# Stability and instability results for sign-changing solutions to second-
order
critical elliptic equations

Bruno Premoselli  Bruno Premoselli, Université Libre de Bruxelles, Service
d’analyse, CP 218, Boulevard du Triomphe, B-1050 Bruxelles, Belgique.
<EMAIL_ADDRESS>and  Jérôme Vétois  Jérôme Vétois, Department of
Mathematics and Statistics, McGill University, 805 Sherbrooke Street West,
Montreal, Quebec H3A 0B9, Canada<EMAIL_ADDRESS>
(Date: February 28, 2022)

###### Abstract.

On a smooth, closed Riemannian manifold $\left(M,g\right)$ of dimension $n\geq
3$, we consider the stationary Schrödinger equation
$\Delta_{g}u+h_{0}u=\left|u\right|^{2^{*}-2}u$, where
$\Delta_{g}:=-\text{div}_{g}\nabla$, $h_{0}\in C^{1}\left(M\right)$ and
$2^{*}:=\frac{2n}{n-2}$. We prove that, up to perturbations of the potential
function $h_{0}$ in $C^{1}\left(M\right)$, the sets of sign-changing solutions
that are bounded in $H^{1}\left(M\right)$ are precompact in the $C^{2}$
topology. We obtain this result under the assumptions that $\left(M,g\right)$
is locally conformally flat, $n\geq 7$ and
$h_{0}\neq\frac{n-2}{4\left(n-1\right)}\operatorname{Scal}_{g}$ at all points
in $M$, where $\operatorname{Scal}_{g}$ is the scalar curvature of the
manifold. We then provide counterexamples in every dimension $n\geq 3$ showing
the optimality of these assumptions.

Published in Journal de Mathématiques Pures et Appliquées 167 (2022),
257–-293.

The first author was supported by the FNRS CdR grant J.0135.19, the Fonds
Thélam and an ARC Avancé 2020 grant. The second author was supported by the
Discovery Grant RGPIN-2016-04195 from the Natural Sciences and Engineering
Research Council of Canada.

##  1\. Introduction

Let $\left(M,g\right)$ be a smooth, closed (i.e. compact and without boundary)
Riemannian manifold of dimension $n\geq 3$. We consider the stationary
Schrödinger equation

$\Delta_{g}u+h_{0}u=\left|u\right|^{2^{*}-2}u\quad\text{in }M,$ (1.1)

where $\Delta_{g}:=-\text{div}_{g}\nabla$ is the Laplace–Beltrami operator on
functions, $h_{0}\in C^{1}\left(M\right)$ and $2^{*}=\frac{2n}{n-2}$ is the
critical exponent for the embeddings of the Sobolev space
$H^{1}\left(M\right)$ into Lebesgue’s spaces. In this article, we are
interested in a question of stability of sign-changing solutions to (1.1)
under perturbations of the potential function $h_{0}$. More precisely, we are
interested in the question of existence of families of functions
$\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{1}\left(M\right)$
and $\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{2}\left(M\right)$ such that $h_{\varepsilon}\to h_{0}$ in
$C^{1}\left(M\right)$ as $\varepsilon\to 0$,
$\left(u_{\varepsilon}\right)_{\varepsilon}$ is bounded in
$H^{1}\left(M\right)$ and blows up at least at some point $\xi_{0}\in M$ as
$\varepsilon\to 0$ (i.e. there exists a family of points
$\left(\xi_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ such that
$\xi_{\varepsilon}\to\xi_{0}$ and
$\left|u_{\varepsilon}\left(\xi_{\varepsilon}\right)\right|\to\infty$ as
$\varepsilon\to 0$) and for each $\varepsilon>0$, $u_{\varepsilon}$ is a sign-
changing solution to the equation

$\Delta_{g}u_{\varepsilon}+h_{\varepsilon}u_{\varepsilon}=\left|u_{\varepsilon}\right|^{2^{*}-2}u_{\varepsilon}\quad\text{in
}M.$ (1.2)

By “sign-changing solutions” we mean solutions that are allowed to change
sign. Stability questions of this type have been extensively studied in recent
years, and most of the focus has been on positive solutions. A possible
reference in book form on stability questions for this type of equations is by
Hebey [He2]. In the case of positive solutions to the Yamabe equation (i.e.
when $u_{\varepsilon}>0$ and
$h_{\varepsilon}=h_{0}=c_{n}\operatorname{Scal}_{g}$ in $M$ for all
$\varepsilon$, where $c_{n}:=\frac{n-2}{4\left(n-1\right)}$ and
$\operatorname{Scal}_{g}$ is the scalar curvature of the manifold), we refer
to the work by Brendle [Br], Brendle and Marques [BM1], Druet [Dr2], Khuri,
Marques and Schoen [KMS], Li and Zhang [LZha1, LZha2], Li and Zhu [LZhu],
Marques [M] and Schoen [Sc2, Sc3] (see also the survey article by Brendle and
Marques [BM2]). (In)Stability results for positive solutions in the case of
more general potential functions can be found, among others, in the articles
by Druet [Dr1], Esposito, Pistoia and Vétois [EPV], Premoselli and Thizy [PT]
and Robert and Vétois [RV4].

As regards the existence of sign-changing blowing-up solutions, a well-studied
case is the Yamabe equation on the standard sphere; see in this case the
historic reference by Ding [Di] and the more recent work by del Pino, Musso,
Pacard and Pistoia [dPMPP1, dPMPP2], Musso and Medina [MM], Musso and Wei [MW]
and Medina, Musso and Wei [MMW]. Few compactness and stability results for
sign-changing solutions of equations of type (1.1) have been obtained so far,
which was mainly due to the lack of a satisfactory asymptotic description for
energy-bounded sequences of solutions to (1.2). Possible references on the
topic include the work by Deng, Musso and Wei [DMW], Pistoia and Vétois [PiV],
Premoselli [P], Premoselli and Vétois [PrV1], Robert and Vétois [RV2, RV3] and
Vétois [V]. Finally, other possible references on the existence and
multiplicity of sign-changing solutions to the Yamabe equation include the
articles by Ammann and Humbert [AH], Clapp [C], Clapp and Fernández [CF],
Clapp, Pistoia and Tavares [CPT], Fernandez and Petean [FP] and Gursky and
Pérez-Ayala [GP]. Needless to say, we do not pretend to any exhaustivity in
these lists.

In this article we obtain new stability results extending the one in Vétois
[V] and we construct explicit counter-examples in all dimensions to show their
sharpness. Our results highlight new and unexpected blow-up phenomena that are
specific to the sign-changing regime.

###  1.1. A stability result for locally conformally flat manifolds

We obtain the following stability result:

######  Theorem 1.1.

Let $(M,g)$ be a smooth, closed, locally conformally flat Riemannian manifold
of dimension $n\geq 7$ and $\left(h_{\varepsilon}\right)_{0\leq\varepsilon\ll
1}$ be a family of functions in $C^{1}\left(M\right)$ such that
$h_{\varepsilon}\to h_{0}$ in $C^{1}\left(M\right)$ and

$h_{0}\left(x\right)\neq
c_{n}\operatorname{Scal}_{g}\left(x\right)\quad\forall x\in M.$ (1.3)

Then every family of solutions $\left(u_{\varepsilon}\right)_{0<\varepsilon\ll
1}$ to (1.2) that is bounded in $H^{1}\left(M\right)$ converges up to a
subsequence in $C^{2}\left(M\right)$.

This result extends previous results obtained by Druet [Dr1] in the case where
$u_{\varepsilon}>0$ in $M$ for all $\varepsilon$ and Vétois [V] in the case
where $h_{0}<c_{n}\operatorname{Scal}_{g}$ in $M$. A simple corollary is that
when the assumptions of Theorem 1.1 are satisfied, for a fixed $C^{1}$
function $h_{0}$ satisfying (1.3), every bounded set in $H^{1}\left(M\right)$
of (possibly sign-changing) solutions to (1.1) is precompact in
$C^{2}\left(M\right)$.

Remark that if we remove the $H^{1}$ bound on
$\left(u_{\varepsilon}\right)_{\varepsilon}$ while assuming (1.3), then this
result does not remain true. Indeed, existence results of families of
solutions with unbounded $H^{1}$ norm have been obtained by Vétois [V] in the
case where $h_{0}<c_{n}\operatorname{Scal}_{g}$ in $M$ and $\left(M,g\right)$
is locally conformally flat and by Chen, Wei and Yan [CWY] when $n\geq 5$,
$h_{0}$ is constant and $h_{0}>n\left(n-2\right)/4$ (the solutions obtained in
[CWY] are positive; see also the article by Vétois and Wang [VW] for the $n=4$
counterpart and the article by Premoselli and Vétois [PrV1] for a construction
of sign-changing blowing-up solutions in this case). In the case where
$h_{0}\equiv c_{n}\operatorname{Scal}_{g}$, at least on the standard sphere of
dimension $n\geq 3$, it follows from Ding’s result [Di] that an analogue of
Theorem 1.1 cannot hold true.

The statement of Theorem 1.1 is very similar to the analogous result obtained
by Druet [Dr1] for positive solutions. However, in the sign-changing case,
very general sign-changing bubbles can a priori arise in the asymptotic
behavior of energy-bounded families of solutions, hence it is actually quite
surprising that the condition (1.3) remains sufficient to ensure the pre-
compactness of such families. This is crucially related to our conformal
flatness assumption as is made clear in the proof. In the case of a non-
locally conformally flat manifold, it is likely that local necessary
conditions for the blow-up of families of solutions to (1.2) would involve the
potential function $h_{0}$ together with an interplay between the local
geometry of the manifold and the arising sign-changing bubbles. It is
therefore still unclear what would be a natural generalization of Theorem 1.1
to the non-locally conformally flat setting.

The assumptions $n\geq 7$ and (1.3) in Theorem 1.1 are sharp. We prove this in
Theorems 1.3 and 1.4 below. To better understand these counterexamples it is
instructive to restrict the focus of Theorem 1.1 to specific blow-up
configurations. In this case, more precise information on the blow-up can be
obtained. In the next result, we consider solutions to (1.2) of the form

$u_{\varepsilon}=u_{0}-\left(\frac{\sqrt{n\left(n-2\right)}\mu_{\varepsilon}}{\mu_{\varepsilon}^{2}+d_{g}\left(\cdot,\xi_{\varepsilon}\right)^{2}}\right)^{\frac{n-2}{2}}+\operatorname{o}\left(1\right)\text{
in }H^{1}\left(M\right)\text{ as }\varepsilon\to 0,$ (1.4)

where $u_{0}$ is a solution to (1.1), $d_{g}$ is the geodesic distance with
respect to the metric $g$ and
$\left(\mu_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ and
$\left(\xi_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ are two families of
positive numbers and points in $M$, respectively, such that
$\mu_{\varepsilon}\to 0$ and $\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to
0$ for some point $\xi_{0}\in M$. In this case, the location of the blow-up
point $\xi_{0}$ is constrained. More precisely, we obtain the following:

######  Theorem 1.2.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\geq 3$, $h_{0}\in C^{0,\vartheta}\left(M\right)$,
$\vartheta\in\left(0,1\right)$, and $u_{0}$ be a solution to (1.1). Assume
that $\varphi_{0}\neq 0$ for all points in M, where

$\varphi_{0}:=\left\\{\begin{aligned} &u_{0}&&\text{if
}n\in\left\\{3,4,5\right\\}\\\
&h_{0}+2u_{0}-c_{n}\operatorname{Scal}_{g}&&\text{if }n=6\\\
&h_{0}-c_{n}\operatorname{Scal}_{g}&&\text{if }n\geq 7.\end{aligned}\right.$
(1.5)

Then there does not exist any families of functions
$\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{0,\vartheta}\left(M\right)$ and
$\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{2}\left(M\right)$,
positive numbers $\left(\mu_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ and
points $\left(\xi_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $M$ satisfying
(1.2) and (1.4) and such that $\mu_{\varepsilon}\to 0$,
$\xi_{\varepsilon}\to\xi_{0}$ and $h_{\varepsilon}\to h_{0}$ in
$C^{0,\vartheta}\left(M\right)$ as $\varepsilon\to 0$.

In the case of positive solutions (and possibly with multiple bubbles), this
result is due to Druet [Dr1]. In particular, it follows from Theorem 1.2 that
if a family $\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ of solutions
to (1.2) blows up as in (1.4), then $\varphi_{0}\left(\xi_{0}\right)=0$, where
$\varphi_{0}$ is as in (1.5). In dimensions $3,4$ and $5$, for configurations
of type (1.4), the weak limit $u_{0}$ thus vanishes at the blow-up point
$\xi_{0}$. Since $u_{0}$ is now a sign-changing solution to (1.1), this does
not imply that $u_{0}$ vanishes on the whole manifold, unlike in the case of
positive solutions. Note also that the assumption (1.4) allows us, in Theorem
1.2, to lift the local conformal flatness assumption from Theorem 1.1 since
there cannot be any cancellation phenomena between several sign-changing
bubbles in this case.

