\dfrac{2^{N}u_{0}(0)K_{N}^{-N}\omega_{N-1}\lambda(t)^{\frac{N-2}{2}}}{N^{\frac{N+2}{4}}(N-2)^{\frac{N-2}{4}}\omega_{N}}+O(\lambda(t)^{\frac{N}{2}}).$

The last integral in (4.3) has been computed by standard considerations, see
for instance [PV22a, equation $(4.6)$].

Finally, let us consider $I_{1,\varepsilon,t}$. We get

$\int_{B_{R}}|\nabla B_{\lambda(t)}|^{2}dx=K_{N}^{-N}+O(\lambda(t)^{N-2}),$

$\int_{B_{R}}B_{\lambda(t)}^{2}dx=\begin{cases}O(\lambda(t)),&N=3,\\\
\frac{3}{2}K_{4}^{-4}\lambda(t)^{2}|\ln\lambda(t)|+O(\lambda(t)^{2}),&N=4,\\\
\frac{4(N-1)}{N(N-2)(N-4)}K_{N}^{-N}\lambda^{2}(t)+O(\lambda(t)^{N-2}),&N\geq
5,\end{cases}$

and, with (4.9) and (4.10),

$\int_{B_{R}}F_{\varepsilon}(B_{\lambda(t)})dx=\dfrac{N-2}{2N}K_{N}^{-N}(\lambda(t)^{\frac{N-2}{2}\varepsilon}+\dfrac{2\beta_{N}}{N-2}\varepsilon)+O(\lambda(t)^{N-(N-2)\varepsilon})+O(\varepsilon^{2}),$

where $K_{N}$ is defined as in (2.6) and

$\beta_{N}=2^{N-3}(N-2)^{2}\dfrac{\omega_{N-1}}{\omega_{N}}\int_{0}^{\infty}\dfrac{r^{\frac{N-2}{2}}\ln(1+r)}{(1+r)^{N}}dr+\dfrac{(N-2)^{2}}{4N}(1-2N\ln\sqrt{N(N-2)}).$

So in view of the previous estimates and (4.5), (4.9) and (4.10), we find

(4.19) $\displaystyle I_{1,\varepsilon,t}$
$\displaystyle=\dfrac{K_{N}^{-N}}{N}\Bigg{(}1-\beta_{N}\varepsilon-\Big{(}\dfrac{N-2}{2}\Big{)}^{2}\varepsilon\ln\lambda(t)+O(\varepsilon^{2})$
$\displaystyle+\left\\{\begin{aligned} &O(\lambda(t))&\text{ if }N=3\\\
&3\lambda(t)^{2}|\ln\lambda(t)|+O(\lambda(t)^{2})&\text{ if }N=4\\\
&\frac{2(N-1)}{(N-2)(N-4)}\lambda^{2}(t)+O(\lambda(t)^{N-2})&\text{ if }N\geq
5\end{aligned}\right\\}\Bigg{)}$

Assume first that $N=3$. Combining (4.17), (4.3) and (4.19), we obtain, with
(4.5),

(4.20) $\displaystyle J_{\varepsilon}(u_{0}+B_{\lambda(t)})$
$\displaystyle=\frac{1}{3}\int_{B_{R}}u_{0}^{6}dx+\dfrac{\varepsilon}{6}\int_{B_{R}}u_{0}^{6}(\ln
u_{0}-\dfrac{1}{6})dx+\dfrac{1}{3}K_{3}^{-3}\left(1-\beta_{3}\varepsilon\right)$
$\displaystyle-\frac{1}{6}K_{3}^{-3}\varepsilon\ln|\varepsilon|-\frac{1}{12}K_{3}^{-3}\varepsilon\ln
t+c_{5}(3,u_{0})|\varepsilon|t+o(\varepsilon).$

Assume now that $N\geq 5$. Combining again (4.17), (4.3) and (4.19), we
obtain, with (4.5)

