# A priori estimates for finite-energy sign-changing blowing-up solutions of
critical elliptic PDEs of second order

Bruno Premoselli  Bruno Premoselli, Université Libre de Bruxelles, Service
d’Analyse CP 214, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
<EMAIL_ADDRESS>
###### Abstract.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$. The
$C^{0}$-theory for positive solutions is now well understood and provides
sharp pointwise estimates for finite-energy blowing-up positive solutions of
nonlinear critical equations of Schrödinger-Yamabe type in $M$. There was a
lack of such a theory for sign-changing solutions. We prove the $C^{0}$-theory
for sign-changing solutions in this paper. Our proof combines an a priori
bubble-tree analysis with a finite-dimensional reduction.

##  1\. Introduction and statement of the results

###  1.1. Statement of the results

Let $(M,g)$ be a smooth and closed Riemannian manifold of dimension $n\geq 3$,
where closed means compact without boundary. Let $(h_{\alpha})_{\alpha}$ be a
sequence of bounded functions in $M$. We are interested in this work in the
following Schrödinger-Yamabe type equation in $M$:

$\triangle_{g}u+h_{\alpha}u=|u|^{2^{*}-2}u,$

where $\triangle_{g}=-\textrm{div}_{g}(\nabla\cdot)$ is the Laplace-Beltrami
operator acting on functions and $2^{*}=\frac{2n}{n-2}$ is the critical
exponent for the embedding of the Sobolev space $H^{1}(M)$ into Lebesgue
spaces. This equation originated a vast amount of work and has been the target
of active investigation for decades.

Let $(u_{\alpha})_{\alpha}$, with $u_{\alpha}\in H^{1}(M)$, be a sequence of
solutions of

$\triangle_{g}u_{\alpha}+h_{\alpha}u_{\alpha}=|u_{\alpha}|^{2^{*}-2}u_{\alpha}$
(1.1)

in $M$. Unless specified otherwise, the $u_{\alpha}$’s are allowed to change
sign. Since $h_{\alpha}$ is bounded the regularity result of Trudinger [53]
ensures that $u_{\alpha}\in C^{1,\beta}(M)$ for all $0<\beta<1$. Following
standard terminology we say that $(u_{\alpha})_{\alpha}$ _blows-up with finite
energy_ if

$\|u_{\alpha}\|_{H^{1}(M)}\leq C\quad\textrm{ and
}\quad\lim_{\alpha\to+\infty}\|u_{\alpha}\|_{L^{\infty}(M)}=+\infty$ (1.2)

for some $C>0$ independent of $\alpha$. The asymptotic behavior of
$(u_{\alpha})_{\alpha}$ in the energy space $H^{1}(M)$ has been known since
Struwe’s seminal work [50]: assume e.g. that $h_{\alpha}$ converges in
$L^{2}(M)$ towards $h$. Then, up to a subsequence, $(u_{\alpha})_{\alpha}$
decomposes as

$u_{\alpha}=u_{0}+\sum_{i=1}^{k}V_{i,\alpha}+\phi_{\alpha},$ (1.3)

where $\|\phi_{\alpha}\|_{H^{1}(M)}\to 0$ as $\alpha\to+\infty$, $u_{0}$ is a
weak solution of the limiting equation

$\triangle_{g}u_{0}+hu_{0}=|u_{0}|^{2^{*}-2}u_{0}$

in $M$, $k\geq 1$ is an integer and, for all $1\leq i\leq k$, $V_{i,\alpha}$
is a _sign-changing bubble_. More precisely, there exists a nontrivial,
possibly sign-changing finite-energy solution $V_{i}$ of the euclidean Yamabe
equation in $\mathbb{R}^{n}$:

$\triangle_{\xi}V_{i}=|V_{i}|^{2^{*}-2}V_{i}$

and there exist a sequence $(x_{i,\alpha})_{\alpha}$ of points of $M$ and a
sequence $(\mu_{i,\alpha})_{\alpha}$ of positive numbers with
$\mu_{i,\alpha}\to 0$ as $\alpha\to+\infty$ such that

$V_{i,\alpha}(x)=\mu_{i,\alpha}^{1-\frac{n}{2}}V_{i}\Big{(}\frac{1}{\mu_{i,\alpha}}\exp_{x_{i,\alpha}}^{-1}(x)\Big{)}\quad\textrm{
for }x\in B_{g}(x_{i,\alpha},\frac{i_{g}}{2}),$

where $i_{g}$ is the injectivity radius of $(M,g)$ and $\exp_{x_{i,\alpha}}$
is the exponential map for $g$ at $x_{i,\alpha}$. For the expression of
$V_{i,\alpha}$ on the whole of $M$ see (2.14) below. The sequences
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$,
moreover satisfy

$\mu_{i,\alpha}\to 0\quad\textrm{ and
}\quad\frac{\mu_{i,\alpha}}{\mu_{j,\alpha}}+\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}+\frac{d_{g}(x_{i,\alpha},x_{j,\alpha})^{2}}{\mu_{i,\alpha}\mu_{j,\alpha}}\to+\infty\quad\textrm{
for all }i\neq j$ (1.4)

as $\alpha\to+\infty$, we have $u_{\alpha}\rightharpoonup u_{0}$ in $H^{1}(M)$
and the energy of $u_{\alpha}$ is quantified:

$\|u_{\alpha}\|_{H^{1}(M)}^{2}=\|u_{0}\|_{H^{1}(M)}^{2}+\sum_{i=1}^{k}\|\nabla
V_{i}\|_{L^{2}(\mathbb{R}^{n})}^{2}+o(1)$

as $\alpha\to+\infty$. When all the functions $u_{\alpha}$ are assumed to be
_positive_ the profiles $V_{i,\alpha}$ are equal to the so-called positive
standard bubbles

$B_{i,\alpha}(x)=\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{\big{(}\mu_{i,\alpha}^{2}+\frac{d_{g}(x_{i,\alpha},x)^{2}}{n(n-2)}\big{)}^{\frac{n-2}{2}}},\quad
x\in M.$

The $B_{i,\alpha}$’s carry the same energy up to an error term and, when
$u_{\alpha}\geq 0$, the energy of $u_{\alpha}$ is quantified as

$\|u_{\alpha}\|_{H^{1}(M)}^{2}=\|u_{0}\|_{H^{1}(M)}^{2}+kE_{n}+o(1),$

where $E_{n}$ is a dimensional constant (see Section 2 below for more
details). In the sign-changing regime the situation is more intricate as we do
not have one sole energy level for the $V_{i}$’s (see e.g. Ding [16]). We let,
for all $x\in M$,

$B_{0,\alpha}(x)=\left\\{\begin{aligned} &0&\textrm{ if }u_{0}\equiv 0\textrm{
and }\ker(\triangle_{g}+h)=\\{0\\}\\\ &1&\textrm{
otherwise}\end{aligned}\right.,\\\ $

where we keep the notations of (1.3).

An important issue is to know whether the description (1.3) holds true in the
$C^{0}$ space, and thus whether or not pointwise estimates on
$(u_{\alpha})_{\alpha}$ inherited from (1.3) hold. This, the $C^{0}$ theory
for positive solutions, has been known to hold true for positive solutions of
(1.1) for almost two decades and has had spectacular applications, but no
analogous results were obtained in the sign-changing case. In this work we
answer this question positively when $u_{\alpha}$ is allowed to change sign.
Our main result is as follows :

######  Theorem 1.1.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$, let
$(h_{\alpha})_{\alpha}$ be a sequence of functions converging in
$L^{\infty}(M)$ to some function $h$ in $M$ and let $(u_{\alpha})_{\alpha}$ be
a sequence of solutions of (1.1) that satisfies (1.2). Let $u_{0}$ and
$V_{1,\alpha},\dots,V_{k,\alpha}$ be as in the Struwe decomposition (1.3) of
$u_{\alpha}$, where the $V_{i,\alpha}$’s are given by (2.14) below. Then, up
to passing to a subsequence for $(u_{\alpha})_{\alpha}$, we have

$\frac{u_{\alpha}-u_{0}-\sum_{i=1}^{k}V_{i,\alpha}}{\sum_{i=0}^{k}B_{i,\alpha}}\to
0$

in $L^{\infty}(M)$ as $\alpha\to+\infty$.

The statement of Theorem 1.1 reformulates as follows: if
$(u_{\alpha})_{\alpha}$ is any sequence of solutions of (1.1) that blows-up
with finite-energy then, up to passing to a subsequence on the
$(u_{\alpha})_{\alpha}$, for any sequence $(x_{\alpha})_{\alpha}$ of points of
$M$

$u_{\alpha}(x_{\alpha})=u_{0}(x_{\alpha})+\sum_{i=1}^{k}V_{i,\alpha}(x_{\alpha})+o\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha})\Big{)}$

holds as $\alpha\to+\infty$. In other words, the difference between
$u_{\alpha}$, the weak limit and the whole set of (possibly sign-changing)
bubbles in Struwe’s decomposition is globally pointwise small with respect to
the sum of the positive bubbles. In Theorem 1.1 no assumption on $h$ is
required: in particular one does not need to assume that $\triangle_{g}+h$ has
no kernel. No assumption on the limiting solutions $V_{i}$, $1\leq i\leq k$,
of the Yamabe equation in $\mathbb{R}^{n}$ is required either.

Theorem 1.1 is the first comprehensive result of this type for finite-energy
_sign-changing_ solutions of (1.1) in dimensions $n\geq 3$. For _positive_
solutions, however, the subject has a rich history, and pointwise blow-up
estimates for solutions of (1.1) have been known for decades to be a
fundamental technical tool to prove both existence results and _a priori_
bounds.

If no bound on the energy is assumed, sharp blow-up estimates for positive
solutions of (1.1) have been used to prove compactness and stability results.
A remarkable success is the compactness of positive solutions of the Yamabe
equation on manifolds which are not conformally diffeomorphic to the standard
sphere, that corresponds to the case
$h_{\alpha}\equiv\frac{n-2}{4(n-1)}S_{g}$, where $S_{g}$ is the scalar
curvature of $(M,g)$: see Li-Zhu [30], Druet [17], Marques [31], Li-Zhang
[28], Khuri-Marques-Schoen [26] for compactness statements until dimension
$24$ and Brendle [6] and Marques-Brendle [7] for counter-examples when $n\geq
25$. We also mention the stability result for positive solutions of (1.1) of
Druet [18] when $n\geq 3$ and $h<\frac{n-2}{4(n-1)}S_{g}$ in $M$ (for the
definition of what we mean by “stability” here see Hebey [25]). Global degree
arguments based on a sharp pointwise blow-up description of positive solutions
have also been used to prove involved existence and compactness results for
the Nirenberg problem on the sphere, see e.g. the celebrated results of Chen-
Lin [9] or Li [29]. It is worth mentioning that in most of these results one
needs to prove, in the course of the analysis, that possible concentration
points are isolated, thus reducing the problem to understanding isolated one-
bubble blow-up.

This is however specific to the framework of the aforementioned results: even
under the assumption (1.2) of finite-energy blow-up, and as the counter-
examples of Robert-Vétois [48] show, concentration points for sequences
$(u_{\alpha})_{\alpha}$ of positive solutions of (1.1) need not be isolated.
One then needs a pointwise description like that of Theorem 1.1 to describe
the interactions between non-isolated bubbling profiles. Theorem 1.1 for
_positive_ solutions of (1.1) satisfying (1.2) was first proven in Druet-
Hebey-Robert [20], and the proof was subsequently shortened in Druet-Hebey
[19] (see also Hebey [25]). It was then successfully applied to obtain _a
priori_ bounds for finite-energy solutions of (1.1). Druet [17] proved for
instance that, if $n=4,5$ or $n\geq 7$, $h_{\alpha}\to h$ in $C^{1}(M)$ and
$h$ satisfies $h(x)\neq\frac{n-2}{4(n-1)}S_{g}(x)$ for all $x\in M$, every
positive finite-energy sequence of solutions of (1.1) is uniformly bounded in
$C^{2}(M)$ (for details on the six-dimensional case see Druet-Hebey [19]). The
case $h_{\alpha}\equiv h>\frac{n-2}{4(n-1)}S_{g}$ is worth mentioning: no
finite-energy positive blowing-up solutions exist by Druet’s [17] result in
this case but sequences of positive solutions of (1.1) whose energy goes to
infinity have been constructed on $\mathbb{S}^{n}$ by Chen-Wei-Yan [11] for
$n\geq 5$ (see also Vétois-Wang [55] for $n=4$). This intermediate case shows
the relevance of developing analytical tools like Theorem 1.1 to specifically
deal with the finite-energy case. Let us also mention that analogous results
to Theorem 1.1 exist in dimension $2$ for positive solutions of Liouville-type
and Moser-Trudinger-type equations, and have been used to obtain existence and
quantification results. See for instance Chen-Lin [10], Li-Shafrir [27],
Druet-Thizy [21] and the references therein.

For _sign-changing_ solutions of (1.1) the situation is much less understood.
If no bound on the energy is assumed the situation is dimmer than for positive
solutions. As shown by Premoselli-Vétois [43], one can only expect compactness
in dimensions $n\geq 7$, when $h<\frac{n-2}{4(n-1)}S_{g}$ and when the
negative part of the sequence $(u_{\alpha})_{\alpha}$ is a priori bounded. The
counterpart of Druet [18] fails for sign-changing solutions in all the other
cases, as proven by Ding [16] for the Yamabe equation, Premoselli-Vétois [43]
and Vétois [54]. The situation for sign-changing finite-energy blow-up is also
considerably more involved than in the positive case: finite-energy sign-
changing solutions of the Yamabe solutions in $\mathbb{R}^{n}$ are not
classified, hence quantification of the energy is lost, and the complexity of
the pointwise interaction between bubbles $V_{i,\alpha}$ increases. Few sign-
changing finite-energy compactness results are known and this was mainly due,
up to now, to the lack of precise pointwise blow-up estimates as those of
Theorem 1.1. We mention for instance the finite-energy compactness result of
Vétois [54] when $n\geq 7$, $h<\frac{n-2}{4(n-1)}$ and $(M,g)$ is locally
conformally flat, that relies on arguments from Devillanova-Solimini [15] in
the Euclidean case. In another context let us also mention Ghoussoub-Mazumdar-
Robert [24], where the authors obtain stability results and sharp pointwise
blow-up estimates for sign-changing solutions of Hardy-Schrödinger equations
with boundary singularity, a case in which the bubbles are centered at the
singularity. Theorem 1.1 fills this technical gap and provides a pointwise
description of sign-changing blowing-up sequences of (1.1) that is as precise
as in the positive case. We believe that it will pave the way for a better
understanding of finite-energy sign-changing blow-up and for more
sophisticated compactness results.

###  1.2. Strategy of proof and organisation of the paper

When investigating sign-changing solutions one quickly runs into technical
difficulties since a direct adaptation of the techniques developed for
_positive_ ones is ineffective: Harnack inequalities and maximum principles
fail, for instance, and the kernel of the limiting operator $\triangle_{g}+h$
has to be taken into account. We develop here a new and robust strategy of
proof that is quite different from the original proofs in Druet-Hebey-Robert
[20] and Druet-Hebey [19] and from the sign-changing setting of Ghoussoub-
Mazumdar-Robert [24]. Our proof combines a bubble-tree analysis with ideas
from finite-dimensional reduction methods, in the spirit e.g. of Ambrosetti-
Malchiodi [1, 2], Bahri [4], Rey [44], Rey-Wei [45] or Wei [56].

The paper is organised as follows. In Section 2 we collect a few results on
sign-changing solutions of the Yamabe equation in $\mathbb{R}^{n}$, prove
decay estimates for solutions of the linearised equation and introduce some
notations. Section 3 is the core of the analysis of the paper. We linearize
equation (1.1) at an approximate solution that is close to
$u_{0}+\sum_{i=1}^{k}V_{i,\alpha}$ (more general configurations have to be
taken into account) and we prove Theorem 3.2 below. It states that, assuming
that the right-hand side satisfies explicit pointwise estimates, solutions of
the linearized equation inherit them and are globally pointwise small with
respect to the sum of the positive bubbles $\sum_{i=0}^{k}B_{i,\alpha}$. This
is an _a priori_ result that can be seen as a linearised version of Theorem
1.1, and applies in particular when the right-hand side behaves like the error
of the approximate solution $u_{0}+\sum_{i=1}^{k}V_{i,\alpha}$ given by

$\big{(}\triangle_{g}+h_{\alpha}\big{)}\big{(}u_{0}+\sum_{i=1}^{k}V_{i,\alpha}\big{)}-\big{|}u_{0}+\sum_{i=1}^{k}V_{i,\alpha}\big{|}^{2^{*}-2}\big{(}u_{0}+\sum_{i=1}^{k}V_{i,\alpha}\big{)}.$

The proof of Theorem 3.2 goes through an _a priori_ bubble-tree analysis,
where sharp estimates are obtained by an inductive bootstrap procedure in the
bubble-tree. It requires a full _a priori_ analysis and is not a simple
invertibility result in weighted spaces. In Section 4 we first perform a
nonlinear perturbative argument (Proposition 4.2) and prove that (1.1) can be
uniquely solved, up to kernel elements, by a function that satisfies explicit
pointwise estimates; the analysis takes place in strong spaces and heavily
relies on Theorem 3.2. As a consequence we prove a slightly more general
result than Theorem 1.1, Theorem 4.3 below, that shows that approximate
solutions of (1.1) obtained by Lyapunov-Schmidt-like techniques satisfy the
same pointwise bounds as in Theorem 1.1. This is a result which if of interest
in itself. Finally the Appendix collects a few technical results.

The method of proof that we develop in Sections 3 and 4 is particularly robust
and applies to more general equations than (1.1). The invertibility result of
Section 3, Theorem 3.2, allows us to bypass most of the technical difficulties
that arise when proving blow-up pointwise estimates for critical nonlinear
equations, since it only requires to be able to write down a representation
formula for a linear problem. The approach developed in this paper adapts for
instance to higher-order critical nonlinear equations involving linear
operators that look like $(\triangle_{g})^{p}$ for some integer $p\geq 1$ at
first-order. It also seems particularly suited to deal with highly nonlinear
equations.

Acknowledgments: The author was supported by a FNRS CdR grant J.0135.19, by
the Fonds Thélam and by an ARC Avancé 2020 grant.

##  2\. Preliminary material

###  2.1. Finite-energy nodal solutions of the Yamabe equation in
$\mathbb{R}^{n}$.

Let $n\geq 3$ and denote by $D^{1,2}(\mathbb{R}^{n})$ the completion of
$C^{\infty}_{c}(\mathbb{R}^{n})$ with respect to the norm
$u\mapsto\int_{\mathbb{R}^{n}}|\nabla u|^{2}dx$, endowed with the associated
scalar product. Let $V\in D^{1,2}(\mathbb{R}^{n})$ be a solution of the nodal
Yamabe equation in $\mathbb{R}^{n}$:

$\triangle_{\xi}V=|V|^{2^{*}-2}V,$ (2.1)

where $\xi$ is the Euclidean metric and
$\triangle_{\xi}=-\sum_{i=1}^{n}\partial_{i}^{2}$. Unless specified otherwise
$V$ is allowed to change sign. If $V$ is positive it is equal by Obata [37],
up to translations and rescalings, to the so-called positive standard bubble

$B_{0}(x)=\left(1+\frac{|x|^{2}}{n(n-2)}\right)^{-\frac{n-2}{2}},$ (2.2)

and is in particular an extremal for the euclidean sharp Sobolev inequality.
By Caffarelli-Gidas-Spruck [8] this result remains true for all positive
solutions of (2.1), without the assumption that they belong to
$D^{1,2}(\mathbb{R}^{n})$. The positive standard bubbles satisfy

$\int_{\mathbb{R}^{n}}|\nabla B_{0}|^{2}dx=K_{n}^{-n},$

where $K_{n}$, an explicit dimensional constant, is the sharp constant for the
embedding of $D^{1,2}(\mathbb{R}^{n})$ into $L^{2^{*}}(\mathbb{R}^{n})$. By
contrast, any sign-changing solution $V$ of (2.1) satisfies

$\int_{\mathbb{R}^{n}}|\nabla V|^{2}dx>2K_{n}^{-n}$

as is easily seen by splitting $V$ into its positive and negative part. A
result of Weth [57] shows the existence of a dimensional constant
$\varepsilon_{n}>0$ so that every sign-changing solution of (2.1) satisfies
$\int_{\mathbb{R}^{n}}|\nabla V|^{2}dx\geq 2K_{n}^{-n}+\varepsilon_{n}$.