Our proofs of Theorems 1.1 and 1.2 rely on the pointwise estimates for sign-
changing solutions recently obtained by Premoselli [P]. The proof of Theorem
1.1 that we give here is conceptually different from the proof in case
$h_{0}<c_{n}\operatorname{Scal}_{g}$ in the article by Vétois [V], which was
based on subtle comparison arguments inspired from those in the Euclidean case
by Devillanova and Solimini [DS].

In what follows, we provide counterexamples showing the optimality of the
conditions (1.3) and $n\geq 7$ in Theorem 1.1. These counterexamples are
obtained by using the Lyapunov–Schmidt reduction method. We obtain two
different types of counterexamples, the first in dimensions $n\geq 6$ and the
second in dimensions $n\in\left\\{3,4,5\right\\}$, which are inspired from
Theorem 1.2.

###  1.2. Counterexamples: the case of dimensions $n\geq 6$

In the case where $n\geq 6$, we construct solutions of (1.2) of the form

$u_{\varepsilon}=u_{0}-\left(\frac{\sqrt{n\left(n-2\right)}\mu_{\varepsilon}}{\mu_{\varepsilon}^{2}+d_{g}\left(\cdot,\xi_{\varepsilon}\right)^{2}}\right)^{\frac{n-2}{2}}+\operatorname{o}\left(1\right)\text{
in }H^{1}\left(M\right)\text{ as }\varepsilon\to 0,$

where $u_{0}$ is a solution to (1.1), $d_{g}$ is the geodesic distance with
respect to the metric $g$ and
$\left(\mu_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ and
$\left(\xi_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ are two families of
positive numbers and points in $M$, respectively, such that
$\mu_{\varepsilon}\to 0$ and $\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to
0$ for some point $\xi_{0}\in M$. As Theorem 1.2 shows, a necessary condition
for such solutions to exist is that $\varphi_{0}\left(\xi_{0}\right)=0$, where
$\varphi_{0}$ is as in (1.5).

We say that $u_{0}$ is nondegenerate if there does not exist any functions
$\psi\in C^{2}\left(M\right)\backslash\left\\{0\right\\}$ such that

$\Delta_{g}\psi+h_{0}\psi=\left(2^{*}-1\right)\left|u_{0}\right|^{2^{*}-2}\psi\quad\text{in
}M.$

We obtain the following result, which, except in the case where $n=6$ and
$u_{0}\left(\xi_{0}\right)>0$, is a straightforward adaptation of a result by
Robert and Vétois [RV4]:

######  Theorem 1.3.

Let $(M,g)$ be a smooth, closed Riemannian manifold of dimension $n\geq 6$,
$h_{0},h\in C^{2}\left(M\right)$, $u_{0}$ be a nondegenerate solution to (1.1)
and $\varphi_{0}$ be the function defined in (1.5). Assume that there exists a
point $\xi_{0}\in M$ such that the following two conditions hold true:

  1. (i)

$\varphi_{0}\left(\xi_{0}\right)=\left|\nabla\varphi_{0}\left(\xi_{0}\right)\right|=0$
and $\left\\{\begin{aligned} &D^{2}\varphi_{0}\left(\xi_{0}\right)<0\text{ if
}n=6\text{ and }u_{0}\left(\xi_{0}\right)>0\\\
&D^{2}\varphi_{0}\left(\xi_{0}\right)\text{ is nondegenerate
otherwise,}\end{aligned}\right.$

  2. (ii)

$K_{0}\left(\xi_{0}\right)\times\left\\{\begin{aligned}
&\left[h-2\left(\Delta_{g}+h_{0}-2\left|u_{0}\right|\right)^{-1}\left(hu_{0}\right)\right]\left(\xi_{0}\right)&&\text{if
}n=6\\\ &h\left(\xi_{0}\right)&&\text{if }n\geq 7\end{aligned}\right\\}>0$,
where

$K_{0}:=\left\\{\begin{aligned}
&\Delta_{g}\varphi_{0}+\frac{c_{6}}{6}\left|\operatorname{Weyl}_{g}\right|_{g}^{2}&&\hbox{if
}n=6\text{ and }u_{0}\left(\xi_{0}\right)<0\\\ &u_{0}&&\hbox{if }n=6\text{ and
}u_{0}\left(\xi_{0}\right)\geq 0\\\ &-u_{0}&&\hbox{if }7\leq n\leq 9\\\
&\Delta_{g}\varphi_{0}+\frac{c_{10}}{6}\left|\operatorname{Weyl}_{g}\right|_{g}^{2}-672u_{0}&&\hbox{if
}n=10\\\
&\Delta_{g}\varphi_{0}+\frac{c_{n}}{6}\left|\operatorname{Weyl}_{g}\right|_{g}^{2}&&\hbox{if
}n\geq 11.\end{aligned}\right.$

Then there exists a family of solutions
$\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ to (1.2) with
$h_{\varepsilon}:=h_{0}+\varepsilon h$, which satisfies (1.4) with
$\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to 0$.

Theorem 1.3 shows in particular that the assumption (1.3) in Theorem 1.1 is
sharp when $n\geq 7$. Note that we do not assume here that $\left(M,g\right)$
is locally conformally flat. We refer to Section 2 for extensions of Theorem
1.3 and further results in the case where $n\geq 6$. Our results in this case
are very close to those obtained in the case of positive solutions in the
article by Robert and Vétois [RV4]. This is not the case for our results in
dimensions $n\in\left\\{3,4,5\right\\}$, which we discuss in what follows.

###  1.3. Counterexamples: the case of dimensions $n\in\left\\{3,4,5\right\\}$

We now consider the case where $n\in\left\\{3,4,5\right\\}$. Our
counterexamples in this case are again of the form (1.4), but this time the
solution $u_{0}$ changes sign and the point $\xi_{0}$ is such that
$u_{0}\left(\xi_{0}\right)=0$. Here again, the condition
$u_{0}\left(\xi_{0}\right)=0$ is necessary for the existence of solutions of
type (1.4) to (1.1) when $n\in\left\\{3,4,5\right\\}$ in view of Theorem 1.2.
Additional difficulties arise in this low-dimensional case. They are related
on the one hand to the degeneracy issues in the Lyapunov–Schmidt procedure and
the construction of suitable initial functions $h_{0}$ and $u_{0}$ and on the
other hand to the higher precision required in the expansions due to the
condition $u_{0}\left(\xi_{0}\right)=0$.

We obtain our result when $n\in\left\\{3,4,5\right\\}$ under some additional
symmetry assumptions. We recall the following definition, which was used for a
different construction in the case of positive solutions by Morabito, Pistoia
and Vaira [MPV]: we say that $\left(M,g\right)$ is symmetric with respect to a
point $\xi_{0}\in M$ if there exists an isometry $\sigma:M\to M$ such that
$\sigma\left(\xi_{0}\right)=\xi_{0}$ and $d\sigma_{\xi_{0}}\left(v\right)=-v$
for all vectors $v\in T_{\xi_{0}}M$. Moreover, if $\left(M,g\right)$ is
symmetric with respect to $\xi_{0}$, then we say that a function $u:M\to M$ is
symmetric with respect to $\xi_{0}$ if $u\circ\sigma=u$ in $M$.

Let us also recall (see for example the article by Li and Zhu [LZhu]) that in
dimension $n=3$, for every smooth metric $g$ on $M$ and every function $h\in
C^{0}\left(M\right)$, if the operator $\Delta_{g}+h$ has empty kernel, then
there exists a function $m_{g,h}\in C^{0}\left(M\right)$, usually called the
mass of the operator $\Delta_{g}+h$, such that the Green’s function $G_{g,h}$
of $\Delta_{g}+h$ satisfies

$G_{g,h}\left(x,\xi\right)=\left(4\pi
d_{g}\left(x,\xi\right)\right)^{-1}+m_{g,h}\left(\xi\right)+\operatorname{o}\left(1\right)$
(1.6)

as $x\to\xi$. A similar result holds true in dimensions
$n\in\left\\{4,5\right\\}$ when the potential function touches the function
$c_{n}\operatorname{Scal}_{g}$ at order 1 at some point on the manifold. More
precisely, a straightforward extension of a result by Robert and Vétois
[RV4]*Proposition 8.1 gives that for every smooth metric $g$ on $M$ and every
function $h\in C^{1,\theta}\left(M\right)$, $\vartheta\in\left(0,1\right)$, if
the operator $\Delta_{g}+h$ has empty kernel and

$\left(h-c_{n}\operatorname{Scal}_{g}\right)\left(\xi\right)=\left|\nabla\left(h-c_{n}\operatorname{Scal}_{g}\right)\left(\xi\right)\right|=0$

for some point $\xi\in M$, then there exists a function $m_{g,h}\in
C^{0}\left(M\right)$ such that

$G_{g,h}\left(x,\xi\right)=\left(\left(n-2\right)\omega_{n-1}\right)^{-1}\Lambda_{\xi}\left(x\right)d_{g_{\xi}}\left(x,\xi\right)^{2-n}+m_{g,h}\left(\xi\right)+\operatorname{o}\left(1\right)$
(1.7)

as $x\to\xi$, where $\omega_{n-1}$ is the volume of the standard unit sphere
of dimension $n-1$ and $g_{\xi}:=\Lambda_{\xi}^{2^{*}-2}g$ is a conformal
metric to $g$ yielding conformal normal coordinates at the point $\xi$ as in
the article by Lee and Parker [LP] (see (3.14)).

With these definitions, we obtain the following:

######  Theorem 1.4.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\in\left\\{3,4,5\right\\}$, $h_{0}\in C^{1,\theta}\left(M\right)$,
$\vartheta\in\left(0,1\right)$, and $u_{0}\not\equiv 0$ be a nondegenerate
solution to (1.1). Assume that there exists a point $\xi_{0}\in M$ such that
$\left(M,g\right)$, $h_{0}$ and $u_{0}$ are symmetric with respect to
$\xi_{0}$ and $u_{0}\left(\xi_{0}\right)=0$. Assume moreover that

  * •

if $n=3$, then $m_{g,\tilde{h}_{0}}\left(\xi_{0}\right)\neq 0$,

  * •

if $n\in\left\\{4,5\right\\}$, then either $h_{0}\left(\xi_{0}\right)\neq
c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)$ or
$[h_{0}\left(\xi_{0}\right)=c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)$
and $m_{g,\tilde{h}_{0}}\left(\xi_{0}\right)\neq 0]$,

where $\tilde{h}_{0}:=h_{0}-\left(2^{*}-1\right)\left|u_{0}\right|^{2^{*}-2}$.
Then for every $h\in C^{0,\vartheta^{\prime}}\left(M\right)$,
$\vartheta^{\prime}\in\left(0,1\right)$, that is symmetric with respect to
$\xi_{0}$ and such that

$\big{[}\big{(}\Delta_{g}+\tilde{h}_{0}\big{)}^{-1}\left(hu_{0}\right)\big{]}\left(\xi_{0}\right)\times\\\
\left\\{\begin{aligned} &-m_{g,\tilde{h}_{0}}\left(\xi_{0}\right)&&\text{if
}n=3\text{ or }[n=4,5\text{ and
}h_{0}\left(\xi_{0}\right)=c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)]\\\
&h_{0}\left(\xi_{0}\right)-c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)&&\text{if
}n\in\left\\{4,5\right\\}\text{ and }h_{0}\left(\xi_{0}\right)\neq
c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\end{aligned}\right\\}>0,$
(1.8)

there exists a family of solutions
$\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ to (1.2) with
$h_{\varepsilon}:=h_{0}+\varepsilon h$, which satisfies (1.4) with
$\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to 0$.

Note that $\tilde{h}_{0}\in C^{1,\theta}\left(M\right)$ for some
$\vartheta\in\left(0,1\right)$ when $n\in\left\\{3,4,5\right\\}$ and the
symmetry assumptions on $\left(M,g\right)$, $h_{0}$ and $u_{0}$ imply that
$\nabla h_{0}\left(\xi_{0}\right)=\nabla
u_{0}\left(\xi_{0}\right)=\nabla\operatorname{Scal}_{g}\left(\xi_{0}\right)=0$,
so the mass $m_{g,\tilde{h}_{0}}$ is well defined when
$h_{0}\left(\xi_{0}\right)=c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)$.
Theorem 1.4 is in stark contrast with the results by Druet [Dr1] for positive
solutions. It shows that, in the sign-changing case, blowing-up solutions can
exist in dimensions $3$, $4$ and $5$ even when (1.3) holds true or the weak
limit $u_{0}$ is not identically zero. Note also that in the case where $n=3$
or [$n\in\left\\{4,5\right\\}$ and
$h_{0}\left(\xi_{0}\right)=c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)$],
the weak limit surprisingly appears in a global constraint via the mass
$m_{g,\tilde{h}_{0}}$. Like in Theorem 1.3, we do not assume in Theorem 1.4
that $\left(M,g\right)$ is locally conformally flat.