(4.21) $\displaystyle J_{\varepsilon}(u_{0}+B_{\lambda(t)})$
$\displaystyle=\frac{1}{N}\int_{B_{R}}u_{0}^{2^{\ast}}dx+\dfrac{\varepsilon}{2^{\ast}}\int_{B_{R}}u_{0}^{2^{\ast}}(\ln
u_{0}-\dfrac{1}{2^{\ast}})dx+\dfrac{1}{N}K_{N}^{-N}\left(1-\beta_{N}\varepsilon\right)$
$\displaystyle+\dfrac{1}{N}K_{N}^{-N}\Bigg{(}-\Big{(}\dfrac{N-2}{2}\Big{)}^{2}\varepsilon\ln\lambda(t)+\dfrac{2(N-1)}{(N-2)(N-4)}\lambda(t)^{2}$
$\displaystyle-\dfrac{2^{N}u_{0}(0)\omega_{N-1}\lambda(t)^{\frac{N-2}{2}}}{(N(N-2))^{\frac{N-2}{4}}\omega_{N}}\Bigg{)}+o(\varepsilon).$

By (4.5) $\lambda(t)$ is a power of $|\varepsilon|t$ when $N\geq 5$. As a
consequence we have $\ln\lambda(t)=C(\ln|\varepsilon|+\ln t)$ for some
dimensional constant $C$,
$\lambda(t)^{2}=o\big{(}\lambda(t)^{\frac{3}{2}}\big{)}$ if $N=5$ and
$\lambda(t)^{\frac{N-2}{2}}=o\big{(}\lambda(t)^{2}\big{)}$ if $N\geq 7$ as
$\varepsilon\to 0$. Proposition 4.5 then follows from (4.5),(4.9) and (4.10)
when $N\geq 5$.

We now assume that $N=4$. Assume first that $u_{0}>0$. Then
$\lambda(t)=t|\varepsilon|$, so that
$\lambda(t)^{2}\ln\lambda(t)=o\big{(}\lambda(t)\big{)}$ as $\varepsilon\to 0$.
Combining (4.17), (4.3) and (4.19) with (4.5) we thus get

(4.22) $\displaystyle J_{\varepsilon}(u_{0}+B_{\lambda(t)})$
$\displaystyle=\frac{1}{4}\int_{B_{R}}u_{0}^{4}dx+\dfrac{\varepsilon}{4}\int_{B_{R}}u_{0}^{4}(\ln
u_{0}-\dfrac{1}{4})dx+\dfrac{1}{4}K_{4}^{-4}\left(1-\beta_{4}\varepsilon\right)$
$\displaystyle-\dfrac{1}{4}K_{4}^{-4}\varepsilon\ln|\varepsilon|-\dfrac{1}{4}K_{4}^{-4}\Big{(}\varepsilon\ln
t+c_{1}(4,u_{0})|\varepsilon|t\Big{)}+o(\varepsilon).$

Assume finally that $N=4$ and $u_{0}\equiv 0$. In this case
$I_{2,\varepsilon,t}$ and $I_{3,\varepsilon,t}$ vanish and (4.19), (4.5),
(4.9) and (4.10) show that

(4.23) $\displaystyle I_{\varepsilon}(t)$
$\displaystyle=\dfrac{K_{4}^{-4}}{4}\big{(}1-\beta_{4}\varepsilon\big{)}+O(\varepsilon^{2}|\ln\varepsilon|^{2})+O(\lambda(t)^{2})+\frac{1}{4}K_{4}^{-4}\Big{(}\varepsilon+3\lambda(t)^{2}\Big{)}\ln\frac{1}{\lambda(t)}.$

Since
$\lambda(t)=\big{(}\frac{|\varepsilon|}{\ln\frac{1}{|\varepsilon|}}\big{)}^{\frac{1}{2}}t$
by (4.5), we have

$\displaystyle\Big{(}\varepsilon+3\lambda(t)^{2}\Big{)}\ln\frac{1}{\lambda(t)}$
$\displaystyle=\big{(}\varepsilon+3|\varepsilon|\frac{t^{2}}{\ln\frac{1}{|\varepsilon|}}\big{)}\big{(}\frac{1}{2}\ln\frac{1}{|\varepsilon|}+\frac{1}{2}\ln\ln\frac{1}{|\varepsilon|}+\ln\frac{1}{t}\big{)}$
$\displaystyle=\hat{c}_{3}|\varepsilon|\ln\frac{1}{|\varepsilon|}+\hat{c}_{4}|\varepsilon|\ln\ln\frac{1}{|\varepsilon|}+\big{(}\varepsilon\ln\frac{1}{t}+|\varepsilon|\frac{3}{2}t^{2}\big{)}+o(|\varepsilon|)$

uniformly for $t\in[a,b]$. This concludes the proof of Proposition 4.5. ∎