We describe in this subsection the behavior at infinity for solutions of (2.1)
and of the linearized equation. We first recall the following simple result:

######  Lemma 2.1.

Let $V\in D^{1,2}(\mathbb{R}^{n})$ be a solution of (2.1).

  * •

There exists a constant $C=C(n,V)$ such that, for all $x\in\mathbb{R}^{n}$,

$|V(x)|+(1+|x|)|\nabla V(x)|\leq C(1+|x|)^{2-n}.$ (2.3)

  * •

There exists $\lambda\in\mathbb{R}$ such that

$V(x)=\frac{\lambda}{|x|^{n-2}}+O(|x|^{1-n})\quad\textrm{ as }|x|\to+\infty,$
(2.4)

and it satisfies

$(n-2)\omega_{n-1}\lambda=\int_{\mathbb{R}^{n}}|V|^{2^{*}-2}Vdx,$ (2.5)

where $\omega_{n-1}$ is the area of the standard sphere $\mathbb{S}^{n-1}$.

If $V$ has constant sign $\lambda=\pm(n(n-2))^{\frac{n-2}{2}}$ by (2.2).

###### Proof.

First, by the regularity theory for critical elliptic equations (see e.g.
Trudinger [53] or also Hebey [25], Theorem $2.15$), $V$ is smooth in
$\mathbb{R}^{n}$, and (2.3) is true for $x\in\overline{B(0,1)}$. For $x\neq 0$
let $V^{*}(x)=|x|^{2-n}V\big{(}\frac{x}{|x|^{2}}\big{)}$ be the Kelvin
transform of $V$. It is a well-known fact that $V^{*}$ still satisfies (2.1)
on $\mathbb{R}^{n}\backslash\\{0\\}$. The Kelvin transform leaves the
$L^{2^{*}}(\mathbb{R}^{n})$ norm invariant and hence $V^{*}\in
L^{2^{*}}(\mathbb{R}^{n})$. Standard removal of singularities results (see
e.g. the arguments in Caffarelli-Gidas-Spruck [8], Lemma $2.1$) then show that
$V^{*}$ is a distributional solution of (2.1) in the whole $\mathbb{R}^{n}$
and hence belongs to $D^{1,2}(\mathbb{R}^{n})$. Regularity theory shows again
that $V^{*}$ smoothly extends at the origin. Hence there exists a constant
$C=C(n,V)>0$ such that

$\|V^{*}\|_{L^{\infty}(B(0,1))}+\|\nabla V^{*}\|_{L^{\infty}(B(0,1))}\leq C,$

which, by definition of $V^{*}$, proves (2.3) for
$x\in\mathbb{R}^{n}\backslash\overline{B(0,1)}$.

At the origin $V^{*}$ moreover expands as $V^{*}(z)=V^{*}(0)+O(|z|)$. Letting
$z=\frac{x}{|x|^{2}}$ yields (2.4) with $\lambda=V^{*}(0)$ when
$|x|\to+\infty$. To prove (2.5) we write a representation formula for $V$ in
$\mathbb{R}^{n}$. Let $R>0$ be fixed and choose $|x|\geq 2R$. With (2.3) we
can write that, for any $x\in\mathbb{R}^{n}$,

$\displaystyle V(x)$
$\displaystyle=\int_{\mathbb{R}^{n}}\frac{1}{(n-2)\omega_{n-1}}|x-y|^{2-n}|V(y)|^{2^{*}-2}V(y)dy$
$\displaystyle=\int_{B(0,R)}\frac{1}{(n-2)\omega_{n-1}}|x-y|^{2-n}|V(y)|^{2^{*}-2}V(y)dy+O(\frac{1}{R^{2}}(1+|x|)^{2-n}).$

Multiplying by $|x|^{n-2}$ both sides of the equality and letting first
$|x|\to+\infty$ and then $R\to+\infty$ proves (2.5). ∎

If $V\in D^{1,2}(\mathbb{R}^{n})$ solves (2.1) we let

$K_{V}=\Big{\\{}h\in
D^{1,2}(\mathbb{R}^{n}),\quad\triangle_{\xi}h=(2^{*}-1)|V|^{2^{*}-2}h\Big{\\}}$
(2.6)

be the set of solutions of the linearised equation at $V$. A simple
integration by parts yields $V\in K_{V}^{\perp}$. The following result proves
decay at infinity for the elements of $K_{V}$:

######  Lemma 2.2.

Let $Z\in K_{V}$. There exists $C=C(n,V,Z)$ such that

$|Z(x)|+(1+|x|)|\nabla Z(x)|\leq C(1+|x|)^{2-n}$ (2.7)

for all $x\in\mathbb{R}^{n}$.

###### Proof.

Let, for $x\in\mathbb{R}^{n}$, $U(x)^{\frac{4}{n-2}}=4(1+|x|^{2})^{-2}$. Let
$\pi:\mathbb{S}^{n}\backslash\\{N\\}\to\mathbb{R}^{n}$ be the stereographic
projection from the North pole and let, for $\varphi\in
D^{1,2}(\mathbb{R}^{n})$ and for $y\in\mathbb{S}^{n}$,
$\tilde{\varphi}(y)=\frac{\varphi}{U}(\pi(y))$. The conformal invariance of
the conformal laplacian shows that, for any $y\in\mathbb{S}^{n}$,
$\tilde{\varphi}$ satisfies

$\Big{(}\triangle_{g_{0}}+\frac{n(n-2)}{4}\Big{)}\tilde{\varphi}(y)=U(\pi(y))^{1-2^{*}}\triangle_{\xi}\varphi(\pi(y))$
(2.8)

where $g_{0}$ is the round metric on $\mathbb{S}^{n}$. Recall that $g_{0}$
satisfies $g_{0}=\pi^{*}\big{(}U^{\frac{4}{n-2}}\xi)$. Let $Z\in K_{V}$ and
consider $\tilde{Z}$. The assumption $Z\in D^{1,2}(\mathbb{R}^{n})$
translates, by (2.8), into

$\int_{\mathbb{S}^{n}}|\nabla\tilde{Z}|_{g_{0}}^{2}+\frac{n(n-2)}{4}|\tilde{Z}|^{2}dv_{g_{0}}=\int_{\mathbb{R}^{n}}|\nabla
Z|^{2}dx<+\infty,$

so that $\tilde{Z}\in H^{1}(\mathbb{S}^{n})$. And by (2.6) $\tilde{Z}$
satisfies

$\triangle_{g_{0}}\tilde{Z}+b\tilde{Z}=0\quad\textrm{ in }\mathbb{S}^{n},$
(2.9)

where we have let

$b=\frac{n(n-2)}{4}-(2^{*}-1)\left(\frac{|V(\pi(\cdot))|}{U(\pi(\cdot))}\right)^{2^{*}-2}.$

Elements of $K_{V}$ are actually in one-to-one correspondence with solutions
$\tilde{Z}\in H^{1}(\mathbb{S}^{n})$ of (2.9) via the transformation
$Z\to\tilde{Z}$. Estimate (2.3) implies
$\|b\|_{L^{\infty}(\mathbb{S}^{n})}\leq C(n,V)$. Standard elliptic theory for
(2.9) then shows that $\|\tilde{Z}\|_{C^{1}(\mathbb{S}^{n})}\leq C(n,V)$.
Since, for all $x\in\mathbb{R}^{n}$, $Z(x)=U(x)\tilde{Z}(\pi^{-1}(x))$, this
proves (2.7). ∎

In the following we will note $n_{V}=\dim K_{V}$ and we will denote by
$\Pi_{K_{V}}$ the orthogonal projection on $K_{V}$ in
$D^{1,2}(\mathbb{R}^{n})$. We always have $n_{V}>0$, and this follows e.g.
from the invariance of solutions of (2.1) by scaling. In general little more
can be said on the dimension of $K_{V}$ as in (2.6). It is known that for
_positive_ solutions of (2.1) $n_{V}=n+1$ holds (see Bianchi-Egnell [5]) and
that positive solutions are non-degenerate in the sense of Duyckaerts-Kenig-
Merle [22]. Constructive examples of non-degenerate nodal solutions of (2.1)
whose value of $n_{V}$ is known have been obtained in Del Pino-Musso-Pacard-
Pistoia [12, 13], Medina-Musso [32], Medina-Musso-Wei [33] and Musso-Wei [36].
No example of _degenerate_ solution of (2.1) is known to this day, but the
existence of such solutions has not been ruled out.

###  2.2. Riemannian sign-changing bubbles

We let until the end of this section $(M,g)$ be a closed manifold (that is,
compact without boundary) of dimension $n\geq 3$ and let $h\in L^{\infty}(M)$.
Let $(x_{\alpha})_{\alpha}$ be a sequence of points in $M$ and
$(\mu_{\alpha})_{\alpha}$ be a sequence of positive numbers and let $V\in
D^{1,2}(\mathbb{R}^{n})$ be a solution of (2.1). Let $\lambda\in\mathbb{R}$ be
given by (2.4). Let $\chi\in C^{\infty}_{c}(\mathbb{R})$ be such that
$\chi\equiv 1$ in $[0,\frac{i_{g}}{2}]$ and $\chi\equiv 0$ in
$\mathbb{R}\backslash[0,i_{g}]$, where $i_{g}$ denotes the injectivity radius
of $g$. We define a riemannian sign-changing bubble
$V^{\mu_{\alpha},x_{\alpha}}$ on $M$, modeled on $V$, as follows:

$\displaystyle\bullet\textrm{ If }\ker(\triangle_{g}+h)=\\{0\\}:$ (2.10)
$\displaystyle V^{\mu_{\alpha},x_{\alpha}}(x)$
$\displaystyle=\chi\big{(}d_{g}(x_{\alpha},x)\big{)}\mu_{\alpha}^{-\frac{n-2}{2}}V\Big{(}\frac{1}{\mu_{\alpha}}\exp_{x_{\alpha}}^{-1}(x)\Big{)}$
$\displaystyle+\big{(}1-\chi\big{(}d_{g}(x_{\alpha},x)\big{)}\big{)}(n-2)\omega_{n-1}\lambda\mu_{\alpha}^{\frac{n-2}{2}}G_{h}(x_{\alpha},x)$
$\displaystyle\bullet\textrm{ If }\ker(\triangle_{g}+h)\neq\\{0\\}:$
$\displaystyle V^{\mu_{\alpha},x_{\alpha}}(x)$
$\displaystyle=\chi\big{(}d_{g}(x_{\alpha},x)\big{)}\mu_{\alpha}^{-\frac{n-2}{2}}V\Big{(}\frac{1}{\mu_{\alpha}}\exp_{x_{\alpha}}^{-1}(x)\Big{)}$

for all $x\in M$, where $\exp_{x_{\alpha}}$ denotes the exponential map at
$x_{\alpha}$ for $g$, and where in the first case $G_{h}$ denotes the Green’s
function of $\triangle_{g}+h$ (see Aubin [3] or Robert [46]). We also let, for
$x\in M$,

$B_{\mu_{\alpha},x_{\alpha}}(x)=\frac{\mu_{\alpha}^{\frac{n-2}{2}}}{\big{(}\mu_{\alpha}^{2}+\frac{d_{g}(x_{\alpha},x)^{2}}{n(n-2)}\big{)}^{\frac{n-2}{2}}}$
(2.11)

be the Riemannian positive standard bubble centered at $x_{\alpha}$. It is
easily seen with (2.3) and (2.10) that we have

$|V^{\mu_{\alpha},x_{\alpha}}(x)|+(\mu_{\alpha}+d_{g}(x_{\alpha},x))|\nabla
V^{\mu_{\alpha},x_{\alpha}}(x)|\leq CB_{\mu_{\alpha},x_{\alpha}}(x)$ (2.12)

for all $x\in M$, where $C$ only depends on $n,(M,g)$ and $V$. Let
$(h_{\alpha})_{\alpha}$ be a sequence of functions in $M$ that converges
towards $h$ in $L^{\infty}(M)$. Straightforward computations using (2.10) show
that there exists $C=C(n,g,V,h)>0$ such that

$\left|\triangle_{g}V^{\mu_{\alpha},x_{\alpha}}+h_{\alpha}V^{\mu_{\alpha},x_{\alpha}}-|V^{\mu_{\alpha},x_{\alpha}}|^{2^{*}-2}V^{\mu_{\alpha},x_{\alpha}}\right|\leq
CB_{\mu_{\alpha},x_{\alpha}}$ (2.13)

everywhere in $M$ (see e.g. Esposito-Pistoia-Vétois [23], proposition 2.2).
Let $k\geq 1$ be an integer and let $(x_{i,\alpha})_{\alpha}$, $1\leq i\leq
k$, be $k$ sequence of points of $M$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq
i\leq k$ be $k$ sequences of positive real numbers. We will let throughout
this paper:

$V_{i,\alpha}=V_{i}^{\mu_{i,\alpha},x_{i,\alpha}},$ (2.14)

where $V_{i}^{\mu_{i,\alpha},x_{i,\alpha}}$ is given by (2.10).

###  2.3. Kernel elements

We endow $H^{1}(M)$ with the scalar product

$\langle u,v\rangle_{H^{1}(M)}=\int_{M}\Big{(}\langle\nabla u,\nabla
v\rangle+uv\Big{)}dv_{g}.$

Let $h\in L^{\infty}(M)$ and let $u_{0}\in C^{1,\beta}(M)$, $0<\beta<1$, be a
solution of

$\triangle_{g}u_{0}+hu_{0}=|u_{0}|^{2^{*}-2}u_{0}.$ (2.15)

Let $k\geq 1$, let $V_{1},\dots,V_{k}$ be solutions of (2.1) and let
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$, be
$k$ sequences respectively of points in $M$ and of positive real numbers with
$\mu_{i,\alpha}\to 0$ as $\alpha\to+\infty$ satisfying (1.4).

For each $1\leq i\leq k$ let $n_{i}=\textrm{dim}K_{V_{i}}$ where $K_{V_{i}}$
is as in (2.6) and let $Z_{i,1},\dots,Z_{i,n_{i}}$ be an orthonormal basis of
$K_{V_{i}}$. As before let $\chi\in C^{\infty}_{c}(\mathbb{R})$ satisfy
$\chi\equiv 1$ in $[0,\frac{i_{g}}{2}]$ and $\chi\equiv 0$ in
$\mathbb{R}\backslash[0,i_{g}]$. For any $\alpha\geq 1$, $1\leq i\leq k$ and
$1\leq j\leq n_{i}$ we let, for $x\in M$,

$Z_{i,j,\alpha}(x)=\chi(d_{g}(x_{i,\alpha},x))\mu_{i,\alpha}^{-\frac{n-2}{2}}Z_{i,j}\Big{(}\frac{1}{\mu_{i,\alpha}}\exp_{x_{i,\alpha}}^{-1}(x)\Big{)}$
(2.16)

and

$K_{i,\alpha}=\textrm{Span}\big{\\{}Z_{i,j,\alpha},1\leq j\leq n_{i}\big{\\}}$

which is a subset of $H^{1}(M)$. We also let

$K_{0}=\Big{\\{}v\in
H^{1}(M),\quad\triangle_{g}v+hv=(2^{*}-1)|u_{0}|^{2^{*}-2}v\Big{\\}},$ (2.17)

and we let $(Z_{0,j})_{1\leq j\leq n_{0}}$ be an orthonormal basis of $K_{0}$
for the $H^{1}(M)$ scalar product. In case $u_{0}\equiv 0$, $K_{0}$ coincides
with the kernel of $\triangle_{g}+h$. Let finally, for $\alpha\geq 1$,

$K_{\alpha}=K_{0}\oplus\bigoplus_{i=1}^{k}K_{i,\alpha}.$ (2.18)

If follows from (2.16) that $Z_{i,j,\alpha}\rightharpoonup 0$ in $H^{1}(M)$ as
$\alpha\to+\infty$. Using (3.8) below we then obtain that, up to passing to a
subsequence for $(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$,
$1\leq i\leq k$, $K_{\alpha}$ is a linear subset of $H^{1}(M)$ of dimension
$\sum_{i=0}^{k}n_{i}$. Note that $n_{0}=\dim K_{0}$ may be equal to $0$, in
which case our convention is that the condition $1\leq j\leq n_{0}$ is empty.

##  3\. Linear blow-up analysis

In this section we linearise equation (1.1) around the first-order terms
$u_{0}+\sum_{i=1}^{k}V_{i,\alpha}$ in the Struwe decomposition (1.3). The main
result of this section, Theorem 3.2, establishes _a priori_ pointwise
estimates for solutions of the linearised equation. This result holds true for
any possible bubbling configuration.

###  3.1. Linear blow-up analysis

Let

$\mathcal{E}=\big{\\{}(\varepsilon_{\alpha})_{\alpha\geq 1}\textrm{ such that
}0<\varepsilon_{\alpha}\leq 1\textrm{ for all }\alpha\geq 1\textrm{ and
}\lim_{\alpha\to+\infty}\varepsilon_{\alpha}=0\big{\\}}.$ (3.1)

Let $(h_{\alpha})_{\alpha}$ be a sequence of functions converging towards $h$
in $L^{\infty}(M)$ as $\alpha\to+\infty$. Let $k\geq 1$ be an integer, let
$u_{0}\in C^{1,\beta}(M)$ for all $0<\beta<1$ (possibly equal to zero) solve
(2.15), let $V_{1},\dots,V_{k}$ in $D^{1,2}(\mathbb{R}^{n})$ be $k$ solutions
of (2.1) and let $(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$,
$1\leq i\leq k$ be $k$ sequences of points in $M$ and positive real numbers
satisfying (1.4). For $i\in\\{1,\dots,k\\}$ and $x\in M$ we define

$\theta_{i,\alpha}(x)=\mu_{i,\alpha}+d_{g}(x_{i,\alpha},x),$ (3.2)
$\Theta_{i,\alpha}(x)=\left\\{\begin{aligned} &\theta_{i,\alpha}(x)&\textrm{
if }n=3\\\
&\theta_{i,\alpha}^{2}(x)\big{|}\ln\theta_{i,\alpha}(x)\big{|}&\textrm{ if
}n=4\\\ &\theta_{i,\alpha}^{2}(x)&\textrm{ if }n\geq 5\end{aligned}\right.$
(3.3)

and

$\displaystyle B_{0,\alpha}(x)$ $\displaystyle=\left\\{\begin{aligned}
&0&\textrm{ if }u_{0}\equiv 0\textrm{ and }\ker(\triangle_{g}+h)=\\{0\\}\\\
&1&\textrm{ otherwise}\end{aligned}\right.$ (3.4) $\displaystyle
B_{i,\alpha}(x)$ $\displaystyle=B_{\mu_{i,\alpha},x_{i,\alpha}}(x)\textrm{ for
}1\leq i\leq n,$

where $B_{\mu_{i,\alpha},x_{i,\alpha}}$ is as in (2.11). By definition
$B_{0,\alpha}\equiv 0\iff u_{0}\equiv 0$ and $K_{0}=\\{0\\}$, where $K_{0}$ is
as in (2.17). We will let, throughout this section and the next one,
$(\mathcal{W}_{\alpha})_{\alpha}$ be a sequence of continuous functions in $M$
satisfying

$\mathcal{W}_{\alpha}=u_{0,\alpha}+\sum_{i=1}^{k}W_{i,\alpha},$ (3.5)

where $u_{0,\alpha}\equiv 0$ if $u_{0}\equiv 0$ and
$\ker(\triangle_{g}+h)=\\{0\\}$ and
$\|u_{0,\alpha}-u_{0}\|_{L^{\infty}(M)}\leq 1$ and
$\|u_{0,\alpha}-u_{0}\|_{L^{\infty}(M)}\to 0$ otherwise, and where, for $1\leq
i\leq k$,

$\left\\{\begin{aligned} &|W_{i,\alpha}(x)|\leq CB_{i,\alpha}(x)\quad\textrm{
for all }x\in M\\\ &|\nabla W_{i,\alpha}(x)|\leq
C\theta_{i,\alpha}(x)^{-1}B_{i,\alpha}(x)\quad\textrm{ for all }x\in M\\\
&\mu_{i,\alpha}^{\frac{n-2}{2}}W_{i,\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\to
V_{i}(y)\quad\textrm{ in }\quad
C^{1}_{loc}(\mathbb{R}^{n})\end{aligned}\right.$

as $\alpha\to+\infty$, for some $C>0$ that only depends on
$n,(M,g),V_{1},\dots V_{k},u_{0}$ and $h$.