We refer to Section 2 for existence results of triples $\left(M,g\right)$,
$u_{0}$ and $h_{0}$ satisfying the assumptions of Theorem 1.4. The non-
degeneracy of the solution $u_{0}$, in particular, is not easily obtained. We
construct our examples in the context of $2$-symmetric manifolds, which is
defined in Section 2. By applying Theorem 1.4 to suitable $2$-symmetric
configurations, we obtain in Theorem 2.3 a general existence result of
blowing-up families of solutions of the form (1.4). Further results are then
obtained in Corollaries 2.4 and 2.5.

###  1.4. Organization of the article

Our article is organized as follows. Section 2 contains extensions of Theorems
1.3 and 1.4 and further results. In Section 3, we prove Theorems 1.1 and 1.2.
In Section 4, we prove Theorem 1.3 and Corollaries 2.1 and 2.2. In Section 5,
we prove Theorems 1.4 and 2.3 and Corollaries 2.4 and 2.5.

##  2\. Additional results

###  2.1. The case of dimensions $n\geq 6$.

By using Theorem 1.3 together with a perturbation argument, we can prove the
following:

######  Corollary 2.1.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\geq 6$, $h_{0}\in C^{p}\left(M\right)$, $1\leq p\leq\infty$, $u_{0}$ be a
positive solution to (1.1) and $\varphi_{0}$ be the function defined in (1.5).
Assume that there exists a point $\xi_{0}\in M$ such that
$\varphi_{0}\left(\xi_{0}\right)=\left|\nabla\varphi_{0}\left(\xi_{0}\right)\right|=0$.
In the case where $n=6$, assume moreover that $\xi_{0}$ is a local maximum
point of the function $\varphi_{0}$. Then there exist families of functions
$\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{p}\left(M\right)$
and $\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{2}\left(M\right)$ satisfying (1.2) and (1.4) with
$\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to 0$ and such that
$h_{\varepsilon}\to h_{0}$ in $C^{p}\left(M\right)$ as $\varepsilon\to 0$.

Remark that in Corollary 2.1 we do not assume that $u_{0}$ is non-degenerate.
We establish this property in the proof by a suitable choice of the
perturbation $\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$. By combining
Corollary 2.1 with well-known results by Aubin [A], Schoen [Sc1] and Trudinger
[T], we then obtain the following:

######  Corollary 2.2.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\geq 6$, $\xi_{0}\in M$, $1\leq p\leq\infty$ and $\varphi\in
C^{p}\left(M\right)$ be such that
$\varphi\left(\xi_{0}\right)=\left|\nabla\varphi\left(\xi_{0}\right)\right|=0$
and the operator $\Delta_{g}+c_{n}\operatorname{Scal}_{g}+\varphi$ is coercive
in $H^{1}\left(M\right)$. In the case where $n=6$, assume moreover that
$\xi_{0}$ is a local maximum point of the function $\varphi_{0}$. Then the
following two assertions hold true:

  1. (i)

If $\varphi<0$ at some point in $M$, then there exist families of functions
$\left(h_{\varepsilon}\right)_{0\leq\varepsilon\ll 1}$ in
$C^{p}\left(M\right)$ and $\left(u_{\varepsilon}\right)_{0\leq\varepsilon\ll
1}$ in $C^{2}\left(M\right)$ satisfying (1.2) and (1.4) with
$\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to 0$ and such that
$h_{\varepsilon}\to h_{0}$ in $C^{p}\left(M\right)$ as $\varepsilon\to 0$,
$u_{0}>0$ in $M$ and $\varphi_{0}=\varphi$ in $M$, where $\varphi_{0}$ is as
in (1.5).

  2. (ii)

If $\left(M,g\right)$ is not conformally diffeomorphic to the standard sphere,
then there exists a positive constant $\varepsilon_{0}$ depending only on
$\left(M,g\right)$ such that if
$\left\|\max\left(\varphi,0\right)\right\|_{L^{1}\left(M\right)}<\varepsilon_{0}$,
then there exist families of functions
$\left(h_{\varepsilon}\right)_{0\leq\varepsilon\ll 1}$ in
$C^{p}\left(M\right)$ and $\left(u_{\varepsilon}\right)_{0\leq\varepsilon\ll
1}$ in $C^{2}\left(M\right)$ satisfying (1.2) and (1.4) with
$\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to 0$ and such that
$h_{\varepsilon}\to h_{0}$ in $C^{p}\left(M\right)$ as $\varepsilon\to 0$,
$u_{0}>0$ in $M$ and $\varphi_{0}=\varphi$ in $M$, where $\varphi_{0}$ is as
in (1.5).

Corollaries 2.1 and 2.2 are proven in Section 4.

###  2.2. The case of dimensions $n\in\\{3,4,5\\}$.

To obtain examples of situations where we can apply Theorem 1.4, we now
introduce some additional symmetry assumptions. We say that $\left(M,g\right)$
is $2$-symmetric with respect to a point $\xi_{0}\in M$ if $M$ is connected
and there exist $3$ isometries $\sigma_{0},\sigma_{1},\sigma_{2}:M\to M$ and a
domain $\Omega\subset M$ such that

$\left\\{\begin{aligned}
&\text{$\sigma_{0}\circ\sigma_{0}=\sigma_{1}\circ\sigma_{1}=\sigma_{2}\circ\sigma_{2}=id_{M}$
in $M$,}\\\ &\text{$\sigma_{0}\circ\sigma_{1}=\sigma_{1}\circ\sigma_{0}$,
$\sigma_{0}\circ\sigma_{2}=\sigma_{2}\circ\sigma_{0}$ and
$\sigma_{1}\circ\sigma_{2}=\sigma_{2}\circ\sigma_{1}$ in $M$,}\\\
&\text{$\xi_{0}\in\Gamma_{\sigma_{0}}\cap\Gamma_{\sigma_{1}}\cap\Gamma_{\sigma_{2}}$
and
$d\left(\sigma_{0}\circ\sigma_{1}\circ\sigma_{2}\right)_{\xi_{0}}\left(v\right)=-v$
for all vectors $v\in T_{\xi_{0}}M$,}\\\
&\text{$\sigma_{0}\left(\Omega\right)=\Omega$ and
$M=\Omega\sqcup\sigma_{1}\left(\Omega\right)\sqcup\sigma_{2}\left(\Omega\right)\sqcup\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\sqcup\left(\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}\right)$}\\\
\end{aligned}\right.$

where $\sqcup$ is the disjoint union and $\Gamma_{\sigma_{i}}:=\left\\{x\in
M:\,\sigma_{i}\left(x\right)=x\right\\}$ for $i\in\left\\{0,1,2\right\\}$.
Remark that it easily follows from these properties that
$\xi_{0}\in\partial\Omega$. Moreover, if $\left(M,g\right)$ is $2$-symmetric
with respect to $\xi_{0}$, then we say that a function $h:M\to M$ is
$2$-symmetric with respect to $\xi_{0}$ if
$h\circ\sigma_{0}=h\circ\sigma_{1}=h\circ\sigma_{2}=h$ in $M$. Clearly, if
$\left(M,g\right)$ and $h$ are $2$-symmetric with respect to $\xi_{0}$, then
by taking $\sigma:=\sigma_{0}\circ\sigma_{1}\circ\sigma_{2}$, we obtain that
$\left(M,g\right)$ and $h$ are symmetric with respect to $\xi_{0}$.

A first example of a $2$-symmetric manifold is given by the standard unit
sphere $\left(\mathbb{S}^{n},g_{0}\right)$ with $n\geq 2$ and

$\left\\{\begin{aligned}
&\xi_{0}:=\left(0,\dotsc,0,1\right),\,\Omega:=\left\\{x\in\mathbb{S}^{n}:\,x_{1}>0\text{
and }x_{2}>0\right\\},\\\
&\sigma_{0}\left(x\right):=\left(x_{1},x_{2},-x_{3},\dotsc,-x_{n},x_{n+1}\right),\,\sigma_{1}\left(x\right):=\left(-x_{1},x_{2},x_{3},\dotsc,x_{n},x_{n+1}\right)\\\
&\text{and
}\sigma_{2}\left(x\right):=\left(x_{1},-x_{2},x_{3},\dotsc,x_{n},x_{n+1}\right)\end{aligned}\right.$

for all points $x:=\left(x_{1},\dotsc,x_{n+1}\right)\in\mathbb{S}^{n}$.
Another example is given by the product of spheres
$\mathbb{S}^{n_{1}}\times\mathbb{S}^{n_{2}}\subset\mathbb{R}^{n_{1}+1}\times\mathbb{R}^{n_{2}+1}$
equipped with the standard product metric and where

$\left\\{\begin{aligned}
&\xi_{0}:=\left(\left(0,\dotsc,0,1\right),\left(0,\dotsc,0,1\right)\right),\\\
&\Omega:=\left\\{\left(x,y\right)\in\mathbb{S}^{n_{1}}\times\mathbb{S}^{n_{2}}:\,x_{1}>0\text{
and }y_{1}>0\right\\},\\\
&\sigma_{0}\left(x,y\right):=\left(\left(x_{1},-x_{2},\dotsc,-x_{n_{1}},x_{n_{1}+1}\right),\left(y_{1},-y_{2},\dotsc,-y_{n_{1}},y_{n_{1}+1}\right)\right),\\\
&\sigma_{1}\left(x,y\right):=\left(\left(-x_{1},x_{2},\dotsc,x_{n_{1}},x_{n_{1}+1}\right),y\right)\\\
&\text{and
}\sigma_{2}\left(x,y\right):=\left(x,\left(-y_{1},y_{2},\dotsc,y_{n_{1}},y_{n_{1}+1}\right)\right)\end{aligned}\right.$

for all points
$\left(x,y\right):=\left(\left(x_{1},\dotsc,x_{n_{1}+1}\right),\left(y_{1},\dotsc,y_{n_{2}+1}\right)\right)\in\mathbb{S}^{n_{1}}\times\mathbb{S}^{n_{2}}$.

For $2$-symmetric manifolds, we are able to reduce the construction of
functions $h_{0}$ and $u_{0}$ satisfying the assumptions of Theorem 1.4 to
constructing a positive solution $u_{0}$ to the problem

$\left\\{\begin{aligned}
&\Delta_{g}u_{0}+h_{0}u_{0}=u_{0}^{2^{*}-1}\quad\text{in }\Omega\\\ &u_{0}\in
H^{1}_{0,\sigma_{0}}\left(\Omega\right),\end{aligned}\right.$ (2.1)

where

$H^{1}_{0,\sigma_{0}}\left(\Omega\right):=\left\\{u\in
H^{1}_{0}\left(\Omega\right),:\,u\circ\sigma_{0}=u\text{ in }\Omega\right\\}.$

More precisely, we obtain the following:

######  Theorem 2.3.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\in\left\\{3,4,5\right\\}$ and $h_{0}\in C^{p}\left(M\right)$, $1\leq
p\leq\infty$. Assume that there exists a point $\xi_{0}\in M$ such that
$\left(M,g\right)$ and $h_{0}$ are $2$-symmetric with respect to $\xi_{0}$ and
there exists a positive solution $u_{0}\in
H^{1}_{0,\sigma_{0}}\left(\Omega\right)$ to (2.1). Then we can extend $u_{0}$
as a sign-changing solution to (1.1) and there exist families of functions
$\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{p}\left(M\right)$
and $\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{2}\left(M\right)$ satisfying (1.2) and (1.4) with
$\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to 0$ and such that
$h_{\varepsilon}\to h_{0}$ in $C^{p}\left(M\right)$ as $\varepsilon\to 0$.

By combining Theorem 2.3 with a similar result as in the article by Aubin [A],
we obtain the following:

######  Corollary 2.4.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\in\left\\{3,4,5\right\\}$ and $h_{0}\in C^{p}\left(M\right)$, $1\leq
p\leq\infty$. Assume that there exists a point $\xi_{0}\in M$ such that
$\left(M,g\right)$ and $h_{0}$ are $2$-symmetric with respect to $\xi_{0}$.
Assume moreover that the operator $\Delta_{g}+h_{0}$ is coercive in
$H^{1}_{0,\sigma_{0}}\left(\Omega\right)$ and
$h_{0}\left(\xi_{1}\right)<c_{n}\operatorname{Scal}_{g}\left(\xi_{1}\right)$
at some point $\xi_{1}\in\Omega\cap\Gamma_{\sigma_{0}}$ (remark that
$\Omega\cap\Gamma_{\sigma_{0}}\neq\emptyset$; see Proposition 5.1). Then there
exists a solution $u_{0}\not\equiv 0$ to (1.1) such that
$u_{0}\left(\xi_{0}\right)=0$ and there exist families of functions
$\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{p}\left(M\right)$
and $\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{2}\left(M\right)$ satisfying (1.2) and (1.4) with
$\xi_{\varepsilon}\to\xi_{0}$ as $\varepsilon\to 0$ and such that
$h_{\varepsilon}\to h_{0}$ in $C^{p}\left(M\right)$ as $\varepsilon\to 0$.