We will repeatedly make use of the following notations:

  * •

If $(a_{\alpha})_{\alpha}$ and $(b_{\alpha})_{\alpha}$ are sequence of real
numbers (or functions) with $b_{\alpha}>0$ we will write

$|a_{\alpha}|\lesssim b_{\alpha}\textrm{ or, equivalently,
}a_{\alpha}=O(b_{\alpha})$ (3.6)

if there exists a positive constant $C$ that only depends on $n,(M,g),k$,
$V_{1},\dots,V_{k},u_{0}$ and $h$ such that $|a_{\alpha}|\leq Cb_{\alpha}$. In
particular such a constant $C$ is independent of $\alpha$ and of the choice of
$(\mu_{i,\alpha})_{\alpha}$, $(x_{i,\alpha})_{\alpha}$, $1\leq i\leq k$. For
sequences of real numbers we will denote $a_{\alpha}=o(b_{\alpha})$ if
$\frac{a_{\alpha}}{b_{\alpha}}\to 0$ as $\alpha\to+\infty$.

  * •

We will denote by $\sigma_{\alpha}$ any sequence of positive numbers
converging to $0$ that only depends on the sequences
$(\|h_{\alpha}-h\|_{L^{\infty}(M)})_{\alpha}$, $(x_{i,\alpha})_{\alpha}$ and
$(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$. Schematically:

$\sigma_{\alpha}=\sigma_{\alpha}\Big{(}\|h_{\alpha}-h\|_{L^{\infty}(M)},(x_{i,\alpha})_{1\leq
i\leq k},(\mu_{i,\alpha})_{1\leq i\leq
k}\Big{)},\quad\lim_{\alpha\to+\infty}\sigma_{\alpha}=0.$ (3.7)

The notation $\sigma_{\alpha}$ might denote, from one line to the other,
different sequences still satisfying (3.7). In some cases, as for instance in
the proof of Proposition 3.5 below, several sequences of this type will have
to be distinguished, and they will be numbered as
$\sigma_{\alpha}^{1},\sigma_{\alpha}^{2},\dots$.

If $i,j,p,q$ are integers satisfying $0\leq i,j\leq k$, $1\leq p\leq n_{i}$
and $1\leq q\leq n_{j}$ we have for instance

$\langle
Z_{i,p,\alpha},Z_{j,q,\alpha}\rangle_{H^{1}(M)}=\delta_{ij}\delta_{pq}+O(\sigma_{\alpha}),$
(3.8)

and this follows from (2.16). For simplicity in (3.8) we have let, when $i=0$,
$Z_{0,j,\alpha}=Z_{0,j}$, $1\leq j\leq n_{0}$, where $Z_{0,j}$ spans $K_{0}$.
Similarly, by (2.15) we have

$\|\triangle_{g}u_{0}+h_{\alpha}u_{0}-|u_{0}|^{2^{*}-2}u_{0}\|_{L^{\infty}(M)}\leq\sigma_{\alpha}.$

Let now, for all $s\in\mathbb{R}$,

$f(s)=|s|^{2^{*}-2}s$ (3.9)

and let $(\mathcal{W}_{\alpha})_{\alpha}$ be a sequence of continuous
functions satisfying (3.5). Our goal is to obtain pointwise estimates on
solutions of the linearized equation of (1.1) at $\mathcal{W}_{\alpha}$. We
first recall the standard linear theory in $H^{1}(M)$. For $\varphi\in
K_{\alpha}^{\perp}$, where $K_{\alpha}$ is as in (2.18), let
$L_{\alpha}:K_{\alpha}^{\perp}\to K_{\alpha}^{\perp}$ be given by

$L_{\alpha}\varphi=\Pi_{K_{\alpha}^{\perp}}\Big{[}\varphi-(\triangle_{g}+1)^{-1}\Big{(}f^{\prime}(\mathcal{W}_{\alpha})\varphi+(1-h_{\alpha})\varphi\Big{)}\Big{]}$

where $\Pi_{K_{\alpha}^{\perp}}$ denotes the orthogonal projection on
$K_{\alpha}^{\perp}$ for the $H^{1}(M)$ scalar product. The following is a
well-known result:

######  Proposition 3.1.

There exists a constant $C=C(n,g,V_{1},\dots,V_{k},u_{0},h)$ such that for
$\alpha$ large enough

$\frac{1}{C}\|\varphi\|_{H^{1}(M)}\leq\|L_{\alpha}\varphi\|_{H^{1}(M)}\leq
C\|\varphi\|_{H^{1}(M)}\quad\textrm{ for all }\varphi\in K_{\alpha}^{\perp}.$

In particular $L_{\alpha}:K_{\alpha}^{\perp}\to K_{\alpha}^{\perp}$ is a
bicontinuous isomorphism.

The proof of Proposition 3.1 follows from standard arguments that can be found
e.g. in Robert-Vétois [49], Esposito-Pistoia-Vétois [23] or Musso-Pistoia
[35]. It shows in particular that for any $\alpha\geq 1$, if $R_{\alpha}$ is a
continuous function in $M$, there exists a unique $\varphi_{\alpha}\in
K_{\alpha}^{\perp}$ and unique real numbers $\lambda_{i,j}^{\alpha}$, $0\leq
i\leq k$, $1\leq j\leq n_{i}$, such that

$\triangle_{g}\varphi_{\alpha}+h_{\alpha}\varphi_{\alpha}-f^{\prime}(\mathcal{W}_{\alpha})\varphi_{\alpha}=R_{\alpha}+\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}.$

By standard elliptic theory $\varphi_{\alpha}\in C^{1}(M)$. The main result of
this section describes how $\varphi_{\alpha}$ inherits pointwise estimates
from $R_{\alpha}$:

######  Theorem 3.2.

Let $(M,g)$ be a closed Riemannian manifold of dimension $n\geq 3$, let
$(h_{\alpha})_{\alpha}$ be a sequence of functions converging towards $h$ in
$L^{\infty}(M)$, $k\geq 1$ be an integer, $u_{0}\in C^{1,\beta}(M)$ for
$0<\beta<1$ be a solution of (2.15), $V_{1},\dots,V_{k}$ in
$D^{1,2}(\mathbb{R}^{n})$ be solutions of (2.1) and let
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ be
$k$ sequences satisfying (1.4). Let $(\mathcal{W}_{\alpha})_{\alpha}$ be a
sequence of continuous functions in $M$ satisfying (3.5). There exist
$C_{0}=C_{0}(n,g,V_{1},\dots,V_{k},u_{0},h)>0$ and there exists a sequence
$(\sigma_{\alpha})_{\alpha}$ as in (3.7) such that, up to passing to a
subsequence for $(\sigma_{\alpha})_{\alpha}$, the following holds:

Let $(\eta_{\alpha})_{\alpha},(\tau_{\alpha})_{\alpha}$ be sequences in
$\mathcal{E}$, $(\gamma_{\alpha})_{\alpha}$ be a bounded sequence of real
numbers and $(R_{\alpha})_{\alpha}$ be a sequence of continuous functions in
$M$ satisfying

$|R_{\alpha}|\leq\tau_{\alpha}B_{0,\alpha}+\eta_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}+\gamma_{\alpha}\Big{(}\sum_{i=1}^{k}B_{i,\alpha}+\sum\limits_{\underset{i\neq
j}{i,j=0..k}}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}$ (3.10)

in $M$. Let $\varphi_{\alpha}\in K_{\alpha}^{\perp}$ be the unique solution of

$\triangle_{g}\varphi_{\alpha}+h_{\alpha}\varphi_{\alpha}-f^{\prime}(\mathcal{W}_{\alpha})\varphi_{\alpha}=R_{\alpha}+\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}$
(3.11)

given by Proposition 3.1. Then $\varphi_{\alpha}$ satisfies:

$|\varphi_{\alpha}(x)|\leq
C_{0}\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\sum_{i=0}^{k}B_{i,\alpha}(x)+C_{0}\gamma_{\alpha}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i}(x)$
(3.12)

for all $\alpha$ and for all $x\in M$. Also, for all $\alpha$,

$\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}|\lambda_{i,j}^{\alpha}|\leq
C_{0}\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}.$
(3.13)

The quantities $\mathcal{E},B_{0,\alpha},B_{i,\alpha}$ and $\Theta_{i,\alpha}$
have been defined in (3.1), (3.3) and (3.4). As before we have let
$Z_{0,j,\alpha}=Z_{0,j}$, $1\leq j\leq n_{0}$, where $Z_{0,j}$ spans $K_{0}$.
Note that by (3.7), passing to a subsequence for $\sigma_{\alpha}$ amounts to
passing to a common subsequence for $(x_{i,\alpha})_{\alpha}$,
$(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$, and $(h_{\alpha})_{\alpha}$.

Theorem 3.2 is a generalisation of Proposition 3.1 where the $H^{1}$ control
is improved into effective pointwise estimates. When $\mathcal{W}_{\alpha}$ is
given by $\mathcal{W}_{\alpha}=u_{0}+\sum_{i=1}^{k}V_{i,\alpha}$, (3.10) is
satisfied by
$R_{\alpha}=\triangle_{g}\mathcal{W}_{\alpha}+h_{\alpha}\mathcal{W}_{\alpha}-|\mathcal{W}_{\alpha}|^{2^{*}-2}\mathcal{W}_{\alpha}$,
and this follows from (2.13). Theorem 3.2 has thus to be interpreted as a
linearised version of Theorem 1.1. Here is an alternative way to see (3.12):
assume, for simplicity, that $\triangle_{g}+h$ has no kernel and let $G_{h}$
be its Green’s function in $M$. Direct computations using (3.10) show that
there exists a sequence $(\sigma_{\alpha})_{\alpha}$ as in (3.7) such that for
all $x\in M$

$\displaystyle\int_{M}G_{h}(x,\cdot)R_{\alpha}dv_{g}$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\sum_{i=0}^{k}B_{i,\alpha}(x)$
$\displaystyle+\gamma_{\alpha}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i}(x)$

holds (see (3.29) below for more details). Theorem 3.2 thus essentially says
that the solution $\varphi_{\alpha}$ of (3.11) behaves as if it formally
satisfied the equation

$\triangle_{g}u+hu=R_{\alpha}.$

A similar interpretation remains true if $\triangle_{g}+h$ has non-zero
kernel, where the projection of $\varphi_{\alpha}$ on the kernel of
$\triangle_{g}+h$ has to be taken into account. The method of proof of Theorem
3.2 is flexible and if a more precise estimate than (3.10) is available on
$R_{\alpha}$ then one recovers a better estimate on $\varphi_{\alpha}$ than
(3.12): roughly speaking one expects $\varphi_{\alpha}$ to satisfy the same
pointwise estimates as $(\triangle_{g}+h)^{-1}R_{\alpha}$. Sharper estimates
than (3.12) have for instance been obtained in Premoselli [41] when $n\geq 7$
and $\mathcal{W}_{\alpha}$ looks like a tower of positive bubbles (possibly
with alternating signs). We also mention Deng-Sun-Wei [14] who obtained by
different arguments pointwise estimates on solutions of the linearised Yamabe
equation in $\mathbb{R}^{n}$ at specific configurations of sums of finitely
many positive bubbles. An important remark here is that our proof of Theorem
3.2 does not require any non-degeneracy assumption in the sense of Duyckaerts-
Kenig-Merle [22] on the possibly sign-changing bubbles $V_{1},\dots,V_{k}$:
any finite-energy solution of (2.1) is allowed in (3.5). This is the main
reason why we cannot expect, in this paper, to get a better control than
(3.12) on $\varphi_{\alpha}$. We refer to Remark 4.7 below for a more detailed
discussion on this point.

The following four subsections are devoted to the proof of Theorem 3.2.
Subsection 3.2 introduces the notion of radius of influence of a nodal bubble
and proves pointwise estimates at different scales of the bubble-tree.
Subsection 3.3 and 3.4 are devoted to the core of the proof, where improved
estimates in the spirit of (3.12) are proven on the area of influence of each
bubble by an inductive argument. Finally subsection 3.5 concludes the proof of
Theorem 3.2.

###  3.2. Radius of influence and bubble-tree computations

Let $k\geq 1$, $V_{1},\dots,V_{k}$ in $D^{1,2}(\mathbb{R}^{n})$ be $k$
solutions of (2.1), $u_{0}$ be a solution of (2.15) and
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ be
$k$ sequences of points in $M$ and of positive real numbers satisfying (1.4).
Let $(\mathcal{W}_{\alpha})_{\alpha}$ be as in (3.5). Recall that
$u_{0},V_{1},\dots,V_{k}$ can change sign. In this subsection we introduce the
definition of the radius of influence of a nodal bubble and prove a few
related results. The exposition is strongly inspired from Druet [17] and
Druet-Hebey [19] (see also Hebey [25], Chapter $8$) where this notion was
developed for positive solutions of (1.1). Remarkably, it remains extremely
useful even to describe the pointwise interactions of nodal bubbles. In the
following we will sometimes refer to the sum

$\mathcal{W}_{\alpha}=u_{0,\alpha}+\sum_{i=1}^{k}W_{i,\alpha}$

as a _bubble-tree_. We follow notations (3.6) and (3.7) in this section.

For $i\in\\{1,\dots,k\\}$ we define

$\mathcal{A}_{i}=\Big{\\{}j\in\\{1,\dots,k\\}\backslash\\{i\\}\textrm{ such
that }\mu_{i,\alpha}=O(\mu_{j,\alpha})\Big{\\}}.$ (3.14)

These are the bubbles in the family $V_{1,\alpha},\dots,V_{k,\alpha}$ that
concentrate at a slower or comparable speed than the $i$-th bubble – the so-
called _lower_ bubbles. If $j\in\mathcal{A}_{i}$, we define the radius of
interaction of the $i$-th bubble with the lower $j$-th bubble as

$s_{i,j,\alpha}=\Big{(}\frac{\mu_{i,\alpha}}{\mu_{j,\alpha}}\frac{d_{g}(x_{i,\alpha},x_{j,\alpha})^{2}}{n(n-2)}+\mu_{i,\alpha}\mu_{j,\alpha}\Big{)}^{\frac{1}{2}}.$
(3.15)

The radius of influence of the $i$-th bubble is then defined as follows

$r_{i,\alpha}=\left\\{\begin{aligned}
&\min\Big{(}\min_{j\in\mathcal{A}_{i}}s_{i,j,\alpha},\frac{i_{g}}{2}\Big{)}&\textrm{
if }u_{0}\equiv 0\textrm{ and }\ker(\triangle_{g}+h)=\\{0\\}\\\
&\min\Big{(}\min_{j\in\mathcal{A}_{i}}s_{i,j,\alpha},\sqrt{\mu_{i,\alpha}}\Big{)}&\textrm{
otherwise }\\\ \end{aligned}\right..$ (3.16)

If $\mathcal{A}_{i}=\emptyset$ we have in particular

$r_{i,\alpha}=\left\\{\begin{aligned} &\frac{i_{g}}{2}&\textrm{ if
}u_{0}\equiv 0\textrm{ and }\ker(\triangle_{g}+h)=\\{0\\}\\\
&\sqrt{\mu_{i,\alpha}}&\textrm{ otherwise }\\\ \end{aligned}\right..$

The radius $r_{i,\alpha}$ is defined so that
$r_{i,\alpha}\leq\sqrt{\mu_{i,\alpha}}$ whenever $B_{0,\alpha}\equiv 1$ (see
(3.4)), so that $B_{i,\alpha}\gtrsim B_{0,\alpha}$ on
$B_{g}(x_{i,\alpha},r_{i,\alpha})$. By (1.4) we have that
$\frac{s_{i,j,\alpha}}{\mu_{i,\alpha}}\to+\infty$ for all
$j\in\mathcal{A}_{i}$ and hence
$\frac{r_{i,\alpha}}{\mu_{i,\alpha}}\to+\infty$ as $\alpha\to+\infty$. Define
now

$\Lambda=\max_{i\neq
j,\mu_{i,\alpha}\asymp\mu_{j,\alpha}}\limsup_{\alpha\to+\infty}\frac{\mu_{i,\alpha}}{\mu_{j,\alpha}}$

where, for $i\neq j$, $\mu_{i,\alpha}\asymp\mu_{j,\alpha}$ means that there
exists $C>0$ independent of $\alpha$ such that
$\frac{1}{C}\mu_{i,\alpha}\leq\mu_{j,\alpha}\leq C\mu_{i,\alpha}$ up to a
subsequence. Assume that there exist $i\neq j$ such that
$\mu_{i,\alpha}\asymp\mu_{j,\alpha}$. The structure relation (1.4) shows that
$\frac{d_{g}(x_{i,\alpha},x_{j,\alpha})}{\mu_{i,\alpha}}\to+\infty$ and we
thus have by (3.15) and (3.16) that for $\alpha$ large enough

$r_{i,\alpha}+r_{j,\alpha}\leq s_{i,j,\alpha}+s_{j,i,\alpha}\leq
2\sqrt{\frac{\Lambda}{n(n-2)}}d_{g}(x_{i,\alpha},x_{j,\alpha}).$

Let $R_{0}>2$ be large enough so that, up to a subsequence,

$\displaystyle\textrm{ for all }i\neq j\textrm{ such that
}\mu_{i,\alpha}\asymp\mu_{j,\alpha},\quad\frac{2r_{i,\alpha}}{R_{0}}\leq\frac{1}{4}d_{g}(x_{i,\alpha},x_{j,\alpha}).$
(3.17)

In particular

$B_{g}\Big{(}x_{i,\alpha},\frac{2r_{i,\alpha}}{R_{0}}\Big{)}\cap
B_{g}\Big{(}x_{j,\alpha},\frac{2r_{j,\alpha}}{R_{0}}\Big{)}=\emptyset$

for all $i\neq j$ with $\mu_{i,\alpha}\asymp\mu_{j,\alpha}$. The following
lemma follows from the definitions of $r_{i,\alpha}$ and $R_{0}$ and its proof
is omitted:

######  Lemma 3.3.

Let $i\in\\{1,\dots,k\\}$ and $j\in\mathcal{A}_{i}$. Then

$B_{j,\alpha}(x)\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{s_{i,j,\alpha}^{n-2}}\quad\textrm{
and }\quad|\nabla
B_{j,\alpha}(x)|\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{s_{i,j,\alpha}^{n-1}}\quad\textrm{
for all }x\in B_{g}\Big{(}x_{i,\alpha},\frac{2r_{i,\alpha}}{R_{0}}\Big{)},$

where $B_{j,\alpha}$ is as in (3.4). In particular,

$|V_{j,\alpha}|\lesssim B_{i,\alpha}\quad\textrm{ and }\quad|\nabla
V_{j,\alpha}|\lesssim\theta_{i,\alpha}^{-1}B_{i,\alpha}$

in $B_{g}\Big{(}x_{i,\alpha},\frac{2r_{i,\alpha}}{R_{0}}\Big{)}$, where
$V_{j,\alpha}$ is as in (2.14) and $\theta_{i,\alpha}$ as in (3.2).

We still define here the radii of influence $r_{i,\alpha}$ by comparing the
heights of the positive bubbles $B_{i,\alpha}$, just as in the positive case
of Druet [17] (see also Druet-Hebey [19]). This is not surprising since,
although $V_{i,\alpha}$ might vanish, its value at a generic point is of the
order of $B_{i,\alpha}$ by (2.12) and (2.14). We will refer to the ball
$B_{g}(x_{i,\alpha},\frac{2r_{i,\alpha}}{R_{0}})$ as _the area of influence of
$V_{i,\alpha}$_.

Any bubble in the bubble-tree also interacts pointwise with bubbles
concentrating strictly faster – the so-called _higher_ bubbles. In our proof
of Theorem 3.2 only the higher bubbles interacting with $B_{i,\alpha}$ in its
area of influence will have to be taken into account. Let
$i\in\\{1,\dots,k\\}$. We define:

$\mathcal{B}_{i}=\Big{\\{}j\in\\{1,\dots,k\\}\backslash\\{i\\}\textrm{ s. t.
}\mu_{j,\alpha}=o(\mu_{i,\alpha})\textrm{ and
}d_{g}(x_{i,\alpha},x_{j,\alpha})\leq\frac{2r_{i,\alpha}}{R_{0}}\Big{\\}}.$
(3.18)

We then have the following lemma:

######  Lemma 3.4.

For any $x\in B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})$ we have

$\displaystyle|\mathcal{W}_{\alpha}(x)|$
$\displaystyle\lesssim\sum_{j=0}^{k}B_{j,\alpha}(x)\lesssim
B_{i,\alpha}(x)+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x)\quad\textrm{ and }$
(3.19) $\displaystyle|\nabla\mathcal{W}_{\alpha}(x)|$ $\displaystyle\lesssim
B_{0,\alpha}(x)+\sum_{j=1}^{k}\theta_{j,\alpha}(x)^{-1}B_{j,\alpha}(x)$
$\displaystyle\lesssim\theta_{i,\alpha}(x)^{-1}B_{i,\alpha}(x)+\sum_{j\in\mathcal{B}_{i}}\theta_{j,\alpha}(x)^{-1}B_{j,\alpha}(x).$

###### Proof.

By (3.5) and (2.12) we have, for any $x\in M$

$|\mathcal{W}_{\alpha}|\lesssim
B_{0,\alpha}(x)+\sum_{j=1}^{k}B_{j,\alpha}(x)\textrm{ and
}|\nabla\mathcal{W}_{\alpha}|\lesssim
B_{0,\alpha}(x)+\sum_{j=1}^{k}\theta_{j,\alpha}(x)^{-1}B_{j,\alpha}(x).$

By (3.2) we have $\theta_{i,\alpha}\lesssim 1$ in $M$ and by definition of
$r_{i,\alpha}$ we always have $B_{i,\alpha}(x)\gtrsim B_{0,\alpha}(x)$ for
$x\in B_{g}(x_{i,\alpha},\frac{2r_{i,\alpha}}{R_{0}})$. Let $j\neq i$. If
$j\in\mathcal{A}_{i}$, Lemma 3.3 shows that $B_{j,\alpha}\lesssim
B_{i,\alpha}$ and $|\nabla
B_{j,\alpha}|\lesssim\theta_{i,\alpha}^{-1}B_{i,\alpha}$ in
$B_{g}(x_{i,\alpha},\frac{2r_{i,\alpha}}{R_{0}})$. Assume now that
$\mu_{j,\alpha}=o(\mu_{i,\alpha})$ but $j\not\in\mathcal{B}_{i}$. Then
$d_{g}(x_{i,\alpha},x_{j,\alpha})>\frac{2r_{i,\alpha}}{R_{0}}$, thus
$d_{g}(x_{i,\alpha},x_{j,\alpha})>>\mu_{j,\alpha}$ and then for all $y\in
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})$

$B_{j,\alpha}(y)\lesssim\frac{\mu_{j,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}\lesssim
B_{i,\alpha}(y)\textrm{ and }|\nabla
B_{j,\alpha}(y)|\lesssim\frac{\mu_{j,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-1}}\lesssim\theta_{i,\alpha}(y)^{-1}B_{i,\alpha}(y).$

∎

Let $i\in\\{1,\dots,k\\}$ and $j\in\mathcal{B}_{i}$. In view of (3.19) it is
important to understand when the profile $B_{j,\alpha}$ becomes dominant with
respect to $B_{i,\alpha}$. It is an easy consequence of (3.15) and
$\mu_{j,\alpha}=o(\mu_{i,\alpha})$ that there exists $C>1$, only depending on
$n$ and $(M,g)$ such that for all $y\in M$,

$\displaystyle\textrm{ if }d_{g}(x_{j,\alpha},y)\leq
s_{j,i,\alpha}\quad\textrm{ then }\quad
B_{j,\alpha}(y)\geq\frac{1}{C}B_{i,\alpha}(y),\quad\textrm{ and }$ (3.20)
$\displaystyle\textrm{ if
}B_{j,\alpha}(y)\geq\frac{1}{C}B_{i,\alpha}(y)\quad\textrm{ then }\quad
d_{g}(x_{j,\alpha},y)\leq Cs_{j,i,\alpha}.$

With (3.19) a consequence is that, for any $x\in
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})\backslash\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},s_{j,i,\alpha})$,

$|\mathcal{W}_{\alpha}(x)|\lesssim\sum_{j=0}^{k}B_{j,\alpha}(x)\lesssim
B_{i,\alpha}(x)$ (3.21)

holds. We will need a similar result for the derivatives of $V_{i,\alpha}$.
Let $i\in\\{1,\dots,k\\}$ and $j\in\mathcal{B}_{i}$. We define a new radius
$\rho_{j,i,\alpha}$ as follows:

$\displaystyle\rho_{j,i,\alpha}$
$\displaystyle=2\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{n-2}{2(n-1)}}\big{(}d_{g}(x_{i,\alpha},x_{j,\alpha})+\mu_{i,\alpha}\big{)}.$
(3.22)

If $j\in\mathcal{B}_{i}$, direct computations show again that there exists
$C=C(n,g)>0$ such that

$\displaystyle\textrm{ if }d_{g}(x_{j,\alpha},y)\leq\rho_{j,i,\alpha}\textrm{
then
}\theta_{j,\alpha}(y)^{-1}B_{j,\alpha}(y)\geq\frac{1}{C}\theta_{i,\alpha}(y)^{-1}B_{i,\alpha}(y),\quad\textrm{
and }$ (3.23) $\displaystyle\textrm{ if
}\theta_{j,\alpha}(y)^{-1}B_{j,\alpha}(y)\geq\frac{1}{C}\theta_{i,\alpha}(y)^{-1}B_{i,\alpha}(y)\textrm{
then }d_{g}(x_{j,\alpha},y)\leq C\rho_{j,i,\alpha}.$

Note that, by definition, $\rho_{j,i,\alpha}=o(r_{i,\alpha})$ for
$j\in\mathcal{B}_{i}$. Direct computations with (3.15) show that
$\rho_{j,i,\alpha}=2\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{-\frac{1}{2(n-1)}}s_{j,i,\alpha}$.
Since $j\in\mathcal{B}_{i}$ we have $\mu_{j,\alpha}=o(\mu_{i,\alpha})$ and
hence $s_{j,i,\alpha}=o(\rho_{j,i,\alpha})$. With (3.19), (3.21) and (3.23) we
obtain

$\displaystyle|\mathcal{W}_{\alpha}(x)|+\theta_{i,\alpha}(x)$
$\displaystyle|\nabla\mathcal{W}_{\alpha}(x)|\lesssim B_{i,\alpha}(x)$ (3.24)
$\displaystyle\quad\textrm{ for all }x\in
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})\backslash\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha}).$

If finally $i\in\\{1,\dots,k\\}$ we define $\mathcal{C}_{i}$ to be the subset
of $\mathcal{B}_{i}$ given by

$\mathcal{C}_{i}=\big{\\{}j\in\mathcal{B}_{i}\textrm{ such that
}d_{g}(x_{i,\alpha},x_{j,\alpha})=O(\mu_{i,\alpha})\big{\\}}.$ (3.25)

When $j\in\mathcal{C}_{i}$ one has
$s_{j,i,\alpha}=O(\sqrt{\mu_{i,\alpha}\mu_{j,\alpha}})$ and
$\rho_{j,i,\alpha}=O(\mu_{j,\alpha}^{\frac{n-2}{2(n-1)}}\mu_{i,\alpha}^{\frac{n}{2(n-1)}})$.

###  3.3. Proof of Theorem 3.2 part $1$: an inductive estimate

Let $(h_{\alpha})_{\alpha}$ be a sequence of functions converging towards $h$
in $L^{\infty}(M)$, $k\geq 1$, $V_{1},\dots,V_{k}$ in
$D^{1,2}(\mathbb{R}^{n})$ be $k$ solutions of (2.1), $u_{0}$ be a solution of
(2.15) and $(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq
i\leq k$ be $k$ sequences of points in $M$ and of positive real numbers
satisfying (1.4). Let $(\mathcal{W}_{\alpha})_{\alpha}$ be as in (3.5), let
$(\eta_{\alpha})_{\alpha}$, $(\tau_{\alpha})_{\alpha}$,
$(\gamma_{\alpha})_{\alpha}$ and $(R_{\alpha})_{\alpha}$ be as in the
statement of Theorem 3.2, and let $\varphi_{\alpha}$ be the unique solution of
(3.11). We keep in the whole proof of Theorem 3.2 the notations of subsections
$3.2$ and $3.3$ and we will repeatedly use (3.6) and (3.7).

We first prove (3.13) which is easily obtained. Direct computations using
(3.10) give

$\|R_{\alpha}\|_{L^{\frac{2n}{n+2}}(M)}\lesssim\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}$

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7), so that by
Proposition 3.1 $\varphi_{\alpha}$ satisfies

$\displaystyle\|\varphi_{\alpha}\|_{H^{1}(M)}$
$\displaystyle\lesssim\|(\triangle_{g}+1)^{-1}(R_{\alpha})\|_{H^{1}(M)}$
$\displaystyle\lesssim\|R_{\alpha}\|_{L^{\frac{2n}{n+2}}(M)}\lesssim\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}.$

Integrating (3.11) against $Z_{i,j,\alpha}$ for all $0\leq i\leq k$ and $1\leq
j\leq n_{i}$ then yields, using (3.8) and the latter inequalities, that

$\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}|\lambda_{i,j}^{\alpha}|\lesssim\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha},$
(3.26)

which proves (3.13).

For $\alpha\geq 1$ we now let:

$a_{\alpha}=\left|\left|\frac{\varphi_{\alpha}}{\sum_{i=0}^{k}B_{i,\alpha}}\right|\right|_{L^{\infty}(M)}+\left|\left|\frac{\nabla\varphi_{\alpha}}{B_{0,\alpha}+\sum_{i=1}^{k}\theta_{i,\alpha}^{-1}B_{i,\alpha}}\right|\right|_{L^{\infty}(M)}$
(3.27)

where the $B_{i,\alpha}$ are as in (3.4) and $\theta_{i,\alpha}$ as in (3.2).
The proof of Theorem 3.2 goes through a top-down induction in the bubble-tree.
Starting from the highest bubbles we improve the naive control given by (3.27)
on the area of influence of each bubble. We show that, up to some boundary
terms, $\varphi_{\alpha}$ is negligible with respect to $B_{i,\alpha}$ on
$B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$. These improved estimates are
then propagated to the lower bubbles, and the induction parameter is the
height in the bubble-tree. In this way we obtain refined estimates on
$\varphi_{\alpha}$ over the whole $M$ where $a_{\alpha}$ only appears in
negligible terms. These estimates self-improve and yield (3.12).

Up to renumbering the bubbles $V_{1,\alpha},\dots,V_{k,\alpha}$ and up to
passing to a subsequence for $(x_{i,\alpha})_{\alpha}$ and
$(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$, we assume that

$\mu_{1,\alpha}\geq\dots\geq\mu_{k,\alpha}$

and we separate the family of bubbles $V_{1,\alpha},\dots,V_{k,\alpha}$ in
subfamilies concentrating at comparable speeds. Precisely, we define
inductively

$E_{1}=\big{\\{}i\in\\{1,\dots,k\\}\textrm{ such that
}\mu_{i,\alpha}\asymp\mu_{1,\alpha}\big{\\}}=\big{\\{}1,\dots,N_{1}\big{\\}}$

for some $1\leq N_{1}\leq k$ and, assuming that we have defined $N_{q}$ for
$q\geq 1$, we let

$E_{q+1}=\big{\\{}i\geq N_{q}+1\textrm{ such that
}\mu_{i,\alpha}\asymp\mu_{N_{q}+1,\alpha}\big{\\}}=\big{\\{}N_{q}+1,\dots,N_{q+1}\\}$

for some $N_{q}+1\leq N_{q+1}\leq k$. There exists an integer $1\leq p\leq k$
such that

$\\{1,\dots,k\\}=E_{1}\cup\dots\cup E_{p}.$ (3.28)

In a given subset $E_{j}$ all the bubbles concentrate with comparable speeds.
We say that bubbles in $E_{j}$ have height $j$ in the bubble-tree. Note that
defining $E_{1},\dots,E_{p}$ as in (3.28) does not involve $\varphi_{\alpha}$
or the sequences $(\eta_{\alpha})_{\alpha}$, $(\tau_{\alpha})_{\alpha}$,
$(\gamma_{\alpha})_{\alpha}$ and $(R_{\alpha})_{\alpha}$ but only the
configuration $V_{1,\alpha},\dots,V_{k,\alpha}$.

Let $\ell\in\\{1,\dots,p\\}$, where $p$ is as in (3.28). We consider the
following two statements:

Property $(H_{\ell,1})$: There exists a sequence
$(\sigma_{\alpha}^{\ell})_{\alpha}$ as in (3.7) such that for all $q\geq\ell$,
for all $i\in E_{q}$ and for all $x\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$ we have

$\displaystyle|\varphi_{\alpha}(x)|$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x)+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x)\Big{)}+a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}$
($H_{\ell,1}$)
$\displaystyle+\gamma_{\alpha}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i}(x)$

where $\mathcal{B}_{i}$ is as in (3.18), $\theta_{i,\alpha}$ as in (3.2) and
$\Theta_{i,\alpha}$ as in (3.3).

Property $(H_{\ell,2})$: There exists a sequence
$(\sigma_{\alpha}^{\ell})_{\alpha}$ as in (3.7) such that for all $q\geq\ell$,
for all $i\in E_{q}$ and for all $x\in M$ we have

$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})}d_{g}(x,\cdot)^{2-n}\big{(}\sum_{i=0}^{k}B_{i,\alpha}\big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
($H_{\ell,2}$)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x)+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x)\Big{)}\quad\textrm{
and }$
$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})}d_{g}(x,\cdot)^{1-n}\big{(}\sum_{i=0}^{k}B_{i,\alpha}\big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{0,\alpha}(x)+\sum_{i=1}^{k}\theta_{i,\alpha}(x)^{-1}B_{i,\alpha}(x)\Big{)}.$

Note that if $q\in\\{1,\dots,p\\}$, $i\in E_{q}$ and $x\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$, then by (3.19) we have

$B_{i,\alpha}(x)+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x)\lesssim\sum_{j\in
E_{s},s\geq q}B_{j,\alpha}(x)\lesssim
B_{i,\alpha}(x)+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x).$

The right-hand side of ($H_{\ell,1}$) thus states that $\varphi_{\alpha}$ is
negligible with respect to the sum of all the (positive) bubbles in
$\cup_{q\geq\ell}E_{q}$. The induction process is given by the following
Proposition, that we prove in the next subsection:

######  Proposition 3.5.

For any $1\leq\ell\leq p$, $(H_{\ell,1})$ and $(H_{\ell,2})$ are true.

###  3.4. Proof of Proposition 3.5.

We prove Proposition 3.5 by reverse induction on $\ell$. The proofs of the
initial case $\ell=p$ and of the induction step are very similar, hence we
prove them together. Let $\ell\in\\{1,\dots,p\\}$. If $\ell\leq p-1$ we assume
that $(H_{\ell+1,1})$ and $(H_{\ell+1,2})$ hold true, while if $\ell=p$ we do
not assume anything. Let $i\in E_{\ell}$ where $E_{\ell}$ is as in (3.28).