In view of Theorem 1.1 and Druet’s results [Dr1] for positive solutions, we
are particularly interested, when $n\in\\{3,4,5\\}$, in obtaining blowing-up
counterexamples when $h_{0}\geq c_{n}\operatorname{Scal}_{g}$ in $M$. For
this, we assume moreover that there exists a group $G$ of isometries of $M$
such that

$\left\\{\begin{aligned} &\text{the set
}\left\\{\sigma\left(\xi\right):\,\sigma\in G\right\\}\text{ is infinite for
all points }\xi\in\overline{\Omega},\\\ &h_{0}\circ\sigma=h_{0}\text{ in
}\Omega\text{ and }\sigma\left(\Omega\right)=\Omega\text{ for all isometries
}\sigma\in G.\end{aligned}\right.$ (2.2)

For example, given any $2$-symmetric manifold $\left(M^{n_{1}},g_{1}\right)$
of dimension $n_{1}\geq 2$, letting $\left(\mathbb{S}^{n_{0}},g_{0}\right)$ be
the standard unit sphere of dimension $n_{0}\geq 1$, it is not difficult to
see that the product manifold $\left(\mathbb{S}^{n_{0}}\times
M^{n_{1}},g_{0}\times g_{1}\right)$ is $2$-symmetric when equipped with

$\left\\{\begin{aligned}
&\xi_{0}:=\left(\left(0,\dotsc,0,1\right),\xi_{0}^{n_{1}}\right),\,\Omega:=\mathbb{S}^{n_{0}}\times\Omega^{n_{1}},\\\
&\sigma_{0}\left(x,y\right):=\left(\left(-x_{1},\dotsc,-x_{n_{0}},x_{n_{0}+1}\right),\sigma_{0}^{n_{1}}\left(y\right)\right),\\\
&\sigma_{1}\left(x,y\right):=\left(x,\sigma_{1}^{n_{1}}\left(y\right)\right)\text{
and
}\sigma_{2}\left(x,y\right):=\left(x,\sigma_{2}^{n_{1}}\left(y\right)\right)\end{aligned}\right.$

for all points $x:=\left(x_{1},\dotsc,x_{n_{0}+1}\right)\in\mathbb{S}^{n_{0}}$
and $y\in M^{n_{1}}$, where $\xi_{0}^{n_{1}}$, $\Omega^{n_{1}}$,
$\sigma_{0}^{n_{1}}$, $\sigma_{1}^{n_{1}}$ and $\sigma_{2}^{n_{1}}$ correspond
to $M^{n_{1}}$ and $\xi_{0}$, $\Omega$, $\sigma_{0}$, $\sigma_{1}$ and
$\sigma_{2}$ correspond to $\mathbb{S}^{n_{0}}\times M^{n_{1}}$. In this case,
letting $\operatorname{Isom}\left(\mathbb{S}^{n_{0}},g_{0}\right)$ be the
group of isometries of $\left(\mathbb{S}^{n_{0}},g_{0}\right)$, we can define
a group

$G=\left\\{\left\\{\begin{array}[]{ccc}\mathbb{S}^{n_{0}}\times
M^{n_{1}}&\to&\mathbb{S}^{n_{0}}\times M^{n_{1}}\\\
\left(x,y\right)&\mapsto&\left(\sigma\left(x\right),y\right)\end{array}\right\\}:\,\sigma\in\operatorname{Isom}\left(\mathbb{S}^{n_{0}},g_{0}\right)\right\\}$

of isometries of $\mathbb{S}^{n_{0}}\times M^{n_{1}}$ which clearly satisfies
(2.2).

By using Theorem 2.3 together with this additional assumption, we obtain the
following:

######  Corollary 2.5.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\in\left\\{3,4,5\right\\}$ and $h_{0}\in C^{p}\left(M\right)$, $1\leq
p\leq\infty$. Assume that there exists a point $\xi_{0}\in M$ such that
$\left(M,g\right)$ and $h_{0}$ are $2$-symmetric with respect to $\xi_{0}$.
Assume moreover that the operator $\Delta_{g}+h_{0}$ is coercive in
$H^{1}_{0,\sigma_{0}}\left(\Omega\right)$ and there exists a group $G$ of
isometries of $M$ satisfying (2.2). Then there exists a solution
$u_{0}\not\equiv 0$ to (1.1) such that $u_{0}\left(\xi_{0}\right)=0$ and there
exist families of functions $\left(h_{\varepsilon}\right)_{0<\varepsilon\ll
1}$ in $C^{p}\left(M\right)$ and
$\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{2}\left(M\right)$
satisfying (1.2) and (1.4) with $\xi_{\varepsilon}\to\xi_{0}$ as
$\varepsilon\to 0$ and such that $h_{\varepsilon}\to h_{0}$ in
$C^{p}\left(M\right)$ as $\varepsilon\to 0$.

Theorems 1.4 and 2.3 and Corollaries 2.4 and 2.5 are proven in Section 5.

##  3\. Proof of Theorem 1.1

This section is devoted to the proof of Theorems 1.1 and 1.2. We first prove
Theorem 1.1.

###### Proof of Theorem 1.1.

Assume by contradiction that there exists a family of solutions
$\left(u_{\varepsilon}\right)_{0\leq\varepsilon\ll 1}$ to (1.2) that is
bounded in $H^{1}\left(M\right)$ and such that no subsequences of
$\left(u_{\varepsilon}\right)_{\varepsilon}$ converge in
$C^{2}\left(M\right)$. By standard elliptic theory, we obtain that up to a
subsequence, $\left(u_{\varepsilon}\right)_{\varepsilon}$ blows up as
$\varepsilon\to 0$. We can then apply the main results in the articles by
Premoselli [P] and Struwe [St], which give that there exist a subsequence
$\left(u_{\alpha}\right)_{\alpha\in\mathbb{N}}$ of
$\left(u_{\varepsilon}\right)_{\varepsilon}$, a solution $u_{0}\in
C^{2}\left(M\right)$ to (1.1), a number $k\in\mathbb{N}$ and for every
$i\in\left\\{1,\dotsc,k\right\\}$, a sequence of positive numbers
$\left(\mu_{i,\alpha}\right)_{\alpha\in\mathbb{N}}$, a sequences of points
$\left(\xi_{i,\alpha}\right)_{\alpha\in\mathbb{N}}$ in $M$ and a nontrivial
solution $V_{i}\in D^{1,2}\left(\mathbb{R}^{n}\right)$ to the equation

$\Delta_{\delta_{0}}V_{i}=\left|V_{i}\right|^{2^{*}-2}V_{i}\quad\text{in
}\mathbb{R}^{n},$

where $\delta_{0}$ is the Euclidean metric, such that

$u_{\alpha}=u_{0}+\sum_{i=1}^{k}V_{i,\alpha}+\operatorname{o}\left(1\right)\quad\text{in
}H^{1}\left(M\right)$

and

$\left|u_{\alpha}-u_{0}-\sum_{i=1}^{k}V_{i,\alpha}\right|=\operatorname{o}\left(1+\sum_{i=1}^{k}B_{i,\alpha}\right)\quad\text{
uniformly in }M$ (3.1)

as $\alpha\to\infty$, where

$B_{i,\alpha}\left(x\right):=\left(\frac{\sqrt{n\left(n-2\right)}\mu_{i,\alpha}}{\mu_{i,\alpha}^{2}+d_{g}\left(x,\xi_{i,\alpha}\right)^{2}}\right)^{\frac{n-2}{2}}\quad\forall
x\in M$

and the bubbling profiles $V_{i,\alpha}$ are defined as follows:

  * •

In the case where $u_{0}\not\equiv 0$ or
$\ker\left(\Delta_{g}+h_{0}\right)\neq\left\\{0\right\\}$, we let

$V_{i,\alpha}\left(x\right):=\chi\left(d_{g}\left(\xi_{i,\alpha},x\right)\right)\mu_{i,\alpha}^{-\frac{n-2}{2}}V_{i}\left(\frac{1}{\mu_{i,\alpha}}\exp_{\xi_{i,\alpha}}^{-1}\left(x\right)\right)\quad\forall
x\in M,$

where $\exp_{\xi_{i,\alpha}}$ is the exponential map at the point
$\xi_{i,\alpha}$ with respect to the metric $g$, we identify
$T_{\xi_{i,\alpha}}M$ with $\mathbb{R}^{n}$ and $\chi$ is a smooth,
nonnegative cutoff function in $\left[0,\infty\right)$ such that $\chi=1$ in
$\left[0,r_{0}/2\right)$ and $\chi=0$ in $\left[r_{0},\infty\right)$ for some
number $r_{0}>0$ smaller than the injectivity radius of the metric $g$.

  * •

In the case where $u_{0}\equiv 0$ and
$\ker(\Delta_{g}+h_{0})=\left\\{0\right\\}$, we define

$F\left(\xi_{i,\alpha},x\right):=\left(n-2\right)\omega_{n-1}d_{g}\left(\xi_{i,\alpha},x\right)^{n-2}G_{g,h_{0}}\left(\xi_{i,\alpha},x\right)\quad\forall
x\in M,$

where $\omega_{n-1}$ is the volume of the standard unit sphere of dimension
$n-1$ and $G_{g,h_{0}}$ is the Green’s function of the operator
$\Delta_{g}+h_{0}$. We then let

$V_{i,\alpha}\left(x\right):=\chi\left(d_{g}\left(\xi_{i,\alpha},x\right)\right)F\left(\xi_{i,\alpha},x\right)\mu_{i,\alpha}^{-\frac{n-2}{2}}V_{i}\left(\frac{1}{\mu_{i,\alpha}}\exp_{\xi_{i,\alpha}}^{-1}\left(x\right)\right)\\\
+\left(1-\chi\left(d_{g}\left(\xi_{i,\alpha},x\right)\right)\right)\left(n-2\right)\omega_{n-1}\lambda_{i}\mu_{i,\alpha}^{\frac{n-2}{2}}G_{g,h_{0}}\left(\xi_{i,\alpha},x\right)\quad\forall
x\in M,$

where $\exp_{\xi_{i,\alpha}}$ and $\chi$ are as in the first case and
$\lambda_{i}\in\mathbb{R}$ is such that

$V_{i}\left(y\right)=\lambda_{i}\left|y\right|^{2-n}+\operatorname{O}\left(\left|y\right|^{1-n}\right)\textrm{
as }|y|\to+\infty,$

where $y\in\mathbb{R}^{n}$ and $\left|y\right|:=\left|y\right|_{\delta_{0}}$.

Moreover, the sequences $\left(\mu_{i,\alpha}\right)_{\alpha\in\mathbb{N}}$
and $\left(\xi_{i,\alpha}\right)_{\alpha\in\mathbb{N}}$ satisfy

$\frac{d_{g}\left(\xi_{i,\alpha},\xi_{j,\alpha}\right)^{2}}{\mu_{i,\alpha}\mu_{j,\alpha}}+\frac{\mu_{i,\alpha}}{\mu_{j,\alpha}}+\frac{\mu_{j,\alpha}}{\mu_{i,\alpha}}\longrightarrow\infty$
(3.2)

as $\alpha\to\infty$, for all $i,j\in\left\\{1,\dotsc,k\right\\}$ such that
$i\neq j$. By renumbering and passing to a subsequence if necessary, we may
assume that $\mu_{1,\alpha}\geq\dotsb\geq\mu_{k,\alpha}$ for all
$\alpha\in\mathbb{N}$ and $\xi_{1,\alpha}\to\xi_{1,0}$ as $\alpha\to\infty$
for some point $\xi_{1,0}\in M$. Since $\left(M,g\right)$ is locally
conformally flat, by choosing $r_{0}$ smaller if necessary, we may assume
moreover that there exists a positive function $\Lambda\in
C^{\infty}\left(M\right)$ such that the metric $\hat{g}:=\Lambda^{2^{*}-2}g$
is flat in $B_{g}\left(\xi_{1,\alpha},r_{0}\right)$ for all
$i\in\left\\{1,\dotsc,k\right\\}$ and $\alpha\in\mathbb{N}$, where
$B_{g}\left(\xi_{1,\alpha},r_{0}\right)$ is the ball of center
$\xi_{1,\alpha}$ and radius $r_{0}$ with respect to the metric $g$. By using
the conformal invariance property of the conformal Laplacian together with the
change of variable
$x=\widehat{\exp}_{\xi_{1,\alpha}}\left(\mu_{1,\alpha}x\right)$ for $y\in
B\left(0,r_{0}/\mu_{1,\alpha}\right)$, where $\widehat{\exp}_{\xi_{1,\alpha}}$
is the exponential map at the point $\xi_{1,\alpha}$ with respect to the
metric $\hat{g}$ and $B\left(0,r_{0}/\mu_{1,\alpha}\right)$ is the ball of
center 0 and radius $r_{0}/\mu_{1,\alpha}$ with respect to the Euclidean
metric, the equation (1.2) can then be rewritten as