Step $1$: improved estimates in
$B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$. Let $G_{h}$ be the Green’s
function of $\triangle_{g}+h$ in $M$. It is by definition orthogonal to the
kernel of $\triangle_{g}+h$ and by standard results (see e.g. Aubin [3] or
Robert [46]) there exists $C>0$ only depending on $(M,g)$ and $h$ such that
$|G_{h}(x,y)|\leq Cd_{g}(x,y)^{2-n}$ and $|\nabla_{y}G_{h}(x,y)|\leq
Cd_{g}(x,y)^{1-n}$ for all $x\neq y$ in $M$. First, using (A.1) and (A.4)
below and (3.10) it is easily seen that there exists a sequence
$(\sigma_{\alpha})_{\alpha}$ as in (3.7) such that for any sequence
$(x_{\alpha})_{\alpha}\in M$

$\displaystyle\int_{M}G_{h}(x_{\alpha},\cdot)R_{\alpha}dv_{g}$ (3.29)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}\gamma_{\alpha}\big{)}\sum_{i=0}^{k}B_{i,\alpha}(x_{\alpha})+\gamma_{\alpha}\sum_{i=1}^{k}\Theta_{i,\alpha}(x_{\alpha})B_{i}(x_{\alpha})$

holds. We write a representation formula for $\varphi_{\alpha}$ in the whole
of $M$. Since $M$ has no boundary we have, for any sequence
$(x_{\alpha})_{\alpha}\in B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$:

$\displaystyle\varphi_{\alpha}(x_{\alpha})-\Pi_{K_{h}}(\varphi_{\alpha})(x_{\alpha})$
$\displaystyle=\int_{M}G_{h}(x_{\alpha},\cdot)(\triangle_{g}+h)\varphi_{\alpha}dv_{g},$

where we have let $K_{h}=\ker(\triangle_{g}+h)$ and where $\Pi_{K_{h}}$
denotes the orthogonal projection on $K_{h}$ for the $H^{1}(M)$ scalar
product. We first claim that

$\big{|}\Pi_{K_{h}}(\varphi_{\alpha})(x_{\alpha})\big{|}\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}a_{\alpha}$
(3.30)

holds. Indeed, integrating by parts and using (3.27) yields

$\big{|}\Pi_{K_{h}}(\varphi_{\alpha})(x_{\alpha})\big{|}\lesssim\int_{M}|\varphi_{\alpha}|dv_{g}\lesssim
a_{\alpha}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\mu_{j,\alpha}^{\frac{n-2}{2}}a_{\alpha}.$

If $B_{0,\alpha}\not\equiv 0$ we have by (3.16)
$r_{i,\alpha}\leq\sqrt{\mu_{i,\alpha}}$ and hence
$a_{\alpha}B_{0,\alpha}(x_{\alpha})\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}a_{\alpha}$.
Since $r_{i,\alpha}\lesssim 1$ we also have
$\mu_{j,\alpha}^{\frac{n-2}{2}}a_{\alpha}\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}a_{\alpha}$
for all $j$ such that $\mu_{j,\alpha}\lesssim\mu_{i,\alpha}$. Finally, if
$\mu_{j,\alpha}>>\mu_{i,\alpha}$, and since $x_{\alpha}\in
B_{g}(x_{i,\alpha},\frac{2r_{i,\alpha}}{R_{0}})$, we use Lemma 3.3 to write
that

$a_{\alpha}\mu_{j,\alpha}^{\frac{n-2}{2}}\lesssim
a_{\alpha}B_{j,\alpha}(x_{\alpha})\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}a_{\alpha}.$

These considerations prove (3.30). We now claim that

$\displaystyle\int_{M\backslash
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\big{(}\sum_{i=0}^{k}B_{i,\alpha})^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.31)
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}+\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}a_{\alpha}$

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7). Let
$\tilde{\mathcal{B}}_{i}\subset\mathcal{B}_{i}$ be defined by

$\tilde{\mathcal{B}}_{i}=\Big{\\{}j\in\mathcal{B}_{i}\textrm{ such that
}d_{g}(x_{i,\alpha},x_{j,\alpha})\leq\frac{5r_{i,\alpha}}{4R_{0}}\Big{\\}}.$

Let also

$\mathcal{D}_{i}=\Big{\\{}j\in\\{1,\dots,k\\}\textrm{ such that
}\mu_{j,\alpha}=o(\mu_{i,\alpha})\textrm{ and
}d_{g}(x_{i,\alpha},x_{j,\alpha})>\frac{5r_{i,\alpha}}{4R_{0}}\Big{\\}}.$

We can write:

$\displaystyle\int_{M\backslash
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\big{(}\sum_{i=0}^{k}B_{i,\alpha})^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.32) $\displaystyle\lesssim\int_{M\backslash
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}a_{\alpha}d_{g}(x_{\alpha},\cdot)^{2-n}\times$
$\displaystyle\quad\quad\Big{(}B_{i,\alpha}^{2^{*}-1}+\sum_{j\in\tilde{\mathcal{B}}_{i}}B_{j,\alpha}^{2^{*}-1}+\sum_{j\in\mathcal{D}_{i}}B_{j,\alpha}^{2^{*}-1}+\sum_{j\in\mathcal{A}_{i}}B_{j,\alpha}^{2^{*}-1}\Big{)}dv_{g}$
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}+a_{\alpha}\sum_{j\in\mathcal{D}_{i}}B_{j,\alpha}(x_{\alpha})$
$\displaystyle+a_{\alpha}\sum_{j\in\mathcal{A}_{i}}B_{j,\alpha}(x_{\alpha}),$

where the last line follows from (A.7) below. With Lemma 3.3, and since
$x_{\alpha}\in B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$, we have
$a_{\alpha}\sum_{j\in\mathcal{A}_{i}}B_{j,\alpha}(x_{\alpha})\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}a_{\alpha}$.
If $j\in\mathcal{D}_{i}$ and since $x_{\alpha}\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$, we have
$d_{g}(x_{j,\alpha},x_{\alpha})\gtrsim r_{i,\alpha}$, so that

$a_{\alpha}\sum_{j\in\mathcal{D}_{i}}B_{j,\alpha}(x_{\alpha})\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}a_{\alpha}.$

Together with (3.32) this proves (3.31). Using (3.26), (3.29), (3.30), (3.31)
and (A.1) below the representation formula gives, for any sequence
$(x_{\alpha})_{\alpha}$ in $B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$,

$\displaystyle|\varphi_{\alpha}(x_{\alpha})|$ (3.33)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}$
$\displaystyle+a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}+\gamma_{\alpha}\sum_{j=1}^{k}\Theta_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$
$\displaystyle+\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}|\mathcal{W}_{\alpha}|^{2^{*}-2}|\varphi_{\alpha}|dv_{g}.$

Note that the sum $\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x)$ vanishes when
$\ell=p$. To estimate the integral in (3.33) we remark that

$\displaystyle B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})\subseteq$
$\displaystyle\Bigg{[}\Big{(}B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})\backslash
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})\Big{)}\backslash\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\Bigg{]}$
$\displaystyle\bigcup\Bigg{[}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})\backslash\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\Bigg{]}$
$\displaystyle\bigcup_{j\in\mathcal{B}_{i}}\Big{(}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})\Big{)}\cap
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})\Big{)}$
$\displaystyle\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})$

where $\rho_{j,i,\alpha}$ is as in (3.22). Let
$y\in\Big{(}B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})\backslash
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})\Big{)}\backslash\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})$.
By (3.24) we have

$|\mathcal{W}_{\alpha}(y)|\lesssim
B_{i,\alpha}(y)\lesssim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}.$

Straightforward computations with (3.27) then yield

$\displaystyle\int\limits_{\big{(}B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})\backslash
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})\big{)}\backslash\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})}$
$\displaystyle
d_{g}(x_{\alpha},\cdot)^{2-n}|\mathcal{W}_{\alpha}|^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.34)
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}B_{i,\alpha}(x_{\alpha}).$

By definition of $\mathcal{W}_{\alpha}$ we have
$|\mathcal{W}_{\alpha}|\lesssim\sum_{i=0}^{k}B_{i,\alpha}$. Since $i\in
E_{\ell}$, if $j\in\mathcal{B}_{i}$ we have $j\in E_{q}$ for some
$q\geq\ell+1$, where $E_{\ell}$ and $E_{q}$ are as in (3.28). The induction
property $(H_{\ell+1,2})$ together with (3.24) then shows that there exists a
sequence $(\sigma_{\alpha}^{\ell+1})_{\alpha}$ satisfying (3.7) such that

$\displaystyle\int\limits_{\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}|\mathcal{W}_{\alpha}|^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.35)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell+1}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}.$

The latter integral only has to be computed $\ell\leq p-1$, otherwise it
vanishes. Remark now that, by (3.19) and (3.27), we have

$|\mathcal{W}_{\alpha}|^{2^{*}-2}|\varphi_{\alpha}|\lesssim
a_{\alpha}B_{i,\alpha}^{2^{*}-1}+a_{\alpha}\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}^{2^{*}-1}$

in
$\bigcup_{j\in\mathcal{B}_{i}}\Big{(}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})\Big{)}\cap
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})$. On the one hand,
straightforward computations using (A.7) below give

$\displaystyle\int_{\bigcup_{j\in\mathcal{B}_{i}}\Big{(}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})\Big{)}\cap
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}a_{\alpha}\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}^{2^{*}-1}dv_{g}$
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})$

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7). On the other
hand, if $j\in\mathcal{B}_{i}$,

$\displaystyle\int_{B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})\cap
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x,\cdot)^{2-n}a_{\alpha}B_{i,\alpha}^{2^{*}-1}dv_{g}$
$\displaystyle\leq\int_{B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})}d_{g}(x,\cdot)^{2-n}a_{\alpha}B_{i,\alpha}^{2^{*}-1}dv_{g}$
$\displaystyle\lesssim\left\\{\begin{aligned}
&\left(\frac{\rho_{j,i,\alpha}}{\mu_{i,\alpha}}\right)^{2}a_{\alpha}B_{i,\alpha}(x)&\textrm{
if }d_{g}(x_{i,\alpha},x_{j,\alpha})=O(\mu_{i,\alpha})\\\
&\left(\frac{\mu_{i,\alpha}}{d_{g}(x_{i,\alpha},x_{j,\alpha})}\right)^{2}a_{\alpha}B_{i,\alpha}(x)&\textrm{
otherwise }\end{aligned}\right.$
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}B_{i,\alpha}(x).$

We have used that, by (3.22), we have $\rho_{j,i,\alpha}=o(\mu_{i,\alpha})$ in
the first case and $\rho_{j,i,\alpha}=o(d_{g}(x_{i,\alpha},x_{j,\alpha}))$ in
the second case. Combining the last two estimates shows that

$\displaystyle\int_{\bigcup_{j\in\mathcal{B}_{i}}\Big{(}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})\Big{)}\cap
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}|\mathcal{W}_{\alpha}|^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.36)
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}$

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7). Estimate
(3.36) shows that in the representation formula the “neck regions”
$B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})$
for $j\in\mathcal{B}_{i}$ only contribute to $\varphi_{\alpha}$ with a term
that is negligible compared to
$B_{i,\alpha}+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}$. This shows why we only
need to prove the induction property on
$B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$ in ($H_{\ell,1}$). We finally
let

$\Omega_{i,\alpha}=B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})\backslash\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha}).$
(3.37)

With (3.24) and Giraud’s lemma we obtain

$\int_{\Omega_{i,\alpha}}d_{g}(x_{\alpha},\cdot)^{2-n}|\mathcal{W}_{\alpha}|^{2^{*}-2}|\varphi_{\alpha}|dv_{g}\lesssim\left(\frac{\mu_{i,\alpha}}{\theta_{i,\alpha}(x_{\alpha})}\right)^{2}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}.$
(3.38)

Gathering (3.34), (3.35), (3.36) and (3.38) in (3.33) we have thus shown that
there exists a sequence $(\sigma_{\alpha})_{\alpha}$ as in (3.7) such that,
for any sequence $(x_{\alpha})_{\alpha}\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$,

$\displaystyle|\varphi_{\alpha}(x_{\alpha})|\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}$
(3.39)
$\displaystyle+a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}+\gamma_{\alpha}\sum_{j=1}^{k}\Theta_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})+\left(\frac{\mu_{i,\alpha}}{\theta_{i,\alpha}(x_{\alpha})}\right)^{2}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$

and

$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.40)
$\displaystyle\lesssim\left(\frac{\mu_{i,\alpha}}{\theta_{i,\alpha}(x_{\alpha})}\right)^{2}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$
$\displaystyle+\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}$

hold, where $\Omega_{i,\alpha}$ is as in (3.37). We can now use (3.39) to
compute again the right-hand side of (3.38) with a greater precision. After a
finite number of iterations we obtain that there exists a sequence
$(\sigma_{\alpha}^{\ell,1})_{\alpha}$ as in (3.7) such that

$\displaystyle|\varphi_{\alpha}(x_{\alpha})|\lesssim
a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}+\gamma_{\alpha}\sum_{j=1}^{k}\Theta_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$
(3.41)
$\displaystyle+\Big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,1}(\gamma_{\alpha}+a_{\alpha})+\mu_{i,\alpha}^{\frac{n-2}{2}}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}\Big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)},$

and

$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.42)
$\displaystyle\lesssim\Big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,1}(\gamma_{\alpha}+a_{\alpha})+\mu_{i,\alpha}^{\frac{n-2}{2}}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}\Big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}$

hold for any sequence $(x_{\alpha})_{\alpha}\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$.

Step $2$: Gradient estimates on $\Omega_{i,\alpha}$. First, using (3.10),
(A.2) and (A.5) below we have, for any sequence $(x_{\alpha})_{\alpha}$ in
$M$,

$\displaystyle\int_{M}d_{g}(x_{\alpha},\cdot)^{1-n}R_{\alpha}dv_{g}$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)}$
$\displaystyle+\gamma_{\alpha}\sum_{j=1}^{k}\Xi_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha}),$

where $\Xi_{i,\alpha}$ is as in (A.3) below. As in Step $1$ we write a
representation formula for $\varphi_{\alpha}$ in $M$, that we differentiate
once with respect to the variable $x$. The details are very similar to those
of Step $1$ and we omit most of them. We then obtain that, for any sequence
$(x_{\alpha})_{\alpha}\in B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$:

$\displaystyle|\nabla\varphi_{\alpha}(x_{\alpha})|$ (3.43)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)}$
$\displaystyle+\gamma_{\alpha}\sum_{j=1}^{k}\Xi_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})+a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-1}}$
$\displaystyle+\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{1-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}.$

The integral over $B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})$ is
computed by following the exact arguments that led to (3.34), (3.35), (3.36)
and (3.38), where the induction argument now relies on the second inequality
in ($H_{\ell,2}$). This gives in the end

$\displaystyle|\nabla\varphi_{\alpha}(x_{\alpha})|$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)}$
$\displaystyle+a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-1}}+\gamma_{\alpha}\sum_{j=1}^{k}\Xi_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})+\frac{\mu_{i,\alpha}^{2}}{\theta_{i,\alpha}(x_{\alpha})^{3}}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$

and

$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{1-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}\lesssim\frac{\mu_{i,\alpha}^{2}}{\theta_{i,\alpha}(x_{\alpha})^{3}}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$
$\displaystyle+\Big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}(\gamma_{\alpha}+a_{\alpha})+\mu_{i,\alpha}^{\frac{n-2}{2}}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}\Big{)}\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)}$

for any sequence $(x_{\alpha})_{\alpha}\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$, where $\Omega_{i,\alpha}$ is
as in (3.37) and $(\sigma_{\alpha})_{\alpha}$ is a sequence satisfying (3.7).
As before these two inequalities self-improve after a finite number of
iterations. We thus obtain that there exists a sequence
$(\sigma_{\alpha}^{\ell,2})_{\alpha}$ satisfying (3.7) such that, for any
sequence $(x_{\alpha})_{\alpha}\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$,

$\displaystyle|\nabla\varphi_{\alpha}(x_{\alpha})|\lesssim
a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-1}}+\gamma_{\alpha}\sum_{j=1}^{k}\Xi_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$
(3.44)
$\displaystyle+\Big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,2}(\gamma_{\alpha}+a_{\alpha})+\mu_{i,\alpha}^{\frac{n-2}{2}}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}\Big{)}\times$
$\displaystyle\quad\quad\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)}$

and

$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{1-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.45)
$\displaystyle\lesssim\Big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,2}(\gamma_{\alpha}+a_{\alpha})+\mu_{i,\alpha}^{\frac{n-2}{2}}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}\Big{)}\times$
$\displaystyle\quad\quad\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)}$

hold.

Step $3$: almost-orthogonality. By construction $\varphi_{\alpha}\in
K_{\alpha}^{\perp}$ and so, for any $1\leq m\leq n_{i}$,

$\int_{M}\langle\nabla\varphi_{\alpha},\nabla
Z_{i,m,\alpha}\rangle+\varphi_{\alpha}Z_{i,m,\alpha}dv_{g}=0.$ (3.46)

Straightforward computations using (3.27) yield

$\left|\int_{M}\varphi_{\alpha}Z_{i,m,\alpha}dv_{g}=0\right|\lesssim\sigma_{\alpha}a_{\alpha}$
(3.47)

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7). By definition
of $Z_{i,m,\alpha}$ and by (2.7) direct computations give, with (3.27) and
(A.6) below

$\int_{M\backslash
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})}|\langle\nabla\varphi_{\alpha},\nabla
Z_{i,m,\alpha}\rangle|dv_{g}\lesssim\sigma_{\alpha}a_{\alpha}.$ (3.48)

Let now $j\in\mathcal{B}_{i}$. By (3.23) and (3.27) we have, for any $y\in
B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\cap
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})$,

$|\nabla\varphi_{\alpha}(y)|\lesssim\sum_{j\in\mathcal{B}_{i}}\theta_{j,\alpha}(y)^{-1}B_{j,\alpha}(y).$

Independently, by (2.7) one always has $|\nabla
Z_{i,m,\alpha}|\lesssim\mu_{i,\alpha}^{1-\frac{n}{2}}$ in $M$. Therefore

$\int_{\Big{(}\bigcup_{j\in\mathcal{B}_{i}}B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\Big{)}\cap
B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}|\langle\nabla\varphi_{\alpha},\nabla
Z_{i,m,\alpha}\rangle|dv_{g}\lesssim\sigma_{\alpha}a_{\alpha}.$ (3.49)

Define now

$\hat{\Omega}_{i,\alpha}=\frac{1}{\mu_{i,\alpha}}\exp_{x_{i,\alpha}}^{-1}(\Omega_{i,\alpha})$
(3.50)

where $\Omega_{i,\alpha}$ is as in (3.37) and, for
$y\in\hat{\Omega}_{i,\alpha}$,

$\hat{\varphi}_{\alpha}(y)=\varphi_{\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)},$
(3.51)

where $\exp_{x_{i,\alpha}}$ is the exponential map for $g$ centered at
$x_{i,\alpha}$. For $y\in\hat{\Omega}_{i,\alpha}$ one has

$|\exp_{x_{i,\alpha}}^{*}g(\mu_{i,\alpha}\cdot)-\xi|(y)\lesssim\mu_{i,\alpha}^{2}|y|^{2}$

where $\xi$ is the euclidean metric. Direct computations with (3.27) show that

$\displaystyle\int_{\Omega_{i,\alpha}}\langle\nabla\varphi_{\alpha},\nabla
Z_{i,m,\alpha}\rangle
dv_{g}=\mu_{i,\alpha}^{\frac{n-2}{2}}\int_{\hat{\Omega}_{i,\alpha}}\langle\nabla\hat{\varphi}_{\alpha},\nabla
Z_{i,m}\rangle_{\xi}dx+O(\sigma_{\alpha}a_{\alpha}).$

Combining the latter with (3.46), (3.47), (3.48) and (3.49) finally shows that
there exists a sequence $(\sigma_{\alpha}^{\ell,3})_{\alpha}$ satisfying (3.7)
such that for all $\alpha$,

$\left|\int_{\hat{\Omega}_{i,\alpha}}\langle\nabla\hat{\varphi}_{\alpha},\nabla
Z_{i,m}\rangle_{\xi}dx\right|\lesssim\mu_{i,\alpha}^{1-\frac{n}{2}}\sigma_{\alpha}^{\ell,3}a_{\alpha}$
(3.52)

for all $1\leq m\leq n_{i}$.

Step $4$: equation satisfied by $\hat{\varphi}_{\alpha}$. Let
$j\in\mathcal{B}_{i}$ and let

$\hat{x}_{j,\alpha}=\frac{1}{\mu_{i,\alpha}}\exp_{x_{i,\alpha}}^{-1}(x_{j,\alpha}).$

If $j\in\mathcal{C}_{i}$, where $\mathcal{C}_{i}$ is as in (3.25), we have
$d_{g}(x_{j,\alpha},x_{i,\alpha})=O(\mu_{i,\alpha})$: hence
$|\hat{x}_{i,\alpha}|\lesssim 1$ and $\rho_{j,i,\alpha}=o(\mu_{i,\alpha})$ as
$\alpha\to+\infty$ by (3.22). We let in this case
$\hat{x}_{j,\infty}=\lim_{\alpha\to+\infty}\hat{x}_{j,\alpha}$. Otherwise,
again by (3.22), we have $|\hat{x}_{j,\alpha}|\to+\infty$ and
$\rho_{j,i,\alpha}=o(d_{g}(x_{i,\alpha},x_{j,\alpha}))$. As
$\alpha\to+\infty$, and up to passing to a subsequence, the sets
$\hat{\Omega}_{i,\alpha}$ thus converge to
$\mathbb{R}^{n}\backslash\\{(\hat{x}_{j,\infty})_{j\in\mathcal{C}_{i}}\\}$. We
prove the following lemma:

######  Lemma 3.6.

There exists a sequence $(\sigma_{\alpha}^{\ell,4})_{\alpha}$ satisfying (3.7)
such that

$\big{|}\big{|}\triangle_{g_{i,\alpha}}\hat{\varphi}_{\alpha}\big{|}\big{|}_{L^{\infty}(\hat{\Omega}_{i,\alpha})}\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,4}(\gamma_{\alpha}+a_{\alpha})\big{)}\mu_{i,\alpha}^{1-\frac{n}{2}}+\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$
(3.53)

holds, where $\Omega_{i,\alpha}$ and $\hat{\Omega}_{i,\alpha}$ are as in
(3.37) and (3.50).

In the lemma we have let
$g_{i,\alpha}=\exp_{x_{i,\alpha}}^{*}g(\mu_{i,\alpha}\cdot)$, so that
$g_{i,\alpha}\to\xi$ in $C^{2}_{loc}(\mathbb{R}^{n})$.

###### Proof of Lemma 3.6.

By definition of $\hat{\varphi}_{\alpha}$ one has, for all
$y\in\hat{\Omega}_{i,\alpha}$,

$\triangle_{g_{i,\alpha}}\hat{\varphi}_{\alpha}=\mu_{i,\alpha}^{2}\triangle_{g}\varphi_{\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}.$

The Laplacian of $\varphi_{\alpha}$ is estimated with (3.11). First, by (3.37)
and by (3.20) we have

$B_{j,\alpha}(\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y))\lesssim\mu_{i,\alpha}^{1-\frac{n}{2}}(1+|y|)^{2-n}$
(3.54)

for all $j\neq i$, $j\geq 1$, and for all $y\in\hat{\Omega}_{i,\alpha}$, so
that by (3.21) and (3.27) we have

$\big{|}\mu_{i,\alpha}^{2}h_{\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\hat{\varphi}_{\alpha}(y)\big{|}\lesssim\sigma_{\alpha}\mu_{i,\alpha}^{1-\frac{n}{2}}$
(3.55)

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7) and for all
$y\in\hat{\Omega}_{i,\alpha}$. If $j\geq 1$, straightforward computations show
that for all $1\leq m\leq n_{j}$

$|(\triangle_{g}+1)Z_{j,m,\alpha}|\lesssim
B_{j,\alpha}^{2^{*}-1}+B_{j,\alpha}$

holds, while if $j=0$ we have, for all $1\leq m\leq n_{0}$,
$|(\triangle_{g}+1)Z_{0,m,\alpha}|\lesssim 1$. As a consequence, Lemma 3.3 and
(3.26) show that:

$\displaystyle\Big{|}\mu_{i,\alpha}^{2}\sum\limits_{\underset{m=1..n_{i}}{j=0..k}}\lambda_{j,m}^{\alpha}$
$\displaystyle(\triangle_{g}+1)Z_{j,m,\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\Big{|}$
(3.56)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\mu_{i,\alpha}^{1-\frac{n}{2}}$

for some sequence $(\sigma_{\alpha})_{\alpha}$ as in (3.7) and for all
$y\in\hat{\Omega}_{i,\alpha}$. Using (3.10), Lemma 3.3, (3.20) and (3.54) we
get that

$\displaystyle\mu_{i,\alpha}^{2}\big{|}R_{\alpha}$
$\displaystyle\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\big{|}\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\mu_{i,\alpha}^{1-\frac{n}{2}}$
$\displaystyle+\gamma_{\alpha}\mu_{i,\alpha}^{2}\sum\limits_{\underset{j\neq
i}{j=0..k}}\big{(}B_{j,\alpha}^{2^{*}-2}B_{i,\alpha}+B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}$

holds for all $y\in\hat{\Omega}_{i,\alpha}$. Let now $j\in\\{1,\dots,k\\}$
with $j\neq i$. Assume first that $\mu_{i,\alpha}=O(\mu_{j,\alpha})$, hence
$j\in\mathcal{A}_{i}$. Lemma 3.3 then shows that

$\displaystyle\mu_{i,\alpha}^{2}\big{(}$ $\displaystyle
B_{j,\alpha}^{2^{*}-2}B_{i,\alpha}+B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}$
$\displaystyle\lesssim\left(\frac{\mu_{i,\alpha}}{s_{i,j,\alpha}}\right)^{4}\mu_{i,\alpha}^{1-\frac{n}{2}}(1+|y|)^{2-n}+\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{s_{i,j,\alpha}^{n-2}}(1+|y|)^{-4}$
$\displaystyle\lesssim\sigma_{\alpha}\mu_{i,\alpha}^{1-\frac{n}{2}}$

for all $y\in\hat{\Omega}_{i,\alpha}$, where we used that
$\frac{\mu_{i,\alpha}}{s_{i,j,\alpha}}\to 0$ as $\alpha\to+\infty$. Assume now
that $\mu_{j,\alpha}=o(\mu_{i,\alpha})$. Then we either have
$d_{g}(x_{i,\alpha},x_{j,\alpha})>\frac{2r_{i,\alpha}}{R_{0}}$ or
$j\in\mathcal{B}_{i}$. In the first case, $B_{j,\alpha}$ satisfies

$B_{j,\alpha}(x)\lesssim\frac{\mu_{j,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}\quad\textrm{
on }\Omega_{i,\alpha},$

so that

$\displaystyle\mu_{i,\alpha}^{2}\big{(}$ $\displaystyle
B_{j,\alpha}^{2^{*}-2}B_{i,\alpha}+B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}$
$\displaystyle\lesssim\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{2}\left(\frac{\mu_{i,\alpha}}{r_{i,\alpha}}\right)^{4}\mu_{i,\alpha}^{1-\frac{n}{2}}(1+|y|)^{2-n}+\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{n-2}{2}}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}(1+|y|)^{-4}$
$\displaystyle\lesssim\sigma_{\alpha}\mu_{i,\alpha}^{1-\frac{n}{2}}.$

Assume finally that $j\in\mathcal{B}_{i}$. If
$y\in\hat{\Omega}_{i,\alpha}\cap\\{y\in\hat{\Omega}_{i,\alpha},|y-\hat{x}_{j,\alpha}|>\frac{1}{4}\\}$
then

$B_{j,\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{n-2}{2}}\mu_{i,\alpha}^{1-\frac{n}{2}},$

so that

$\displaystyle\mu_{i,\alpha}^{2}\big{(}$ $\displaystyle
B_{j,\alpha}^{2^{*}-2}B_{i,\alpha}+B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}$
$\displaystyle\lesssim\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{2}\mu_{i,\alpha}^{1-\frac{n}{2}}(1+|y|)^{2-n}+\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{n-2}{2}}\mu_{i,\alpha}^{1-\frac{n}{2}}(1+|y|)^{-4}$
$\displaystyle\lesssim\sigma_{\alpha}\mu_{i,\alpha}^{1-\frac{n}{2}}.$

If $y\in B(\hat{x}_{j,\alpha},\frac{1}{4})\backslash
B(\hat{x}_{j,\alpha},\frac{1}{2}\frac{\rho_{j,i,\alpha}}{\mu_{i,\alpha}})$ we
then have

$B_{i,\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\gtrsim\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{(\mu_{i,\alpha}+d_{g}(x_{i,\alpha},x_{j,\alpha}))^{n-2}}$

and thus by (3.22)

$\frac{B_{j,\alpha}}{B_{i,\alpha}}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{1}{n-1}}.$

As a consequence

$\displaystyle\mu_{i,\alpha}^{2}\big{(}$ $\displaystyle
B_{j,\alpha}^{2^{*}-2}B_{i,\alpha}+B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}$
$\displaystyle\lesssim\left(\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{1}{n-1}}+\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{4}{(n-2)(n-1)}}\right)\mu_{i,\alpha}^{2}B_{i,\alpha}^{2^{*}-1}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}$
$\displaystyle\lesssim\sigma_{\alpha}\mu_{i,\alpha}^{1-\frac{n}{2}}.$

The previous arguments therefore show that

$\displaystyle\mu_{i,\alpha}^{2}\big{|}\gamma_{\alpha}R_{\alpha}$
$\displaystyle\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\big{|}\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\mu_{i,\alpha}^{1-\frac{n}{2}}$
(3.57)

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7). Finally by
(3.24) and (3.54) one has

$\displaystyle\mu_{i,\alpha}^{2}\big{|}\mathcal{W}_{\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\big{|}^{2^{*}-2}|\varphi_{\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}|$
(3.58)
$\displaystyle\lesssim(1+|y|)^{-4}\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$

for any $y\in\hat{\Omega}_{i,\alpha}$. Combining (3.55), (3.56), (3.57) and
(3.58) in (3.11) yields (3.53) and concludes the proof of the lemma. ∎

Step $5$: control of $\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$
and end of the proof of Proposition 3.5.

The following result provides a control on
$\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}$:

######  Lemma 3.7.

There exists a sequence $(\sigma_{\alpha}^{\ell,5})_{\alpha}$ satisfying (3.7)
such that

$\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,5}(\gamma_{\alpha}+a_{\alpha})\big{)}\mu_{i,\alpha}^{1-\frac{n}{2}}$
(3.59)

for all $\alpha\geq 1$.

###### Proof of Lemma 3.7.

Let

$\sigma_{\alpha}^{\ell,5}=\sigma_{\alpha}^{\ell,1}+\sigma_{\alpha}^{\ell,2}+\sigma_{\alpha}^{\ell,3}+\sigma_{\alpha}^{\ell,4}+\mu_{1,\alpha}+\sum_{i=1}^{k}\left(\frac{\mu_{i,\alpha}}{r_{i,\alpha}}\right)^{n-2}+\sum\limits_{\underset{\mu_{i,\alpha}=o(\mu_{j,\alpha})}{i,j\in\\{1,\dots,k\\}}}\left(\frac{\mu_{i,\alpha}}{\mu_{j,\alpha}}\right)^{\frac{n-2}{2}},$
(3.60)

where the sequences
$(\sigma_{\alpha}^{\ell,1})_{\alpha},(\sigma_{\alpha}^{\ell,2})_{\alpha},(\sigma_{\alpha}^{\ell,3})_{\alpha}$
and $(\sigma_{\alpha}^{\ell,4})_{\alpha}$ appear respectively in (3.41),
(3.44), (3.52) and (3.53). The sequence $(\sigma_{\alpha}^{\ell,5})$ obviously
still satisfies (3.7). We claim that (3.59) holds true for this choice of
$\sigma_{\alpha}^{\ell,5}$. We proceed by contradiction and assume that, up to
a subsequence,

$\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}>>\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,5}(\gamma_{\alpha}+a_{\alpha})\big{)}\mu_{i,\alpha}^{1-\frac{n}{2}}$
(3.61)

where $\sigma_{\alpha}^{\ell,5}$ is given by (3.60). Our goal will be to pass
(3.53) to the limit and to get a contradiction with the assumption
$\varphi_{\alpha}\in K_{\alpha}^{\perp}$.

We first claim that

$\sum_{j=1}^{k}(\Theta_{j,\alpha}B_{j,\alpha})\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\sigma_{\alpha}^{\ell,5}\mu_{i,\alpha}^{1-\frac{n}{2}}$
(3.62)

for all $y\in\hat{\Omega}_{i,\alpha}$. Let indeed
$y\in\hat{\Omega}_{i,\alpha}$. We first have, by (3.3) and and (3.60),

$(\Theta_{i,\alpha}B_{i,\alpha})\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\mu_{i,\alpha}^{2-\frac{n}{2}}(1+|y|)^{3-n}\lesssim\sigma_{\alpha}^{\ell,5}\mu_{i,\alpha}^{1-\frac{n}{2}},$

where we used that $\mu_{i,\alpha}\leq\mu_{1,\alpha}$ for all $i\geq 1$. Let
now $j\neq i$. Assume first that $\mu_{i,\alpha}=O(\mu_{j,\alpha})$, so that
$j\in\mathcal{A}_{i}$. By Lemma 3.3 we have
$B_{j,\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\mu_{i,\alpha}^{\frac{n-2}{2}}{r_{i,\alpha}^{n-2}}$
for all $y\in\hat{\Omega}_{i,\alpha}$. Since $\Theta_{j,\alpha}\lesssim 1$
this gives

$(\Theta_{j,\alpha}B_{j,\alpha})\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\left(\frac{\mu_{i,\alpha}}{r_{i,\alpha}}\right)^{n-2}\mu_{i,\alpha}^{1-\frac{n}{2}}\lesssim\sigma_{\alpha}^{\ell,5}\mu_{i,\alpha}^{1-\frac{n}{2}}.$

Assume now that $\mu_{j,\alpha}=o(\mu_{i,\alpha})$. As before, this either
means that $d_{g}(x_{i,\alpha},x_{j,\alpha})>\frac{2r_{i,\alpha}}{R_{0}}$ or
that $j\in\mathcal{B}_{i}$. In the first case, since
$\Theta_{j,\alpha}\lesssim 1$, we obtain that

$(\Theta_{j,\alpha}B_{j,\alpha})\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{n-2}{2}}\left(\frac{\mu_{i,\alpha}}{r_{i,\alpha}}\right)^{n-2}\mu_{i,\alpha}^{1-\frac{n}{2}}\lesssim\sigma_{\alpha}^{\ell,5}\mu_{i,\alpha}^{1-\frac{n}{2}}.$

Assume finally that $j\in\mathcal{B}_{i}$. If $y\in\hat{\Omega}_{i,\alpha}\cap
B(\hat{x}_{j,\alpha},1)$ then
$d_{g}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y),x_{j,\alpha}\big{)}\lesssim\mu_{i,\alpha}$.
Hence, by (3.20),

$(\Theta_{j,\alpha}B_{j,\alpha})\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\mu_{i,\alpha}^{2-\frac{n}{2}}\lesssim\sigma_{\alpha}^{\ell,5}\mu_{i,\alpha}^{1-\frac{n}{2}}.$

If instead
$y\in\hat{\Omega}_{i,\alpha}\cap\\{y\in\hat{\Omega}_{i,\alpha},|y-{x}_{j,\alpha}|>1\\}$
then
$d_{g}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y),x_{j,\alpha}\big{)}\gtrsim\mu_{i,\alpha}$
and hence, since $\Theta_{j,\alpha}\lesssim 1$,

$(\Theta_{j,\alpha}B_{j,\alpha})\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\left(\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\right)^{\frac{n-2}{2}}\mu_{i,\alpha}^{1-\frac{n}{2}}\lesssim\sigma_{\alpha}^{\ell,5}\mu_{i,\alpha}^{1-\frac{n}{2}}.$

Gathering these estimates proves (3.62). Let now, for
$y\in\hat{\Omega}_{i,\alpha}$

$\psi_{\alpha}(y)=\frac{1}{\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}}\hat{\varphi}_{\alpha}(y)$

where $\hat{\varphi}_{\alpha}$ was defined in (3.51). With (3.41), (3.60),
(3.61) and (3.62) we obtain that $\psi_{\alpha}$ satisfies

$|\psi_{\alpha}(y)|\lesssim(1+|y|)^{2-n}+o(1)\quad\textrm{ for all
}y\in\hat{\Omega}_{i,\alpha}.$ (3.63)

Independently, a straightforward adaptation of the arguments that led to
(3.62) shows that for all $y\in\hat{\Omega}_{i,\alpha}$,

$\sum_{j=1}^{k}(\Xi_{j,\alpha}B_{j,\alpha})\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\lesssim\mu_{i,\alpha}^{1-\frac{n}{2}}\sigma_{\alpha}^{\ell,5}$

holds, where $\Xi_{j,\alpha}$ is as in (A.3). Combining the latter with
(3.44), (3.60) and (3.61) we also obtain that

$|\nabla\psi_{\alpha}(y)|\lesssim(1+|y|)^{1-n}+o(1)$ (3.64)

for all $y\in\hat{\Omega}_{i,\alpha}$. If $y\in\hat{\Omega}_{i,\alpha}$,
$\psi_{\alpha}$ also satisfies by (3.53)

$|\triangle_{g_{i,\alpha}}\psi_{\alpha}(y)|\lesssim 1.$

Since $\hat{\Omega}_{i,\alpha}$ converges towards
$\mathbb{R}^{n}\backslash\\{(\hat{x}_{i,\infty})_{\in\mathcal{C}_{i}}\\}$,
standard elliptic theory and (3.63) show that $\psi_{\alpha}$ converges in
$C^{1,\beta}_{loc}(\mathbb{R}^{n}\backslash\\{(\hat{x}_{i,\infty})_{\in\mathcal{C}_{i}}\\})$,
for some $0<\beta<1$, towards $\psi_{\infty}$, where $\mathcal{C}_{i}$ is as
in (3.25). By passing (3.63) and (3.64) to the limit we see that
$\psi_{\infty}$ satisfies

$|\psi_{\infty}(y)|+(1+|y|)|\nabla\psi_{\infty}(y)|\lesssim(1+|y|)^{2-n}\quad\textrm{
for all
}y\in\mathbb{R}^{n}\backslash\\{(\hat{x}_{i,\infty})_{\in\mathcal{C}_{i}}\\}.$
(3.65)

Standard arguments then show that $\psi_{\infty}\in D^{1,2}(\mathbb{R}^{n})$.
With (3.61), (3.64) and the dominated convergence theorem, we can also pass
(3.52) to the limit to obtain

$\int_{\mathbb{R}^{n}}\langle\nabla\psi_{\infty},\nabla Z_{i,m}\rangle dx=0$

for any $1\leq m\leq n_{i}$. Hence, $\psi_{\infty}\in K_{V_{i}}^{\perp}$ where
$K_{V_{i}}$ is as in (2.6).