$\Delta_{\delta_{0}}\hat{u}_{\alpha}+\mu_{1,\alpha}^{2}\hat{h}_{\alpha}\hat{u}_{\alpha}=\left|\hat{u}_{\alpha}\right|^{2^{*}-2}\hat{u}_{\alpha}\quad\text{in
}B\left(0,r_{0}/\mu_{1,\alpha}\right),$

where

$\hat{u}_{\alpha}\left(y\right):=\mu_{1,\alpha}^{\frac{n-2}{2}}\big{(}\Lambda^{-1}u_{\alpha}\big{)}\big{(}\widehat{\exp}_{\xi_{1,\alpha}}\left(\mu_{1,\alpha}y\right)\big{)}$

and

$\hat{h}_{\alpha}\left(y\right):=\big{(}\Lambda^{2-2^{*}}\left(h_{\alpha}-c_{n}\operatorname{Scal}_{g}\right)\big{)}\big{(}\widehat{\exp}_{\xi_{1,\alpha}}\left(\mu_{1,\alpha}y\right)\big{)}$

for all points $y\in B\left(0,r_{0}/\mu_{1,\alpha}\right)$. For every
$\varrho>0$, by using a Pohozaev-type identity in
$B\left(0,\varrho/\sqrt{\mu_{1,\alpha}}\right)$ (see for example the article
by Marques [M]*Formula (2.8)), we then obtain

$\int_{B\left(0,\varrho/\sqrt{\mu_{1,\alpha}}\right)}\left(\hat{h}_{\alpha}+\frac{1}{2}\big{<}\nabla\hat{h}_{\alpha},\cdot\big{>}_{\delta_{0}}\right)\hat{u}_{\alpha}^{2}\,dv_{\delta_{0}}=\mu_{1,\alpha}^{-5/2}\int_{\partial
B\left(0,\varrho/\sqrt{\mu_{1,\alpha}}\right)}\Big{(}\frac{\varrho}{2}\left|\nabla\hat{u}_{\alpha}\right|_{\delta_{0}}^{2}\\\
-\varrho\left(\partial_{\nu}\hat{u}_{\alpha}\right)^{2}-\frac{n-2}{2}\sqrt{\mu_{1,\alpha}}\hat{u}_{\alpha}\partial_{\nu}\hat{u}_{\alpha}-\frac{\varrho}{2^{*}}\left|\hat{u}_{\alpha}\right|^{2^{*}}+\frac{\varrho}{2}\mu_{1,\alpha}^{2}\hat{h}_{\alpha}\hat{u}_{\alpha}^{2}\Big{)}d\sigma_{\delta_{0}},$
(3.3)

where $\nu$ and $d\sigma_{\delta_{0}}$ are the outward unit normal vector and
volume element, respectively, of the Euclidean metric on $\partial
B\left(0,\varrho/\sqrt{\mu_{1,\alpha}}\right)$. To estimate the terms in the
right-hand side of (3.3), we use another change of variable. We define

$\check{u}_{\alpha}\left(y\right):=\big{(}\Lambda^{-1}u_{\alpha}\big{)}\big{(}\widehat{\exp}_{\xi_{1,\alpha}}\left(\sqrt{\mu_{1,\alpha}}\,y\right)\big{)}$

and

$\check{h}_{\alpha}\left(y\right):=\big{(}\Lambda^{2-2^{*}}\left(h_{\alpha}-c_{n}\operatorname{Scal}_{g}\right)\big{)}\big{(}\widehat{\exp}_{\xi_{1,\alpha}}\left(\sqrt{\mu_{1,\alpha}}y\right)\big{)}$

for all points $y\in B\left(0,r_{0}/\sqrt{\mu_{1,\alpha}}\right)$. Then we
have

$\Delta_{\delta_{0}}\check{u}_{\alpha}+\mu_{1,\alpha}\check{h}_{\alpha}\check{u}_{\alpha}=\mu_{1,\alpha}\left|\check{u}_{\alpha}\right|^{2^{*}-2}\check{u}_{\alpha}\quad\text{in
}B\left(0,r_{0}/\sqrt{\mu_{1,\alpha}}\right)$ (3.4)

and

$\int_{\partial
B\left(0,\varrho/\sqrt{\mu_{1,\alpha}}\right)}\bigg{(}\frac{\varrho}{2}\left|\nabla\hat{u}_{\alpha}\right|_{\delta_{0}}^{2}-\varrho\left(\partial_{\nu}\hat{u}_{\alpha}\right)^{2}-\frac{n-2}{2}\sqrt{\mu_{1,\alpha}}\hat{u}_{\alpha}\partial_{\nu}\hat{u}_{\alpha}-\frac{\varrho}{2^{*}}\left|\hat{u}_{\alpha}\right|^{2^{*}}\\\
+\frac{\varrho}{2}\mu_{1,\alpha}^{2}\hat{h}_{\alpha}\hat{u}_{\alpha}^{2}\bigg{)}d\sigma_{\delta_{0}}=\mu_{1,\alpha}^{\frac{n-1}{2}}\int_{\partial
B\left(0,\varrho\right)}\\\
\left(\frac{\varrho}{2}\left|\nabla\check{u}_{\alpha}\right|_{\delta_{0}}^{2}-\varrho\left(\partial_{\nu}\check{u}_{\alpha}\right)^{2}-\frac{n-2}{2}\check{u}_{\alpha}\partial_{\nu}\check{u}_{\alpha}-\frac{\varrho}{2^{*}}\mu_{1,\alpha}\left|\check{u}_{\alpha}\right|^{2^{*}}+\frac{\varrho}{2}\mu_{1,\alpha}\check{h}_{\alpha}\hat{u}_{\alpha}^{2}\right)d\sigma_{\delta_{0}},$
(3.5)

Passing to a subsequence if necessary, we may assume that there exists
$\varrho_{0}>0$ such that for every $i\in\left\\{2,\dotsc,k\right\\}$, either
$d_{\hat{g}}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)=\operatorname{o}\left(\sqrt{\mu_{1,\alpha}}\right)$
as $\alpha\to\infty$ or
$d_{\hat{g}}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)\geq
2\varrho_{0}\sqrt{\mu_{1,\alpha}}$ for all $\alpha\in\mathbb{N}$. We then let
$I_{1}$ be the set of indices $i\in\left\\{1,\dotsc,k\right\\}$ such that
$d_{\hat{g}}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)=\operatorname{o}\left(\sqrt{\mu_{1,\alpha}}\right)$
as $\alpha\to\infty$, $I_{2}:=\left\\{1,\dotsc,k\right\\}\backslash I_{1}$ and

$\varrho_{\alpha}:=2\mu_{1,\alpha}^{-1/2}\max_{i\in
I_{1}}d_{\hat{g}}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)$

for all $\alpha\in\mathbb{N}$. By using (3.1) and remarking that

$\left|V_{i,\alpha}\left(x\right)\right|=\operatorname{O}\left(B_{i,\alpha}\left(x\right)\right)$
(3.6)

uniformly with respect to $x\in M$, $i\in\left\\{1,\dotsc,k\right\\}$ and
$\alpha\in\mathbb{N}$, we then obtain

$\displaystyle\left|\check{u}_{\alpha}\left(y\right)\right|=\operatorname{O}\left(1+\sum_{i=1}^{k}\mu_{i,\alpha}^{\frac{n-2}{2}}d_{\hat{g}}\big{(}\exp_{\xi_{1,\alpha}}\left(\sqrt{\mu_{1,\alpha}}\,y\right),\xi_{i,\alpha}\big{)}^{2-n}\right)\allowdisplaybreaks$
$\displaystyle\qquad=\operatorname{O}\left(1+\sum_{i\in
I_{1}}\left(\frac{\mu_{i,\alpha}}{\mu_{1,\alpha}}\right)^{\frac{n-2}{2}}\left(\left|y\right|-\frac{\varrho_{\alpha}}{2}\right)^{2-n}+\sum_{i\in
I_{2}}\left(\frac{\mu_{i,\alpha}}{\mu_{1,\alpha}}\right)^{\frac{n-2}{2}}\left(2\varrho_{0}-\left|y\right|\right)^{2-n}\right)$
$\displaystyle\qquad=\operatorname{O}\big{(}\left|y\right|^{2-n}\big{)}$ (3.7)

uniformly with respect to $\alpha\in\mathbb{N}$ and $y\in
B\left(0,\varrho_{0}\right)\backslash B\left(0,\varrho_{\alpha}\right)$. By
standard elliptic theory and since $\varrho_{\alpha}\to 0$ and $h_{\alpha}\to
h_{0}$ in $C^{1}\left(M\right)$ as $\alpha\to\infty$, it follows from (3.4)
and (3) that $\left(\check{u}_{\alpha}\right)_{\alpha}$ is bounded in
$C^{1}_{loc}\left(B\left(0,\varrho_{0}\right)\backslash\left\\{0\right\\}\right)$.
In particular, we obtain

$\int_{\partial
B\left(0,\varrho_{0}\right)}\bigg{(}\frac{\varrho_{0}}{2}\left|\nabla\check{u}_{\alpha}\right|_{\delta_{0}}^{2}-\varrho_{0}\left(\partial_{\nu}\check{u}_{\alpha}\right)^{2}-\frac{n-2}{2}\check{u}_{\alpha}\partial_{\nu}\check{u}_{\alpha}-\frac{\varrho_{0}}{2^{*}}\mu_{1,\alpha}\left|\check{u}_{\alpha}\right|^{2^{*}}\\\
+\frac{\varrho_{0}}{2}\mu_{1,\alpha}\check{h}_{\alpha}\hat{u}_{\alpha}^{2}\bigg{)}d\sigma_{\delta_{0}}=\operatorname{O}\left(1\right)$
(3.8)

uniformly with respect to $\alpha\in\mathbb{N}$. We now consider the terms in
the left-hand side of (3.3). Since $h_{\alpha}\to h_{0}$ in
$C^{1}\left(M\right)$ as $\alpha\to\infty$, we obtain

$\hat{h}_{\alpha}+\frac{1}{2}\big{<}\nabla\hat{h}_{\alpha},\cdot\big{>}_{\delta_{0}}=\big{(}\Lambda^{2-2^{*}}\left(h_{0}-c_{n}\operatorname{Scal}_{g}\right)\big{)}\left(\xi_{1,\alpha}\right)+\operatorname{o}\left(1\right)$
(3.9)

as $\alpha\to\infty$, uniformly in
$B\left(0,\varrho_{0}/\sqrt{\mu_{1,\alpha}}\right)$. By putting together
(1.3), (3.3), (3.5), (3.8) and (3.9), we then obtain

$\int_{B\left(0,\varrho_{0}/\sqrt{\mu_{1,\alpha}}\right)}\hat{u}_{\alpha}^{2}\,dv_{\delta_{0}}=\operatorname{O}\left(\mu_{1,\alpha}^{\frac{n-6}{2}}\right)$
(3.10)

uniformly with respect to $\alpha\in\mathbb{N}$. On the other hand, for every
$R>0$, by remarking that
$B_{g}\left(\xi_{1,\alpha},R\mu_{1,\alpha}\right)\subset
B_{\hat{g}}\left(\xi_{1,\alpha},\varrho_{0}\sqrt{\mu_{1,\alpha}}\right)$, we
obtain

$\int_{B\left(0,\varrho_{0}/\sqrt{\mu_{1,\alpha}}\right)}\hat{u}_{\alpha}^{2}\,dv_{\delta_{0}}\geq
C\int_{B\left(0,R\right)}\tilde{u}_{\alpha}^{2}\,dv_{\tilde{g}_{\alpha}}$
(3.11)

for some constant $C>0$ independent of $\alpha$, where

$\tilde{u}_{\alpha}\left(y\right):=\mu_{1,\alpha}^{\frac{n-2}{2}}u_{\alpha}\big{(}\exp_{\xi_{1,\alpha}}\left(\mu_{1,\alpha}y\right)\big{)}\quad\text{and}\quad\tilde{g}_{\alpha}\left(y\right):=\exp_{\xi_{1,\alpha}}^{*}g\left(\mu_{1,\alpha}y\right)$

for all $y\in B\left(0,R\right)$. Since $\mu_{i,\alpha}\leq\mu_{1,\alpha}$ for
all $\alpha\in\mathbb{N}$ and $i\in\left\\{2,\dotsc,k\right\\}$, by using
(3.2) and passing to a subsequence if necessary, we may assume that for every
$i\in\left\\{2,\dotsc,k\right\\}$, we have
$\mu_{1,\alpha}=\operatorname{o}\left(d_{g}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)\right)$
or $\mu_{i,\alpha}=\operatorname{o}\left(\mu_{1,\alpha}\right)$ as
$\alpha\to\infty$. By passing again to a subsequence, we may assume moreover
that there exist two numbers $R_{1},R_{2}>0$ such that $R_{1}<R_{2}$ and for
every $i\in\left\\{2,\dotsc,k\right\\}$, either
$d_{g}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)\leq R_{1}\mu_{1,\alpha}$ for
all $\alpha\in\mathbb{N}$ or
$d_{g}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)\geq R_{2}\mu_{1,\alpha}$ for
all $\alpha\in\mathbb{N}$. We then let $J_{1}$ be the set of indices
$i\in\left\\{2,\dotsc,k\right\\}$ such that
$\mu_{1,\alpha}=\operatorname{o}\left(d_{g}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)\right)$
as $\alpha\to\infty$, $J_{2}$ be the set of indices
$i\in\left\\{2,\dotsc,k\right\\}$ such that
$\mu_{i,\alpha}=\operatorname{o}\left(\mu_{1,\alpha}\right)$ as
$\alpha\to\infty$ and $d_{g}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)\leq
R_{1}\mu_{1,\alpha}$ for all $\alpha\in\mathbb{N}$,
$J_{3}:=\left\\{2,\dotsc,k\right\\}\backslash\left(J_{1}\cup J_{2}\right)$ and
$R^{\prime}_{1},R^{\prime}_{2}>0$ be two numbers such that
$R_{1}<R^{\prime}_{1}<R^{\prime}_{2}<R_{2}$. By using (3.1) and (3.6), we then
obtain

$\displaystyle\left|\tilde{u}_{\alpha}\left(y\right)-V_{1}\left(y\right)\right|$
$\displaystyle=\operatorname{O}\left(\sum_{i=2}^{k}\left(\frac{\mu_{1,\alpha}\mu_{i,\alpha}}{d_{g}\big{(}\exp_{\xi_{1,\alpha}}\left(\mu_{1,\alpha}y\right),\xi_{i,\alpha}\big{)}^{2}}\right)^{\frac{n-2}{2}}\right)+\operatorname{o}\left(1\right)\allowdisplaybreaks$
$\displaystyle=\operatorname{O}\Bigg{(}\sum_{i\in
J_{1}}\left(\frac{\mu_{1,\alpha}\mu_{i,\alpha}}{d_{g}\left(\xi_{1,\alpha},\xi_{i,\alpha}\right)^{2}}\right)^{\frac{n-2}{2}}+\sum_{i\in
J_{2}}\left(\frac{\mu_{i,\alpha}}{\mu_{1,\alpha}}\right)^{\frac{n-2}{2}}\left(\left|y\right|-R_{1}\right)^{2-n}$
$\displaystyle\qquad+\sum_{i\in
J_{3}}\left(\frac{\mu_{i,\alpha}}{\mu_{1,\alpha}}\right)^{\frac{n-2}{2}}\left(R_{2}-\left|y\right|\right)^{2-n}\Bigg{)}+\operatorname{o}\left(1\right)=\operatorname{o}\left(1\right)$
(3.12)

as $\alpha\to\infty$, uniformly with respect to $y\in
B\left(0,R^{\prime}_{2}\right)\backslash B\left(0,R^{\prime}_{1}\right)$. By
remarking that $\tilde{g}_{\alpha}\to\delta_{0}$ uniformly in
$B\left(0,R^{\prime}_{2}\right)$ as $\alpha\to\infty$, it follows from (3)
that

$\int_{B\left(0,R^{\prime}_{2}\right)}\tilde{u}_{\alpha}^{2}\,dv_{\tilde{g}_{\alpha}}\geq\int_{B\left(0,R^{\prime}_{2}\right)\backslash
B\left(0,R^{\prime}_{1}\right)}\tilde{u}_{\alpha}^{2}\,dv_{\tilde{g}_{\alpha}}\longrightarrow\int_{B\left(0,R^{\prime}_{2}\right)\backslash
B\left(0,R^{\prime}_{1}\right)}V_{1}^{2}\,dv_{\delta_{0}}>0$ (3.13)

as $\alpha\to\infty$. Since $\mu_{1,\alpha}\to 0$ as $\alpha\to\infty$, by
putting together (3.10), (3.11) and (3.13), we then obtain a contradiction
when $n\geq 7$. This ends the proof of Theorem 1.1. ∎

We now prove Theorem 1.2, which shows that the conditions
$\varphi_{0}\left(\xi_{0}\right)=0$ in Theorem 1.3 and
$u_{0}\left(\xi_{0}\right)=0$ in Theorem 1.4 are necessary in the case of
solutions of type (1.4). Note that in the statement of Theorem 1.2, precisely
in (1.4), $u_{0}$ can be any sign-changing solution to (1.1).

Let us first introduce a crucial ingredient, namely the conformal normal
coordinates, which we use both in the proof of Theorem 1.2 and in our
constructions of families of blowing-up solutions in the next sections. By
using Lee and Parker’s construction [LP], we obtain the existence of a smooth
family of metrics $\left(g_{\xi}\right)_{\xi\in M}$,
$g_{\xi}=\Lambda_{\xi}^{2^{*}-2}g$, such that for every point $\xi\in M$, the
function $\Lambda_{\xi}$ is smooth, positive and satisfies

$\Lambda_{\xi}\left(\xi\right)=1,\quad\nabla\Lambda_{\xi}\left(\xi\right)=0\quad\text{and}\quad
dv_{g_{\xi}}\left(x\right)=\big{(}1+\operatorname{O}\big{(}\left|x\right|^{N}\big{)}\big{)}dv_{\delta_{0}}\left(x\right)$
(3.14)

in geodesic normal coordinates, where $dv_{g_{\xi}}$ and $dv_{\delta_{0}}$ are
the volume elements of $g_{\xi}$ and the Euclidean metric $\delta_{0}$,
respectively, and $N\in\mathbb{N}$ can be chosen arbitrarily large. In
particular (see [LP]), it follows from (3.14) that

$\operatorname{Scal}_{g_{\xi}}(\xi)=0,\quad\nabla\operatorname{Scal}_{g_{\xi}}(\xi)=0\quad\text{and}\quad\Delta_{g}\operatorname{Scal}_{g_{\xi}}(\xi)=\frac{1}{6}|\operatorname{Weyl}_{g}(\xi)|_{g}^{2}.$
(3.15)

In the case where $\left(M,g\right)$ is locally conformally flat, we may
assume moreover that there exists a number $r_{0}>0$ such that for every point
$\xi\in M$, the metric $g_{\xi}$ is flat in the ball of center $\xi$ and
radius $r_{0}$.

As a first step in the proof of Theorem 1.2, we obtain the following result,
which is essentially contained in the article by Premoselli [P]:

######  Lemma 3.1.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\geq 3$, $h_{0}\in C^{0,\vartheta}\left(M\right)$,
$\vartheta\in\left(0,1\right)$, and $u_{0}$ be a solution to (1.1). Assume
that there exist families of functions
$\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{0,\vartheta}\left(M\right)$ and
$\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{2}\left(M\right)$,
positive numbers $\left(\mu_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ and
points $\left(\xi_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $M$ satisfying
(1.2) and (1.4) and such that $\mu_{\varepsilon}\to 0$,
$\xi_{\varepsilon}\to\xi_{0}$ and $h_{\varepsilon}\to h_{0}$ in
$C^{0,\vartheta}\left(M\right)$ as $\varepsilon\to 0$. Let
$\left(g_{\xi}\right)_{\xi}$, $g_{\xi}=\Lambda_{\xi}^{2^{*}-2}g$, be a family
of conformal metrics to $g$ satisfying (3.14). Then

$u_{\varepsilon}-u_{0}+B_{\varepsilon}=\operatorname{o}\big{(}B_{\varepsilon}+1\big{)}$
(3.16)

as $\varepsilon\to 0$, uniformly in $M$, where

$B_{\varepsilon}\left(x\right):=\Lambda_{\xi_{\varepsilon}}\left(x\right)\chi\left(d_{g_{\xi_{\varepsilon}}}\left(\xi_{\varepsilon},x\right)\right)\left(\frac{\sqrt{n\left(n-2\right)}\mu_{\varepsilon}}{\mu_{\varepsilon}^{2}+d_{g_{\xi_{\varepsilon}}}\left(\xi_{\varepsilon},x\right)^{2}}\right)^{\frac{n-2}{2}}\quad\forall
x\in M$

and $\chi$ is a smooth, nonnegative cutoff function in $\left[0,\infty\right)$
such that $\chi=1$ in $\left[0,r_{0}/2\right)$ and $\chi=0$ in
$\left[r_{0},\infty\right)$ for some number $r_{0}>0$ smaller than the
injectivity radii of the metrics $g_{\xi}$.

###### Proof of Lemma 3.1.

Since $u_{\varepsilon}$ satisfies (1.1) and (1.4), the main result in the
article by Premoselli [P] shows that

$u_{\varepsilon}=u_{0}-\widetilde{B}_{\varepsilon}+\operatorname{o}\left(1+\left(\frac{\mu_{\varepsilon}}{\mu_{\varepsilon}^{2}+d_{g}\left(\xi_{\varepsilon},\cdot\right)^{2}}\right)^{\frac{n-2}{2}}\right)\text{
as }\varepsilon\to 0,$ (3.17)

uniformly in $M$, where $\widetilde{B}_{\varepsilon}$ is defined by

$\widetilde{B}_{\varepsilon}\left(x\right):=\chi\left(d_{g}\left(\xi_{\varepsilon},x\right)\right)\left(\frac{\sqrt{n\left(n-2\right)}\mu_{\varepsilon}}{\mu_{\varepsilon}^{2}+d_{g}\left(\xi_{\varepsilon},x\right)^{2}}\right)^{\frac{n-2}{2}}\quad\forall
x\in M$

if $u_{0}\not\equiv 0$ or
$\ker\left(\Delta_{g}+h_{0}\right)\neq\left\\{0\right\\}$, and

$\widetilde{B}_{\varepsilon}\left(x\right):=\chi\left(d_{g}\left(\xi_{\varepsilon},x\right)\right)F\left(\xi_{\varepsilon},x\right)\left(\frac{\sqrt{n\left(n-2\right)}\mu_{\varepsilon}}{\mu_{\varepsilon}^{2}+d_{g}\left(\xi_{\varepsilon},x\right)^{2}}\right)^{\frac{n-2}{2}}\\\
+\left(1-\chi\left(d_{g}(\xi_{\varepsilon},x)\right)\right)k_{n}\mu_{\varepsilon}^{\frac{n-2}{2}}G_{g,h_{0}}\left(\xi_{\varepsilon},x\right)\quad\forall
x\in M$

otherwise, where

$\displaystyle
k_{n}:=n^{\frac{n-2}{4}}\left(n-2\right)^{\frac{n+2}{4}}\omega_{n-1},\allowdisplaybreaks$
(3.18) $\displaystyle
F\left(\xi_{\varepsilon},x\right):=\left(n-2\right)\omega_{n-1}d_{g}\left(\xi_{\varepsilon},x\right)^{n-2}G_{g,h_{0}}\left(\xi_{\varepsilon},x\right)\quad\forall
x\in M$

and $G_{g,h_{0}}$ is the Green’s function of the operator $\Delta_{g}+h_{0}$.
Since $\Lambda_{\xi_{\varepsilon}}(\xi_{\varepsilon})=1$, we have

$\Lambda_{\xi_{\varepsilon}}(x)=1+\operatorname{O}\left(d_{g}(\xi_{\varepsilon},x)\right)$

and

$d_{g_{\xi_{\varepsilon}}}\left(x,\xi_{\varepsilon}\right)^{2}=d_{g}\left(x,\xi_{\varepsilon}\right)^{2}+\operatorname{O}\left(d_{g}\left(x,\xi_{\varepsilon}\right)^{3}\right)$
(3.19)

uniformly with respect to $\varepsilon>0$ and $x\in M$. Independently, by
definition of $G_{g,h_{0}}$, we have

$F\left(\xi_{\varepsilon},x\right)=1+\operatorname{O}\left(d_{g}\left(x,\xi_{\varepsilon}\right)\right).$

This shows that

$\displaystyle\widetilde{B}_{\varepsilon}(x)$
$\displaystyle=\big{(}1+\operatorname{O}\left(d_{g}\left(x,\xi_{\varepsilon}\right)\right)\big{)}B_{\varepsilon}(x)=B_{\varepsilon}(x)+\operatorname{o}\left(B_{\varepsilon}(x)+1\right)$
(3.20)

as $\varepsilon\to 0$, uniformly with respect to $x\in M$. The estimate (3.16)
then follows from (3.17), (3.19) and (3.20). ∎

By using Lemma 3.1, we then obtain the following:

######  Lemma 3.2.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\geq 3$, $h_{0}\in C^{0,\vartheta}\left(M\right)$,
$\vartheta\in\left(0,1\right)$, and $u_{0}$ be a solution to (1.1). Let
$\left(h_{\varepsilon}\right)_{\varepsilon}$,
$\left(u_{\varepsilon}\right)_{\varepsilon}$,
$\left(\mu_{\varepsilon}\right)_{\varepsilon}$,
$\left(\xi_{\varepsilon}\right)_{\varepsilon}$,
$\left(B_{\varepsilon}\right)_{\varepsilon}$ and $\left(g_{\xi}\right)_{\xi}$
be as in Lemma 3.1. Then

$\int_{B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)}\left(\left<\nabla\hat{u}_{\varepsilon},\cdot\right>_{\delta_{0}}+\frac{n-2}{2}\hat{u}_{\varepsilon}\right)\left(\left(\Delta_{\hat{g}_{\varepsilon}}-\Delta_{\delta_{0}}\right)\hat{u}_{\varepsilon}+\mu_{\varepsilon}^{2}\hat{h}_{\varepsilon}\hat{u}_{\varepsilon}\right)dv_{\delta_{0}}\\\
=\frac{1}{2}n^{\frac{n-2}{4}}\left(n-2\right)^{\frac{n+6}{4}}\omega_{n-1}u_{0}\left(\xi_{0}\right)\mu_{\varepsilon}^{\frac{n-2}{2}}+\operatorname{o}\left(\mu_{\varepsilon}^{\frac{n-2}{2}}\right)$
(3.21)

as $\varepsilon\to 0$, where

$\hat{u}_{\varepsilon}\left(y\right):=\mu_{\varepsilon}^{\frac{n-2}{2}}\big{(}\Lambda_{\xi_{\varepsilon}}^{-1}u_{\varepsilon}\big{)}\left(\exp_{\xi_{\varepsilon}}\left(\mu_{\varepsilon}y\right)\right),\quad\hat{g}_{\varepsilon}\left(y\right):=\exp_{\xi_{\varepsilon}}^{*}g_{\xi_{\varepsilon}}\left(\mu_{\varepsilon}y\right)$
(3.22)

and

$\hat{h}_{\varepsilon}\left(y\right):=\big{(}c_{n}\operatorname{Scal}_{\hat{g}_{\varepsilon}}+\Lambda_{\xi_{\varepsilon}}^{2-2^{*}}\left(h_{\varepsilon}-c_{n}\operatorname{Scal}_{g}\right)\big{)}\left(\exp_{\xi_{\varepsilon}}\left(\mu_{\varepsilon}y\right)\right)$
(3.23)

for all points $y\in B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)$,
$\exp_{\xi_{\varepsilon}}$ is the exponential map at the point
$\xi_{\varepsilon}$ with respect to the metric $g_{\xi_{\varepsilon}}$ and we
identify $T_{\xi_{\varepsilon}}M$ with $\mathbb{R}^{n}$.