We will now show that $\psi_{\infty}\in K_{V_{i}}$ and
$\psi_{\infty}\not\equiv 0$ which will yield the desired contradiction. Let
$y\in\mathbb{R}^{n}\backslash\\{(\hat{x}_{i,\infty})_{\in\mathcal{C}_{i}}\\}$
be fixed. We first claim that, if $j\neq i$,

$\mu_{i,\alpha}^{2}B_{j,\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}^{2^{*}-2}\to
0\quad\textrm{ as }\alpha\to+\infty.$

This easily follows from the arguments developed in the proof of (3.62). With
(3.5) we then have that

$\frac{1}{\|\varphi_{\alpha}\|_{L^{\infty}(\Omega_{i,\alpha})}}\mu_{i,\alpha}^{2}\Big{(}|\mathcal{W}_{\alpha}|^{2^{*}-2}\varphi_{\alpha}\Big{)}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}y)\big{)}\to
V_{i}(y)^{2^{*}-2}\psi_{\infty}(y)$

for all $y\in\hat{\Omega}_{i,\alpha}$ as $\alpha\to+\infty$. Using (3.55),
(3.56), (3.57) and (3.61) we can then pass the equation satisfied by
$\psi_{\alpha}$ to the limit to see that $\psi_{\infty}$ satisfies

$\triangle_{\xi}\psi_{\infty}=(2^{*}-1)|V_{i}|^{2^{*}-2}\psi_{\infty}\quad\textrm{
in }\mathbb{R}^{n}\backslash\\{(\hat{x}_{i,\infty})_{\in\mathcal{C}_{i}}\\}.$

By (3.65) the singularities of $\psi_{\infty}$ are removable, and
$\psi_{\infty}$ extends to a $D^{1,2}(\mathbb{R}^{n})\cap
C^{1,\beta}(\mathbb{R}^{n})$ function in $\mathbb{R}^{n}$, still denoted by
$\psi_{\infty}$, satisfying

$\triangle_{\xi}\psi_{\infty}=(2^{*}-1)|V_{i}|^{2^{*}-2}\psi_{\infty}\quad\textrm{
in }\mathbb{R}^{n}.$

This proves that $\psi_{\infty}\in K_{V_{i}}$ and hence that
$\psi_{\infty}\equiv 0$. It remains to see that $\psi_{\infty}\not\equiv 0$.
By definition $\|\psi_{\alpha}\|_{L^{\infty}(\hat{\Omega}_{i,\alpha})}=1$
holds. If we let, for all $\alpha$, $y_{\alpha}\in\hat{\Omega}_{i,\alpha}$ be
such that $|\psi_{\alpha}(y_{\alpha})|=1$, (3.63) shows that
$|y_{\alpha}|\lesssim 1$ and hence, up to a subsequence, that $y_{\alpha}\to
y_{\infty}\in\mathbb{R}^{n}$. If
$y_{\infty}\in\mathbb{R}^{n}\backslash\\{(\hat{x}_{i,\infty})_{\in\mathcal{C}_{i}}\\}$
the local $C^{1,\beta}$ convergence shows that
$|\psi_{\infty}(y_{\infty})|=1$, a contradiction. If instead
$y_{\alpha}\to\hat{x}_{j,\infty}$ for some $j\in\mathcal{C}_{i}$ we can find,
by (3.64), another sequence of points of $\hat{\Omega}_{i,\alpha}$, that we
call $(w_{\alpha})_{\alpha}$, such that $w_{\alpha}\to
w\in\mathbb{R}^{n}\backslash\\{(\hat{x}_{i,\infty})_{\in\mathcal{C}_{i}}\\}$
and $\liminf_{\alpha\to+\infty}|\psi_{\alpha}(w_{\alpha})|>0$. This proves
again that $\psi_{\infty}\not\equiv 0$, contradicts (3.61) and concludes the
proof of Lemma 3.7. ∎

The remark after (3.36) explained how the “neck regions”

$B_{g}(x_{j,\alpha},\rho_{j,i,\alpha})\backslash\cup_{s\in\mathcal{B}_{i}}B_{g}(x_{s,\alpha},\frac{r_{s,\alpha}}{R_{0}})\quad
j\in\mathcal{B}_{i},$

offer a negligible contribution to the pointwise estimates on
$\varphi_{\alpha}$, hence did not need to be taken into account in the
induction property ($H_{\ell,1}$). The choice of the radius
$\rho_{j,i,\alpha}$ might seem odd at first – a more natural choice in view of
the pointwise relations between bubbles would have been to choose the radius
$s_{j,i,\alpha}$ as in (3.15) – but is crucial in order to ensure the
$C^{1}$-boundedness of $\psi_{\infty}$ in the contradiction argument of Lemma
3.7 (see (3.65)).

###### End of the proof of Proposition 3.5.

We are now in position to conclude the proof of the induction argument. We
plug (3.59) into (3.41), (3.42), (3.44) and (3.45). This shows that, for any
sequence $(x_{\alpha})_{\alpha}\in
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$:

$\displaystyle|\varphi_{\alpha}(x_{\alpha})|\lesssim
a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-2}}+\gamma_{\alpha}\sum_{j=1}^{k}\Theta_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$
$\displaystyle+\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,5}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)},$

where $\Theta_{j,\alpha}$ is as in (3.3), that

$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
(3.66)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,5}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)},$
$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{3r_{i,\alpha}}{2R_{0}})}d_{g}(x_{\alpha},\cdot)^{1-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,5}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)},$

where $\Xi_{j,\alpha}$ is as in (A.3) and that

$\displaystyle|\nabla\varphi_{\alpha}(x_{\alpha})|\lesssim
a_{\alpha}\frac{\mu_{i,\alpha}^{\frac{n-2}{2}}}{r_{i,\alpha}^{n-1}}+\gamma_{\alpha}\sum_{j=1}^{k}\Xi_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$
(3.67)
$\displaystyle+\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,5}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{0,\alpha}(x_{\alpha})+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\Big{)},$

where the sequence $\sigma_{\alpha}^{\ell,5}$ is as in (3.60). The first
inequality remains true for all $i\in E_{\ell}$, and this together with
$(H_{\ell+1,1})$ (in case $\ell\leq p-1$) proves ($H_{\ell,1}$). To conclude
the proof of ($H_{\ell,2}$) it remains to prove that (3.66) remains true for
for any sequence $(x_{\alpha})_{\alpha}$ of points of $M$. First, it is easily
seen that (3.34), (3.35) and (3.36) remain true for any sequence
$(x_{\alpha})_{\alpha}$ of points of $M$, and not only of
$B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$ (remark that (3.35) follows
from $(H_{\ell+1,2})$ in case $\ell\leq p-1$). If $(x_{\alpha})_{\alpha}$ is
any sequence of points in $M$ we now have, using ($H_{\ell,1}$), that

$\displaystyle\int_{B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}|\varphi_{\alpha}|dv_{g}$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{\ell,5}(\gamma_{\alpha}+a_{\alpha})\big{)}\Big{(}B_{i,\alpha}(x_{\alpha})+\sum_{j\in\mathcal{B}_{i}}B_{j,\alpha}(x_{\alpha})\Big{)}.$

The second integral in (3.66) is computed in the same way and (3.66) remains
true for any sequence $(x_{\alpha})_{\alpha}$ in $M$. This proves
($H_{\ell,2}$) and concludes the proof of Proposition 3.5. ∎

###  3.5. Proof of Theorem 3.2 part $2$: end of the proof

We are now in position to end the proof of Theorem 3.2. Let $G_{0}$ be the
Green’s function of the operator $\triangle_{g}+h-f^{\prime}(u_{0})$ in $M$,
where $f$ is as in (3.9). $G_{0}$ is by definition orthogonal to $K_{0}$,
which is defined in (2.17), and there exists $C>0$ depending only on
$n,(M,g),h$ and $u_{0}$ so that $|G_{0}(x,y)|\leq Cd_{g}(x,y)^{2-n}$ for any
$x\neq y$ in $M$ (see Robert [46]). By (3.11) $\varphi_{\alpha}$ satisfies

$\displaystyle\triangle_{g}\varphi_{\alpha}+h\varphi_{\alpha}-f^{\prime}(u_{0})\varphi_{\alpha}$
$\displaystyle=\big{(}f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})\big{)}\varphi_{\alpha}+(h-h_{\alpha})\varphi_{\alpha}$
$\displaystyle+R_{\alpha}+\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}.$

Let $(x_{\alpha})_{\alpha}$ be a sequence of points in $M$. By definition
$\varphi_{\alpha}\in K_{0}^{\perp}$ and so a representation formula for
$\triangle_{g}+h-f^{\prime}(u_{0})$ in $M$ shows, using (3.26), (3.27), (3.29)
and (A.1) below, that there exists a sequence $(\sigma_{\alpha})_{\alpha}$
satisfying (3.7) such that

$\displaystyle|\varphi_{\alpha}(x_{\alpha})|$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\sum_{j=0}^{k}B_{j,\alpha}(x_{\alpha})$
(3.68)
$\displaystyle+\gamma_{\alpha}\sum_{j=1}^{k}\Theta_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$
$\displaystyle+\int_{M}d_{g}(x_{\alpha},\cdot)^{2-n}\big{|}f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})\big{|}|\varphi_{\alpha}|dv_{g}.$

By (3.19) we have

$|f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})|\lesssim\Big{(}\sum_{i=1}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2}\quad\textrm{
in }\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}}).$

Proposition 3.5 (more precisely ($H_{\ell,2}$)) then shows that

$\displaystyle\int_{\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})}$
$\displaystyle
d_{g}(x_{\alpha},\cdot)^{2-n}\big{|}f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})\big{|}|\varphi_{\alpha}|dv_{g}$
(3.69)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}(\gamma_{\alpha}+a_{\alpha})\big{)}\sum_{j=0}^{k}B_{j,\alpha}(x_{\alpha}).$

To estimate the integral over
$M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$
we distinguish two cases. Assume first that $B_{0,\alpha}\equiv 0$. According
to (3.4) this corresponds to the case where $u_{0}\equiv 0$ and
$\triangle_{g}+h$ has no kernel. Then $f^{\prime}(u_{0})\equiv 0$ and on
$M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$
we have

$|f^{\prime}(\mathcal{W}_{\alpha})|\lesssim\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-2}.$

Straightforward computations with (3.27) and (A.7) below then show that

$\displaystyle\int_{M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\big{|}f^{\prime}(\mathcal{W}_{\alpha})\big{|}|\varphi_{\alpha}|dv_{g}$
(3.70) $\displaystyle\lesssim
a_{\alpha}\sum_{j=1}^{k}\left(\frac{\mu_{j,\alpha}}{r_{j,\alpha}}\right)^{2}B_{j,\alpha}(x_{\alpha})=\sigma_{\alpha}a_{\alpha}\sum_{j=1}^{k}B_{i,\alpha}(x_{\alpha})$

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7). Assume now
that $B_{0,\alpha}\equiv 1$, which happens when $u_{0}\not\equiv 0$ or when
$u_{0}\equiv 0$ and $\triangle_{g}+h$ has kernel. By (3.16) and (3.17)
$r_{i,\alpha}\leq\sqrt{\mu_{i,\alpha}}$ holds true for all $1\leq i\leq k$ and
we can decompose
$M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})$
as

$\displaystyle
M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})\subseteq$
$\displaystyle\Big{(}M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})\Big{)}\bigcup\bigcup_{i=1}^{k}\Big{(}B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})\backslash\bigcup_{j=1}^{k}B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})\Big{)}.$

If $i\in\\{1,\dots,k\\}$ we have, by (3.4) and for all $y\in
B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})\backslash\bigcup_{j=1}^{k}B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})$,

$\big{|}f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})\big{|}(y)\lesssim\sum_{i=1}^{k}B_{i,\alpha}(y)^{2^{*}-2}.$

As a consequence and by (3.27) and (A.7) below, for any $i\in\\{1,\dots,k\\}$
we have

$\displaystyle\int\limits_{B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})\backslash\bigcup_{j=1}^{k}B_{g}(x_{j,\alpha},\frac{r_{j,\alpha}}{R_{0}})}d_{g}(x_{\alpha},\cdot)^{2-n}\big{|}f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})\big{|}|\varphi_{\alpha}|dv_{g}$
(3.71)
$\displaystyle\lesssim\int\limits_{B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})\backslash
B_{g}(x_{i,\alpha},\frac{r_{i,\alpha}}{R_{0}})}a_{\alpha}d_{g}(x_{\alpha},\cdot)^{2-n}B_{i,\alpha}^{2^{*}-1}dv_{g}$
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}\sum_{j=1}^{k}B_{j,\alpha}(x_{\alpha}).$

Since $B_{0,\alpha}=1$ we have

$\Theta_{i,\alpha}(y)B_{i,\alpha}(y)\lesssim\sigma_{\alpha}(B_{i,\alpha}(y)+B_{0,\alpha}(y))$
(3.72)

for all $y\in M$. This follows e.g. from (3.3) and the simple inequality

$\Theta_{i,\alpha}B_{i,\alpha}\lesssim\sqrt{\mu_{i,\alpha}}B_{i,\alpha}\mathds{1}_{\\{d_{g}(x_{i,\alpha},\cdot)\leq\sqrt{\mu_{i,\alpha}}\\}}+\sqrt{\mu_{i,\alpha}}\mathds{1}_{\\{d_{g}(x_{i,\alpha},\cdot)\geq\sqrt{\mu_{i,\alpha}}\\}}.$

If $y\in
M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})$ we then
have

$\big{|}f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})\big{|}(y)\lesssim\sum_{i=1}^{k}B_{i,\alpha}(y)+\sum_{i=1}^{k}B_{i,\alpha}(y)^{2^{*}-2}+\sigma_{\alpha}B_{0,\alpha}$

for some sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7), so that
(3.72), (A.1) and (A.7) below give

$\displaystyle\int_{M\backslash\bigcup_{i=1}^{k}B_{g}(x_{i,\alpha},\sqrt{\mu_{i,\alpha}})}$
$\displaystyle
d_{g}(x_{\alpha},\cdot)^{2-n}\big{|}f^{\prime}(\mathcal{W}_{\alpha})-f^{\prime}(u_{0})\big{|}|\varphi_{\alpha}|dv_{g}$
(3.73)
$\displaystyle\lesssim\sigma_{\alpha}a_{\alpha}\sum_{j=0}^{k}B_{j,\alpha}(x_{\alpha}).$

By plugging (3.69), (3.70), (3.71) and (3.73) in (3.68) we have thus proven
that there exists a sequence $(\sigma_{\alpha})_{\alpha}$ satisfying (3.7)
such that

$\displaystyle|\varphi_{\alpha}(x_{\alpha})|$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}(\gamma_{\alpha}+a_{\alpha})\big{)}\sum_{j=0}^{k}B_{j,\alpha}(x_{\alpha})$
(3.74)
$\displaystyle+\gamma_{\alpha}\sum_{j=1}^{k}\Theta_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$

for any sequence $(x_{\alpha})_{\alpha}$ in $M$. Estimates on
$|\nabla\varphi_{\alpha}|$ are obtained in a similar way, just as we did in
(3.44): a straightforward adaptation of the arguments that led to (3.69),
(3.70), (3.71) and (3.73) then shows that

$\displaystyle|\nabla\varphi_{\alpha}(x_{\alpha})|$ (3.75)
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}(\gamma_{\alpha}+a_{\alpha})\big{)}\big{(}B_{0,\alpha}+\sum_{j=1}^{k}\theta_{j,\alpha}(x_{\alpha})^{-1}B_{j,\alpha}(x_{\alpha})\big{)}$
$\displaystyle+\gamma_{\alpha}\sum_{j=1}^{k}\Xi_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$

for the same sequence $(\sigma_{\alpha})_{\alpha}$ as in (3.74) and for any
sequence $(x_{\alpha})_{\alpha}$ of points of $M$. Going back to the
definition of $a_{\alpha}$ in (3.27) we let $(y_{\alpha})_{\alpha}$ be a
sequence of points of $M$ such that

$\left|\frac{\varphi_{\alpha}(y_{\alpha})}{\sum_{i=0}^{k}B_{i,\alpha}(y_{\alpha})}\right|+\left|\frac{\nabla\varphi_{\alpha}(y_{\alpha})}{B_{0,\alpha}(y_{\alpha})+\sum_{i=1}^{k}\theta_{i,\alpha}(y_{\alpha})^{-1}B_{i,\alpha}(y_{\alpha})}\right|=a_{\alpha}.$

Since $\Theta_{j,\alpha}\lesssim 1$ and $\Xi_{j,\alpha}\lesssim 1$ for all
$1\leq j\leq k$ we have
$\sum_{j=1}^{k}\Theta_{j,\alpha}B_{j,\alpha}\lesssim\sum_{i=1}^{k}B_{i,\alpha}$
and
$\sum_{j=1}^{k}\Xi_{j,\alpha}B_{j,\alpha}\lesssim\sum_{j=1}^{k}\theta_{j,\alpha}^{-1}B_{j,\alpha}$
in $M$. Applying (3.74) and (3.75) at $x_{\alpha}=y_{\alpha}$ then yields

$a_{\alpha}\lesssim\tau_{\alpha}+\eta_{\alpha}+\gamma_{\alpha}+\sigma_{\alpha}a_{\alpha}.$
(3.76)

Since $\sigma_{\alpha}\to 0$ by (3.7), _up to passing to a subsequence for the
sequence $(\sigma_{\alpha})_{\alpha}$ only_ (which amounts to passing to a
common subsequence for $(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$,
$1\leq i\leq k$, and $(h_{\alpha})_{\alpha}$), the previous inequality self-
improves and gives

$a_{\alpha}\lesssim\tau_{\alpha}+\eta_{\alpha}+\gamma_{\alpha}.$

Replugging the latter in (3.74) finally gives

$\displaystyle|\varphi_{\alpha}(x_{\alpha})|$
$\displaystyle\lesssim\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}\gamma_{\alpha}\big{)}\sum_{j=0}^{k}B_{j,\alpha}(x_{\alpha})+\gamma_{\alpha}\sum_{j=1}^{k}\Theta_{j,\alpha}(x_{\alpha})B_{j,\alpha}(x_{\alpha})$

for any sequence $(x_{\alpha})_{\alpha}$ in $M$, and concludes the proof of
Theorem 3.2.

$\square$

######  Remark 3.8.

The proof of Theorem 3.2 keeps track of all the sequences appearing in the
estimates by distinguishing the error terms coming from (3.10) from those that
originate from the interaction between the bubbles $V_{i,\alpha}$ themselves.
It is particularly important to single out, in the estimates, the sequences
satisfying (3.7), i.e. that only depend on the choice of
$(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$, and
$(h_{\alpha})_{\alpha}$. See for instance the final self-improving argument of
(3.76) that proves that, up to passing to a subsequence for the sequences
$(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$, and
$(h_{\alpha})_{\alpha}$ only, estimate (3.12) holds true for any of the
sequences $(\tau_{\alpha})_{\alpha},(\eta_{\alpha})_{\alpha}$ and
$(\gamma_{\alpha})_{\alpha}$. This will be used again in the proof of
Proposition 4.2 below.

######  Remark 3.9.

Theorem 3.2 remains true for a slighty more general choice of $K_{\alpha}$ in
(2.18). A careful inspection of the proof shows for instance that when $i\geq
1$ one can replace the $Z_{i,j,\alpha}$ explicitly given by (2.16) by another
family $(Z_{i,j,\alpha})_{\alpha}$ of smooth functions in $M$, $1\leq j\leq
n_{i}$, satisfying

$\big{|}(\triangle_{g}+1)Z_{i,j,\alpha}\big{|}\lesssim
B_{i,\alpha}^{2^{*}-1}+B_{i,\alpha},\quad|Z_{i,j,\alpha}|\lesssim
B_{i,\alpha},\quad|\nabla
Z_{i,j,\alpha}|\lesssim\theta_{i,\alpha}^{-1}B_{i,\alpha}\textrm{ and }$
$\mu_{i,\alpha}^{\frac{n-2}{2}}Z_{i,j,\alpha}\big{(}\exp_{x_{i,\alpha}}(\mu_{i,\alpha}x)\big{)}\to
Z_{i,j}(x)\textrm{ in }C^{0}_{loc}(\mathbb{R}^{n}).$

##  4\. Nonlinear procedure and proof of Theorem 1.1

In this section we prove Theorem 1.1. We first perform a nonlinear
perturbative argument to show that (1.1) possesses, up to kernel elements, a
canonical solution that looks like $\mathcal{W}_{\alpha}$ at first-order and
that comes with explicit pointwise bounds. This is the content of Proposition
4.2, whose proof heavily relies on Theorem 3.2. We then apply this
construction to obtain _a priori_ bounds and prove a slightly more general
result, Theorem 4.3 below, from which Theorem 1.1 follows.