###### Proof of Lemma 3.2.

By using the conformal invariance property of the conformal Laplacian together
with the change of variable
$x=\exp_{\xi_{\varepsilon}}\left(\mu_{\varepsilon}y\right)$ for $y\in
B\left(0,r_{0}/\mu_{\varepsilon}\right)$, where $r_{0}$ is as in Lemma 3.1,
the equation (1.2) can be rewritten as

$\Delta_{\hat{g}_{\varepsilon}}\hat{u}_{\varepsilon}+\mu_{\varepsilon}^{2}\hat{h}_{\varepsilon}\hat{u}_{\varepsilon}=\left|\hat{u}_{\varepsilon}\right|^{2^{*}-2}\hat{u}_{\varepsilon}\quad\text{in
}B\left(0,r_{0}/\mu_{\varepsilon}\right),$

where $\hat{u}_{\varepsilon}$ and $\hat{h}_{\varepsilon}$ are as in (3.22) and
(3.23). By using a Pohozaev-type identity (see for example the article by
Marques [M]*formula (2.7)), we then obtain

$\int_{B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)}\left(\left<\nabla\hat{u}_{\varepsilon},\cdot\right>_{\delta_{0}}+\frac{n-2}{2}\hat{u}_{\varepsilon}\right)\left(\left(\Delta_{\hat{g}_{\varepsilon}}-\Delta_{\delta_{0}}\right)\hat{u}_{\varepsilon}+\mu_{\varepsilon}^{2}\hat{h}_{\varepsilon}\hat{u}_{\varepsilon}\right)dv_{\delta_{0}}=\frac{1}{\sqrt{\mu_{\varepsilon}}}\\\
\times\int_{\partial
B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)}\left(\frac{n-2}{2}\sqrt{\mu_{\varepsilon}}\hat{u}_{\varepsilon}\partial_{\nu}\hat{u}_{\varepsilon}+\left(\partial_{\nu}\hat{u}_{\varepsilon}\right)^{2}-\frac{1}{2}\left|\nabla\hat{u}_{\varepsilon}\right|_{\delta_{0}}^{2}+\frac{1}{2^{*}}\left|\hat{u}_{\varepsilon}\right|^{2^{*}}\right)d\sigma_{\delta_{0}},$
(3.24)

where $\nu$ and $d\sigma_{\delta_{0}}$ are the outward unit normal vector and
volume element, respectively, of the metric induced by $\delta_{0}$ on
$\partial B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)$. Similarly as in
(3.4)-(3.8), to estimate the terms in the right-hand side of (3.24), we use
another change of variable, which gives

$\Delta_{\check{g}_{\varepsilon}}\check{u}_{\varepsilon}+\mu_{\varepsilon}\check{h}_{\varepsilon}\check{u}_{\varepsilon}=\mu_{\varepsilon}\left|\check{u}_{\varepsilon}\right|^{2^{*}-2}\check{u}_{\varepsilon}\quad\text{in
}B\left(0,r_{0}/\sqrt{\mu_{\varepsilon}}\right)$ (3.25)

and

$\int_{\partial
B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)}\left(\frac{n-2}{2}\sqrt{\mu_{\varepsilon}}\hat{u}_{\varepsilon}\partial_{\nu}\hat{u}_{\varepsilon}+\left(\partial_{\nu}\hat{u}_{\varepsilon}\right)^{2}-\frac{1}{2}\left|\nabla\hat{u}_{\varepsilon}\right|_{\delta_{0}}^{2}+\frac{1}{2^{*}}\left|\hat{u}_{\varepsilon}\right|^{2^{*}}\right)d\sigma_{\delta_{0}}\\\
=\mu_{\varepsilon}^{\frac{n-1}{2}}\int_{\partial
B\left(0,1\right)}\left(\frac{n-2}{2}\check{u}_{\varepsilon}\partial_{\nu}\check{u}_{\varepsilon}+\left(\partial_{\nu}\check{u}_{\varepsilon}\right)^{2}-\frac{1}{2}\left|\nabla\check{u}_{\varepsilon}\right|_{\check{g}_{\varepsilon}}^{2}+\frac{\mu_{\varepsilon}}{2^{*}}\left|\check{u}_{\varepsilon}\right|^{2^{*}}\right)d\sigma_{\delta_{0}},$
(3.26)

where

$\check{u}_{\varepsilon}\left(y\right):=\big{(}\Lambda_{\xi_{\varepsilon}}^{-1}u_{\varepsilon}\big{)}\left(\exp_{\xi_{\varepsilon}}\left(\sqrt{\mu_{\varepsilon}}\,y\right)\right)\quad\text{and}\quad\check{g}_{\varepsilon}(y)=\exp_{\xi_{\varepsilon}}^{*}g_{\xi_{\varepsilon}}\left(\sqrt{\mu_{\varepsilon}}\,y\right).$

By standard elliptic theory and since
$\Lambda_{\xi_{\varepsilon}}\left(\xi_{\varepsilon}\right)=1$, it follows from
(3.16) and (3.25) that

$\check{u}_{\varepsilon}\to
u_{0}\left(\xi_{0}\right)-\left(n\left(n-2\right)\right)^{\frac{n-2}{4}}{\left|\cdot\right|^{2-n}}\quad\text{in
}C^{1}_{loc}\left(\mathbb{R}^{n}\backslash\left\\{0\right\\}\right)$ (3.27)

as $\varepsilon\to 0$. It then follows from (3.27) that

$\int_{\partial
B\left(0,1\right)}\left(\frac{n-2}{2}\check{u}_{\varepsilon}\partial_{\nu}\check{u}_{\varepsilon}+\left(\partial_{\nu}\check{u}_{\varepsilon}\right)^{2}-\frac{1}{2}\left|\nabla\check{u}_{\varepsilon}\right|_{\check{g}_{\varepsilon}}^{2}+\frac{\mu_{\varepsilon}}{2^{*}}\left|\check{u}_{\varepsilon}\right|^{2^{*}}\right)d\sigma_{\delta_{0}}\\\
=\frac{1}{2}n^{\frac{n-2}{4}}\left(n-2\right)^{\frac{n+6}{4}}\omega_{n-1}u_{0}\left(\xi_{0}\right)+\operatorname{o}\left(1\right)$
(3.28)

as $\varepsilon\to 0$. The estimate (3.21) then follows from (3.24), (3.26)
and (3.28). ∎

We can now use Lemmas 3.1 and 3.2 to prove Theorem 1.2.

###### Proof of Theorem 1.2.

We proceed by contrapositive. Assume that there exist families of functions
$\left(h_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{0,\vartheta}\left(M\right)$ and
$\left(u_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $C^{2}\left(M\right)$,
positive numbers $\left(\mu_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ and
points $\left(\xi_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in $M$ satisfying
(1.2) and (1.4) and such that $\mu_{\varepsilon}\to 0$,
$\xi_{\varepsilon}\to\xi_{0}$ and $h_{\varepsilon}\to h_{0}$ in
$C^{0,\vartheta}\left(M\right)$ as $\varepsilon\to 0$. Let
$\left(B_{\varepsilon}\right)_{\varepsilon}$ and $\left(g_{\xi}\right)_{\xi}$
be as in Lemma 3.1. We now estimate the terms in the left-hand side of (3.21).
It follows from (3.16) together with standard elliptic estimates that

$\left|\nabla^{i}\left(\hat{u}_{\varepsilon}+U_{0}\right)\left(y\right)\right|=\operatorname{o}\big{(}\left(1+\left|y\right|\right)^{2-n-i}\big{)}\quad\text{for
}i\in\left\\{0,1,2\right\\}$ (3.29)

as $\varepsilon\to 0$, uniformly with respect to $y\in
B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)$, where

$U_{0}\left(y\right):=\left(\frac{\sqrt{n\left(n-2\right)}}{1+\left|y\right|^{2}}\right)^{\frac{n-2}{2}}.$
(3.30)

Since the function $U_{0}$ is radially symmetric about the origin, it follows
from (3.14) that

$\left(\Delta_{\hat{g}_{\varepsilon}}-\Delta_{\delta_{0}}\right)U_{0}\left(y\right)=\operatorname{O}\big{(}\mu_{\varepsilon}^{N}\left|y\right|^{N-1}\left|\nabla
U_{0}\left(y\right)\right|\big{)}=\operatorname{O}\left(\frac{\mu_{\varepsilon}^{N}\left|y\right|^{N}}{\left(1+\left|y\right|\right)^{n}}\right)$
(3.31)

uniformly with respect to $y\in B\left(0,1/\sqrt{\mu}_{\varepsilon}\right)$,
where $\left|y\right|:=\left|y\right|_{\delta_{0}}$. By using (3.29) and
(3.31), we then obtain

$\left(\Delta_{\hat{g}_{\varepsilon}}-\Delta_{\delta_{0}}\right)\hat{u}_{\varepsilon}\left(y\right)=\operatorname{O}\Bigg{(}\frac{\mu_{\varepsilon}^{N-1}\left|y\right|^{N}}{\left(1+\left|y\right|\right)^{n}}+\left|\left(\hat{g}_{\varepsilon}-\delta_{0}\right)\left(y\right)\right||\nabla^{2}\left(\hat{u}_{\varepsilon}+U_{0}\right)\left(y\right)|\\\
+\left|\nabla\hat{g}_{\varepsilon}\left(y\right)\right|\left|\nabla\left(\hat{u}_{\varepsilon}+U_{0}\right)\left(y\right)\right|\Bigg{)}=\operatorname{o}\left(\frac{\mu_{\varepsilon}^{2}\left|y\right|}{\left(1+\left|y\right|\right)^{n-1}}\right)$
(3.32)

as $\varepsilon\to 0$, uniformly with respect to $y\in
B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)$. Since $h_{\varepsilon}\to h_{0}$
in $C^{0,\vartheta}\left(M\right)$ as $\varepsilon\to 0$, by using (3.15) and
(3.29), we obtain

$\widehat{h}_{\varepsilon}\left(y\right)\widehat{u}_{\varepsilon}\left(y\right)=\left(c_{n}\operatorname{Scal}_{g}-h_{0}\right)\left(\xi_{0}\right)U_{0}\left(y\right)+\operatorname{o}\left(U_{0}\left(y\right)\right)$
(3.33)

uniformly with respect to $y\in B\left(0,1/\sqrt{\mu}_{\varepsilon}\right)$.
It follows from (3.29), (3.32) and (3.33) that

$\int_{B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)}\left(\left<\nabla\hat{u}_{\varepsilon},\cdot\right>_{\delta_{0}}+\frac{n-2}{2}\hat{u}_{\varepsilon}\right)\left(\left(\Delta_{\hat{g}_{\varepsilon}}-\Delta_{\delta_{0}}\right)\hat{u}_{\varepsilon}+\mu_{\varepsilon}^{2}\hat{h}_{\varepsilon}\hat{u}_{\varepsilon}\right)dv_{\delta_{0}}\\\
=\left\\{\begin{aligned}
&\operatorname{O}\left(\mu_{\varepsilon}^{2}\int_{B\left(0,1/\sqrt{\mu_{\varepsilon}}\right)}\frac{dy}{\left(1+\left|y\right|\right)^{2n-4}}\right)=\left\\{\begin{aligned}
&\operatorname{O}\left(\mu_{\varepsilon}^{3/2}\right)&&\text{if }n=3\\\
&\operatorname{O}\left(\mu_{\varepsilon}^{2}\ln\left(1/\mu_{\varepsilon}\right)\right)&&\text{if
}n=4\end{aligned}\right.\\\
&\frac{1}{2}n^{\frac{n-2}{2}}\left(n-2\right)^{\frac{n}{2}}a_{n}\left(c_{n}\operatorname{Scal}_{g}-h_{0}\right)\left(\xi_{0}\right)\mu_{\varepsilon}^{2}+\operatorname{o}\left(\mu_{\varepsilon}^{2}\right)\quad\text{if
}n\geq 5\end{aligned}\right.$ (3.34)

as $\varepsilon\to 0$, where

$a_{n}:=\int_{\mathbb{R}^{n}}\frac{\left|y\right|^{2}-1}{\big{(}1+\left|y\right|^{2}\big{)}^{n-1}}\,dy.$
(3.35)