###  4.1. Nonlinear procedure

As in Section 3 we let $(h_{\alpha})_{\alpha}$ be a sequence of functions
converging towards $h$ in $L^{\infty}(M)$ as $\alpha\to+\infty$, $k\geq 1$ be
an integer, $u_{0}\in C^{1,\beta}(M)$ for some $0<\beta<1$ solve

$\triangle_{g}u_{0}+hu_{0}=|u_{0}|^{2^{*}-2}u_{0},$

$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ be
$k$ sequences satisfying (1.4), we let $V_{1},\dots,V_{k}$ in
$D^{1,2}(\mathbb{R}^{n})$ be $k$ solutions of (2.1) and we let
$(\mathcal{W}_{\alpha})_{\alpha}$ be a sequence of continuous functions
satisfying (3.5). We assume in addition that there exists a positive constant
$C_{1}$, only depending on $n,(M,g),V_{1},\dots,V_{k},u_{0}$ and $h$, and a
sequence $(\varepsilon_{\alpha})_{\alpha}\in\mathcal{E}$ such that for all
$\alpha\geq 1$

$\displaystyle\Big{|}\triangle_{g}\mathcal{W}_{\alpha}+h_{\alpha}\mathcal{W}_{\alpha}-$
$\displaystyle|\mathcal{W}_{\alpha}|^{2^{*}-2}\mathcal{W}_{\alpha}\Big{|}\leq
C_{1}(\varepsilon_{\alpha}+\|h_{\alpha}-h\|_{L^{\infty}(M)})B_{0,\alpha}$
(4.1)
$\displaystyle+C_{1}\sum_{i=1}^{k}B_{i,\alpha}+C_{1}\sum\limits_{\underset{i\neq
j}{i,j=0..k}}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}+C_{1}\varepsilon_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}$

in $M$. We keep in this section the notations of Section 3, in particular
(3.6) and (3.7). Our first result constructs a canonical solution of (1.1) in
$H^{1}(M)$, up to kernel elements, that looks like $\mathcal{W}_{\alpha}$ at
first-order:

######  Proposition 4.1.

Let $(\mathcal{W}_{\alpha})$ be as in (3.5) and (4.1). There exists $\delta>0$
such that, up to a passing to a subsequence for
$(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$,
$(h_{\alpha})_{\alpha}$ and $(\varepsilon_{\alpha})_{\alpha}$, there exists a
unique solution

$\varphi_{\alpha}\in K_{\alpha}^{\perp}\cap\\{\varphi\in
H^{1}(M),\|\varphi\|_{H^{1}(M)}\leq\delta\\}$

of

$\Pi_{K_{\alpha}^{\perp}}\Bigg{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}-(\triangle_{g}+1)^{-1}\Big{[}(1-h_{\alpha})(\mathcal{W}_{\alpha}+\varphi_{\alpha})+f\big{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}\big{)}\Big{]}\Bigg{)}=0,$

where $f$ is as in (3.9).

This result is a simple variant of the standard nonlinear procedure found e.g.
in Esposito-Pistoia-Vétois [23] or Robert-Vétois [49] and we omit its proof.
We now prove that the linear theory of Theorem 3.2 provides pointwise
estimates on the remainder constructed in Proposition 4.1:

######  Proposition 4.2.

Let $(h_{\alpha})_{\alpha}$ be a sequence of functions converging towards $h$
in $L^{\infty}(M)$, $k\geq 1$ be an integer, $u_{0}\in C^{1,\beta}(M)$ for
$0<\beta<1$ be a solution of (2.15), $V_{1},\dots,V_{k}$ in
$D^{1,2}(\mathbb{R}^{n})$ be $k$ solutions of (2.1) and let
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ be
$k$ sequences satisfying (1.4). Let $(\mathcal{W}_{\alpha})_{\alpha}$ be a
sequence of continuous functions in $M$ satisfying (3.5) and (4.1) for some
$(\varepsilon_{\alpha})_{\alpha}\in\mathcal{E}$. There exists $C>0$
independent of $\alpha$ and there exists a sequence
$(\sigma_{\alpha})_{\alpha}$ satisfying (3.7) such that, up to passing to a
subsequence for $(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq
i\leq k$, $(h_{\alpha})_{\alpha}$ and $(\varepsilon_{\alpha})_{\alpha}$, the
equation

$\Pi_{K_{\alpha}^{\perp}}\Bigg{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}-(\triangle_{g}+1)^{-1}\Big{[}(1-h_{\alpha})(\mathcal{W}_{\alpha}+\varphi_{\alpha})+f\big{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}\big{)}\Big{]}\Bigg{)}=0$
(4.2)

has a unique solution $\varphi_{\alpha}$ in

$\displaystyle K_{\alpha}^{\perp}\cap\Bigg{\\{}\varphi\in C^{0}(M),$
$\displaystyle|\varphi(x)|\leq
C(\varepsilon_{\alpha}+\sigma_{\alpha})^{\frac{1}{2}}\sum_{i=0}^{k}B_{i,\alpha}(x)$
$\displaystyle+C\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\quad\textrm{
for all }x\in M\Bigg{\\}}.$

This solution $\varphi_{\alpha}$ also satisfies

$\|\varphi_{\alpha}\|_{H^{1}(M)}\leq C(\varepsilon_{\alpha}+\sigma_{\alpha}).$
(4.3)

Recall that $\mathcal{E}$ is as in (3.1), $\Theta_{i,\alpha}$ is as in (3.3)
and $f$ is as in (3.9).

###### Proof.

Let $(h_{\alpha})_{\alpha}$, $u_{0}$, $V_{1},\dots,V_{k}$,
$(x_{i,\alpha})_{\alpha}$ and $(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$ be
as in the statement of Proposition 4.2 and let
$(\mathcal{W}_{\alpha})_{\alpha}$ be a sequence of continuous functions
satisfying (3.5) and (4.1). Throughout this proof we fix a real number
$0<\tau<2^{*}-2$. We claim that there exists a sequence
$(\sigma_{\alpha}^{1})_{\alpha}$ satisfying (3.7) such that for all $x\in M$

$\displaystyle\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)\Big{)}^{2^{*}-1-\tau}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{\tau}+\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{2^{*}-1}$
(4.4)
$\displaystyle+\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)\Big{)}^{1+\tau}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}^{2^{*}-2-\tau}$
$\displaystyle+\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)\Big{)}^{2^{*}-2}\Big{(}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\Big{)}$
$\displaystyle\leq\sigma_{\alpha}^{1}\Big{(}\sum_{i=0}^{k}B_{i,\alpha}(x)+\sum_{i=1}^{k}B_{i,\alpha}(x)^{2^{*}-1}\Big{)}.$

This easily follows from Lemma A.2 proven in the Appendix. Since
$\mathcal{W}_{\alpha}$ satisfies (3.5) Theorem 3.2 applies and, up to passing
to a subsequence for $(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$,
$1\leq i\leq k$ and $(h_{\alpha})_{\alpha}$, shows the existence of $C_{0}>0$
and of a sequence $(\sigma_{\alpha}^{2})_{\alpha}$ satisfying (3.7) such that
for any $(\tau_{\alpha})_{\alpha},(\eta_{\alpha})_{\alpha}\in\mathcal{E}$, any
bounded sequence $(\gamma_{\alpha})_{\alpha}$ and any $(R_{\alpha})_{\alpha}$
satisfying (3.10), (3.12) holds with $\sigma_{\alpha}=\sigma_{\alpha}^{2}$. We
now let $(\varepsilon_{\alpha})_{\alpha}\in\mathcal{E}$ and assume in addition
that $\mathcal{W}_{\alpha}$ satisfies (4.1). Define the following sequence:

$\nu_{\alpha}=\big{(}\sigma_{\alpha}^{1}+\sigma_{\alpha}^{2}+\|h_{\alpha}-h_{0}\|_{L^{\infty}(M)}+\varepsilon_{\alpha}\big{)}^{\frac{1}{2}}$
(4.5)

and define

$\displaystyle\mathcal{S}_{\alpha}=\Big{\\{}\varphi\in C^{0}(M)\textrm{ such
that }$ (4.6) $\displaystyle|\varphi(x)|\leq
2C_{0}C_{1}\nu_{\alpha}\sum_{i=0}^{k}B_{i,\alpha}(x)+2C_{0}C_{1}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i,\alpha}(x)\textrm{
for all }x\in M$ $\displaystyle\textrm{ and
}\int_{M}\varphi(\triangle_{g}+1)Z_{i,j,\alpha}dv_{g}=0\textrm{ for all
}i=0\dots k,j=1\dots n_{i}\Big{\\}},$

where $C_{1}$ is the constant appearing in (4.1). We endow
$\mathcal{S}_{\alpha}$ with the norm

$\|\varphi\|_{*}=\left\|\frac{\varphi}{\nu_{\alpha}\sum_{i=0}^{k}B_{i,\alpha}+\sum_{i=1}^{k}\Theta_{i,\alpha}B_{i,\alpha}}\right\|_{L^{\infty}(M)},\quad\varphi\in\mathcal{S}_{\alpha}.$

For any fixed value of $\alpha$, $(\mathcal{S}_{\alpha},\|\cdot\|_{*})$ is a
Banach space and it is easily seen that if $\varphi\in\mathcal{S}_{\alpha}\cap
C^{1}(M)$ then $\varphi\in K_{\alpha}^{\perp}$ where $K_{\alpha}$ is as in
(2.18). If $\varphi\in C^{0}(M)$ we define $T_{\alpha}(\varphi)\in
K_{\alpha}^{\perp}$ as the unique solution in $K_{\alpha}^{\perp}$ of

$\big{(}\triangle_{g}+h_{\alpha}-f^{\prime}(\mathcal{W}_{\alpha})\big{)}T_{\alpha}(\varphi)=\tilde{R}_{\alpha}+N_{\alpha}(\varphi)+\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}$

for some unique real numbers $\lambda_{i,j}^{\alpha}$, where we have let

$\displaystyle\tilde{R}_{\alpha}$
$\displaystyle=-\triangle_{g}\mathcal{W}_{\alpha}-h_{\alpha}\mathcal{W}_{\alpha}+|\mathcal{W}_{\alpha}|^{2^{*}-2}\mathcal{W}_{\alpha}\quad\textrm{
and }$ (4.7) $\displaystyle N_{\alpha}(\varphi)$
$\displaystyle=f(\mathcal{W}_{\alpha}+\varphi)-f(\mathcal{W}_{\alpha})-f^{\prime}(\mathcal{W}_{\alpha})\varphi.$

The existence of $T_{\alpha}(\varphi)$ is ensured by Proposition 3.1, and by
standard elliptic theory $T_{\alpha}(\varphi)\in C^{1}(M)$. To prove
Proposition 4.2 we will prove that, up to a subsequence, $T_{\alpha}$ is a
contraction on $S_{\alpha}$.

Step $1$: $T_{\alpha}$ stabilizes $\mathcal{S}_{\alpha}$. If
$\varphi\in\mathcal{S}_{\alpha}$ we have

$|N_{\alpha}(\varphi)|\lesssim\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2-\tau}|\varphi|^{1+\tau}+|\varphi|^{2^{*}-1}$

in $M$. By definition of $\mathcal{S}_{\alpha}$ and by (4.4) there exists
$C_{2}>0$ such that

$\displaystyle|N_{\alpha}(\varphi)|\leq
C_{2}(\sigma_{\alpha}^{1}+\nu_{\alpha}^{1+\tau})\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}+C_{2}\sigma_{\alpha}^{1}\sum_{i=0}^{k}B_{i,\alpha}$

in $M$. Together with (4.1) we thus have

$\big{|}\tilde{R}_{\alpha}+N_{\alpha}(\varphi)\big{|}\leq\tau_{\alpha}B_{0,\alpha}+\eta_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}+\gamma_{\alpha}\Big{(}\sum_{i=1}^{k}B_{i,\alpha}+\sum\limits_{\underset{i\neq
j}{i,j=0..k}}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}$

in $M$, where we have let

$\displaystyle\tau_{\alpha}$
$\displaystyle=C_{1}(\varepsilon_{\alpha}+\|h_{\alpha}-h_{0}\|_{L^{\infty}(M)})+C_{2}\sigma_{\alpha}^{1},$
(4.8) $\displaystyle\eta_{\alpha}$
$\displaystyle=C_{1}\varepsilon_{\alpha}+C_{2}(v_{\alpha}^{1+\tau}+\sigma_{\alpha}^{1}),$
$\displaystyle\gamma_{\alpha}$ $\displaystyle=C_{1}+C_{2}\sigma_{\alpha}^{1}.$

A direct application of Theorem 3.2 with
$R_{\alpha}=\tilde{R}_{\alpha}+N_{\alpha}(\varphi)$ then shows that, for any
$x\in M$,

$\displaystyle|T_{\alpha}(\varphi)(x)|\leq
C_{0}\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{2}\gamma_{\alpha}\big{)}\sum_{i=0}^{k}B_{i,\alpha}(x)+C_{0}\gamma_{\alpha}\sum_{i=1}^{k}\Theta_{i,\alpha}(x)B_{i}(x)$

holds with $\tau_{\alpha},\eta_{\alpha},\gamma_{\alpha}$ given by (4.8). Since
$\sigma_{\alpha}^{1}\to 0$ and since, by (4.5), we have
$\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{1}\gamma_{\alpha}=o(\nu_{\alpha})$
as $\alpha\to+\infty$ we can assume, up to passing to a subsequence for
$(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$,
$(h_{\alpha})_{\alpha}$ and $(\varepsilon_{\alpha})_{\alpha}$, that

$\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{1}\gamma_{\alpha}\leq
2C_{1}\nu_{\alpha}\textrm{ and }\gamma_{\alpha}\leq 2C_{1}.$

By definition of $\mathcal{S}_{\alpha}$ in (4.6), this shows that $T_{\alpha}$
stabilises $\mathcal{S}_{\alpha}$.

Step $2$: $T_{\alpha}$ is a contraction on $\mathcal{S}_{\alpha}$. Let
$\varphi_{1},\varphi_{2}\in\mathcal{S}_{\alpha}$. By definition
$T_{\alpha}(\varphi_{1})-T_{\alpha}(\varphi_{2})$ satisfies

$\displaystyle\big{(}\triangle_{g}+h_{\alpha}$
$\displaystyle-f^{\prime}(\mathcal{W}_{\alpha})\big{)}(T_{\alpha}(\varphi_{1})-T_{\alpha}(\varphi_{2}))=\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}\tilde{\lambda}_{i,j,\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}$
(4.9)
$\displaystyle+f(\mathcal{W}_{\alpha}+\varphi_{1})-f(\mathcal{W}_{\alpha}+\varphi_{2})-f^{\prime}(\mathcal{W}_{\alpha})(\varphi_{1}-\varphi_{2}),$

for some real numbers $\tilde{\lambda}_{i,j,\alpha}$. By the mean-value
inequality we have

$\displaystyle\big{|}f(\mathcal{W}_{\alpha}+\varphi_{1})$
$\displaystyle-f(\mathcal{W}_{\alpha}+\varphi_{2})-f^{\prime}(\mathcal{W}_{\alpha})(\varphi_{1}-\varphi_{2})\big{|}$
$\displaystyle\lesssim\Big{(}\sum_{i=0}^{k}B_{i,\alpha}\Big{)}^{2^{*}-2-\tau}(|\varphi_{1}|^{\tau}+|\varphi_{2}|^{\tau})|\varphi_{1}-\varphi_{2}|$

in $M$. By (4.4), by definition of $\mathcal{S}_{\alpha}$ in (4.6) and of the
norm $\|\cdot\|_{*}$ we get that there exists a positive constant $C_{3}$ such
that

$\displaystyle\big{|}f(\mathcal{W}_{\alpha}+\varphi_{1})-f(\mathcal{W}_{\alpha}+\varphi_{2})-f^{\prime}(\mathcal{W}_{\alpha})(\varphi_{1}-\varphi_{2})\big{|}$
$\displaystyle\leq
C_{3}\|\varphi_{1}-\varphi_{2}\|_{*}(\nu_{\alpha}^{1+\tau}+\nu_{\alpha}^{\tau}\sigma_{\alpha}^{1}+\nu_{\alpha}\sigma_{\alpha}^{1}+\sigma_{\alpha}^{1})\Big{(}\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}+\sum_{i=0}^{k}B_{i,\alpha}\Big{)}$

everywhere in $M$. Note, by (4.6), that $\|\varphi_{1}-\varphi_{2}\|_{*}$ is
uniformly bounded. We thus have

$\displaystyle\big{|}f(\mathcal{W}_{\alpha}+\varphi_{1})$
$\displaystyle-f(\mathcal{W}_{\alpha}+\varphi_{2})-f^{\prime}(\mathcal{W}_{\alpha})(\varphi_{1}-\varphi_{2})\big{|}$
$\displaystyle\leq\tau_{\alpha}B_{0,\alpha}+\eta_{\alpha}\sum_{i=1}^{k}B_{i,\alpha}^{2^{*}-1}+\gamma_{\alpha}\Big{(}\sum_{i=1}^{k}B_{i,\alpha}+\sum\limits_{\underset{i\neq
j}{i,j=0..k}}B_{i,\alpha}^{2^{*}-2}B_{j,\alpha}\Big{)}$

in $M$, where we have let

$\tau_{\alpha}=\eta_{\alpha}=\gamma_{\alpha}=C_{3}\|\varphi_{1}-\varphi_{2}\|_{*}\big{(}\nu_{\alpha}^{1+\tau}+\nu_{\alpha}^{\tau}\sigma_{\alpha}^{1}+\nu_{\alpha}\sigma_{\alpha}^{1}+\sigma_{\alpha}^{1}\big{)}.$

A direct application of Theorem 3.2 with
$R_{\alpha}=f(\mathcal{W}_{\alpha}+\varphi_{1})-f(\mathcal{W}_{\alpha}+\varphi_{2})-f^{\prime}(\mathcal{W}_{\alpha})(\varphi_{1}-\varphi_{2})$
then shows that

$\displaystyle\big{|}T_{\alpha}(\varphi_{1})-T_{\alpha}(\varphi_{2})\big{|}\leq
C_{0}\big{(}\tau_{\alpha}+\eta_{\alpha}+\sigma_{\alpha}^{2}\gamma_{\alpha}\big{)}\sum_{i=0}^{k}B_{i,\alpha}+C_{0}\gamma_{\alpha}\sum_{i=1}^{k}\Theta_{i,\alpha}B_{i}$

holds everywhere in $M$. We have $\gamma_{\alpha}=o(1)$ and

$C_{3}\big{(}\nu_{\alpha}^{1+\tau}+\nu_{\alpha}^{\tau}\sigma_{\alpha}^{1}+\nu_{\alpha}\sigma_{\alpha}^{1}+\sigma_{\alpha}^{1}\big{)}=o(\nu_{\alpha})$

as $\alpha\to+\infty$. Up to passing to a subsequence for
$(x_{i,\alpha})_{\alpha},(\mu_{i,\alpha})_{\alpha}$, $1\leq i\leq k$,
$(h_{\alpha})_{\alpha}$ and $(\varepsilon_{\alpha})_{\alpha}$ the latter
inequality becomes, by definition of $\|\cdot\|_{*}$,

$\displaystyle\|T_{\alpha}(\varphi_{1})-T_{\alpha}(\varphi_{2})\|_{*}$
$\displaystyle\leq\frac{1}{2}\|\varphi_{1}-\varphi_{2}\|_{*}$

and shows that $T_{\alpha}$ is a contraction in $S_{\alpha}$.

Picard’s fixed-point theorem now applies to $T_{\alpha}$ on
$(\mathcal{S}_{\alpha},\|\cdot\|_{*})$ and shows the existence of a unique
$\varphi_{\alpha}\in\mathcal{S}_{\alpha}$ such that
$T_{\alpha}(\varphi_{\alpha})=\varphi_{\alpha}$, that is such that

$(\triangle_{g}+h_{\alpha})\big{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}\big{)}=f\big{(}\mathcal{W}_{\alpha}+\varphi_{\alpha}\big{)}+\sum\limits_{\underset{j=1..n_{i}}{i=0..k}}\lambda_{i,j}^{\alpha}(\triangle_{g}+1)Z_{i,j,\alpha}$

for some real numbers $\lambda_{i,j}^{\alpha}$. Elliptic regularity shows that
$\varphi_{\alpha}$ is of class $C^{1}$ and thus belongs to
$K_{\alpha}^{\perp}$. Clearly, for a function
$\varphi\in\mathcal{S}_{\alpha}\cap K_{\alpha}^{\perp}$, solving the latter
equation is equivalent to solving (4.2). It remains to prove (4.3) to conclude
the proof of Proposition 4.2. The equation above rewrites as