By putting together (3.21) and (3.34), we then obtain

$\omega_{n-1}u_{0}\left(\xi_{0}\right)\mu_{\varepsilon}^{\frac{n-2}{2}}+\operatorname{o}\left(\mu_{\varepsilon}^{\frac{n-2}{2}}\right)\\\
=\left\\{\begin{aligned}
&\operatorname{O}\big{(}\mu_{\varepsilon}^{3/2}\big{)}&&\text{if }n=3\\\
&\operatorname{O}\left(\mu_{\varepsilon}^{2}\ln\left(1/\mu_{\varepsilon}\right)\right)&&\text{if
}n=4\\\
&n^{\frac{n-2}{4}}\left(n-2\right)^{\frac{n-6}{4}}a_{n}\left(c_{n}\operatorname{Scal}_{g}-h_{0}\right)\left(\xi_{0}\right)\mu_{\varepsilon}^{2}+\operatorname{o}\left(\mu_{\varepsilon}^{2}\right)&&\text{if
}n\geq 5\end{aligned}\right.$

as $\varepsilon\to 0$. In the case where $n=6$, straightforward computations
give

$n^{\frac{n-2}{4}}\left(n-2\right)^{\frac{n-6}{4}}a_{n}=\frac{\omega_{5}}{2}\,.$
(3.36)

It then follows from (3.35) and (3.36) that
$\varphi_{0}\left(\xi_{0}\right)=0$, where $\varphi_{0}$ is as in (1.5). By
contrapositive, this proves Theorem 1.2. ∎

##  4\. Proofs of Theorem 1.3 and Corollaries 2.1 and 2.2

This section is devoted to our constructions of blowing-up solutions in the
case of dimensions $n\geq 6$.

###### Proof of Theorem 1.3.

The proof of Robert and Vétois [RV4]*Theorem 1.5 continues to apply with
straightforward adaptations in the case where $n\geq 7$ or [$n=6$ and
$u_{0}\left(\xi_{0}\right)<0$]. Therefore, in what follows, we only consider
the case where $n=6$ and $u_{0}\left(\xi_{0}\right)>0$. We let
$\left(g_{\xi}\right)_{\xi\in M}$, $g_{\xi}=\Lambda_{\xi}^{2^{*}-2}g$, be a
family of conformal metrics to $g$ satisfying (3.14). We let $\chi$ be a
smooth, nonnegative cutoff function in $\left[0,\infty\right)$ such that
$\chi=1$ in $\left[0,r_{0}/2\right)$ and $\chi=0$ in
$\left[r_{0},\infty\right)$ for some number $r_{0}>0$ smaller than the
injectivity radii of the metrics $g_{\xi}$. For every $\mu>0$ and $x,\xi\in
M$, we then define

$B_{\mu,\xi}\left(x\right):=\Lambda_{\xi}\left(x\right)\chi\left(d_{g_{\xi}}\left(x,\xi\right)\right)\left(\frac{2\sqrt{6}\,\mu}{\mu^{2}+d_{g_{\xi}}\left(x,\xi\right)^{2}}\right)^{2}.$

Since $u_{0}$ is a nondegenerate solution to (1.1), we obtain the existence of
a family of solutions $\left(w_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{2}\left(M\right)$ to the equations

$\Delta_{g}w_{\varepsilon}+\left(h_{0}+\varepsilon
h\right)w_{\varepsilon}=\left|w_{\varepsilon}\right|w_{\varepsilon}\quad\text{in
}M$ (4.1)

such that $w_{\varepsilon}\to u_{0}$ in $C^{2}\left(M\right)$ as
$\varepsilon\to 0$ and $w_{\varepsilon}$ is differentiable with respect to
$\varepsilon$. By differentiating (4.1), we obtain

$\left.\frac{dw_{\varepsilon}}{d\varepsilon}\right|_{\varepsilon=0}=-\big{(}\Delta_{g}+h_{0}-2\left|u_{0}\right|\big{)}^{-1}\left(hu_{0}\right).$
(4.2)

We then define

$J_{\varepsilon}\left(u\right):=\frac{1}{2}\int_{M}\left(\left|\nabla
u\right|_{g}^{2}+\left(h_{0}+\varepsilon
h\right)u^{2}\right)dv_{g}-\frac{1}{3}\int_{M}\left|u\right|^{3}dv_{g}$

for all $\varepsilon>0$ and $u\in H^{1}\left(M\right)$. By using (4.1)
together with straightforward computations, we obtain

$J_{\varepsilon}\left(w_{\varepsilon}-B_{\mu,\xi}\right)=\frac{1}{6}\int_{\mathbb{R}^{n}}\left|w_{\varepsilon}\right|^{3}\,dv_{g}+J_{\varepsilon}\left(B_{\mu,\xi}\right)+\int_{M}w_{\varepsilon}B_{\mu,\xi}^{2}\,dv_{g}\\\
-\frac{2}{3}\int_{A_{\varepsilon,\mu,\xi}}\left(3w_{\varepsilon}-B_{\mu,\xi}\right)B_{\mu,\xi}^{2}\,dv_{g}-\frac{2}{3}\int_{A^{\prime}_{\varepsilon,\mu,\xi}}\left(3B_{\mu,\xi}-w_{\varepsilon}\right)w_{\varepsilon}^{2}\,dv_{g},$
(4.3)

where

$A_{\varepsilon,\mu,\xi}:=\left\\{x\in M:\,w_{\varepsilon}\left(x\right)\geq
B_{\mu,\xi}\left(x\right)\right\\}$

and

$A^{\prime}_{\varepsilon,\mu,\xi}:=\left\\{x\in
M:\,0<w_{\varepsilon}\left(x\right)<B_{\mu,\xi}\left(x\right)\right\\}.$

By applying [RV4]*Proposition 7.2, we obtain

$J_{\varepsilon}\left(B_{\mu,\xi}\right)+\int_{M}w_{\varepsilon}B_{\mu,\xi}^{2}\,dv_{g}=\frac{1}{6}\int_{\mathbb{R}^{n}}U_{0}^{3}\,dv_{\delta_{0}}+\frac{1}{2}\varphi_{\varepsilon}\left(\xi\right)\mu^{2}\int_{\mathbb{R}^{n}}U_{0}^{2}\,dv_{\delta_{0}}+\operatorname{o}\left(\mu^{3}\right)$
(4.4)

as $\varepsilon,\mu\to 0$, uniformly with respect to $\xi\in M$, where
$\delta_{0}$ is the Euclidean metric, $U_{0}$ is as in (3.30) and

$\varphi_{\varepsilon}:=h_{0}-c_{n}\operatorname{Scal}_{g}+\varepsilon
h+2w_{\varepsilon}.$

Further estimates yield

$\int_{A_{\varepsilon,\mu,\xi}}\left(3w_{\varepsilon}-B_{\mu,\xi}\right)B_{\mu,\xi}^{2}\,dv_{g}=64\sqrt{6}\,\omega_{5}u_{0}\left(\xi_{0}\right)^{3/2}\mu^{3}+\operatorname{o}\left(\mu^{3}\right)$
(4.5)

and

$\int_{A^{\prime}_{\varepsilon,\mu,\xi}}\left(3B_{\mu,\xi}-w_{\varepsilon}\right)w_{\varepsilon}^{2}\,dv_{g}=64\sqrt{6}\,\omega_{5}u_{0}\left(\xi_{0}\right)^{3/2}\mu^{3}+\operatorname{o}\left(\mu^{3}\right)$
(4.6)

as $\varepsilon,\mu\to 0$, uniformly with respect to $\xi\in M$, where
$\omega_{5}$ is the volume of the standard unit sphere of dimension $5$. It
follows from (4.3)–(4.6) that

$J_{\varepsilon}\left(w_{\varepsilon}-B_{\mu,\xi}\right)=\frac{1}{6}\int_{\mathbb{R}^{n}}\left|w_{\varepsilon}\right|^{3}\,dv_{g}+\frac{1}{6}\int_{\mathbb{R}^{n}}U_{0}^{3}\,dv_{\delta_{0}}+\frac{1}{2}\varphi_{\varepsilon}\left(\xi\right)\mu^{2}\int_{\mathbb{R}^{n}}U_{0}^{2}\,dv_{\delta_{0}}\\\
-\frac{256}{3}\sqrt{6}\,\omega_{5}u_{0}\left(\xi_{0}\right)^{3/2}\mu^{3}+\operatorname{o}\left(\mu^{3}\right)$
(4.7)

as $\varepsilon,\mu\to 0$, uniformly with respect to $\xi\in M$. On the other
hand, by using (4.1), we obtain

$\left\|\left(\Delta_{g}+h_{0}+\varepsilon
h-\left|w_{\varepsilon}-B_{\mu,\xi}\right|\right)\left(w_{\varepsilon}-B_{\mu,\xi}\right)\right\|_{L^{3/2}\left(M\right)}\\\
=\left\|\left(\Delta_{g}+h_{0}+\varepsilon
h+2w_{\varepsilon}-B_{\mu,\xi}\right)B_{\mu,\xi}\right\|_{L^{3/2}\left(M\right)}\\\
+2\left\|\left(2w_{\varepsilon}-B_{\mu,\xi}\right)B_{\mu,\xi}\right\|_{L^{3/2}\left(A_{\varepsilon,\mu,\xi}\right)}+2\left\|w_{\varepsilon}^{2}\right\|_{L^{3/2}\left(A^{\prime}_{\varepsilon,\mu,\xi}\right)}.$
(4.8)

By applying [RV4]*Proposition 7.1, we obtain

$\left\|\left(\Delta_{g}+h_{0}+\varepsilon
h+2w_{\varepsilon}-B_{\mu,\xi}\right)B_{\mu,\xi}\right\|_{L^{3/2}\left(M\right)}=\operatorname{O}\left(\mu^{2}\ln\left(1/\mu\right)^{2/3}\right)$
(4.9)

uniformly with respect to $\varepsilon,\mu>0$ and $\xi\in M$. Moreover, we
obtain

$\left\|\left(2w_{\varepsilon}-B_{\mu,\xi}\right)B_{\mu,\xi}\right\|_{L^{3/2}\left(A_{\varepsilon,\mu,\xi}\right)}=\operatorname{O}\left(\left\|B_{\mu,\xi}\right\|_{L^{3/2}\left(A_{\varepsilon,\mu,\xi}\right)}\right)=\operatorname{O}\left(\mu^{2}\ln\left(1/\mu\right)^{2/3}\right)$
(4.10)

and

$\left\|w_{\varepsilon}^{2}\right\|_{L^{3/2}(A^{\prime}_{\varepsilon,\mu,\xi})}=\operatorname{O}\left(\left\|1\right\|_{L^{3/2}(A^{\prime}_{\varepsilon,\mu,\xi})}\right)=\operatorname{O}\left(\mu^{3}\right)$
(4.11)

uniformly with respect to $\varepsilon,\mu>0$ and $\xi\in M$. It follows from
(4.8)–(4.11) that

$\left\|\left(\Delta_{g}+h_{0}+\varepsilon
h-\left|w_{\varepsilon}-B_{\mu,\xi}\right|\right)\left(w_{\varepsilon}-B_{\mu,\xi}\right)\right\|_{L^{3/2}\left(M\right)}=\operatorname{O}\left(\mu^{2}\ln\left(1/\mu\right)^{2/3}\right)$
(4.12)

uniformly with respect to $\varepsilon,\mu>0$ and $\xi\in M$. By applying
[RV2]*Theorem 1.1, it follows from (4.12) that for small $\varepsilon,\mu>0$
and all $\xi\in M$, there exists a function $\psi_{\varepsilon,\mu,\xi}\in
H^{1}\left(M\right)$ such that $\psi_{\varepsilon,\mu,\xi}\to 0$ in
$H^{1}\left(M\right)$ and

$J_{\varepsilon}\left(w_{\varepsilon}-B_{\mu,\xi}+\psi_{\varepsilon,\mu,\xi}\right)=J_{\varepsilon}\left(w_{\varepsilon}-B_{\mu,\xi}\right)+\operatorname{o}\left(\mu^{3}\right)$
(4.13)

as $\varepsilon,\mu\to 0$, uniformly with respect to $\xi\in M$, and

$D_{u}J_{\varepsilon}\left(w_{\varepsilon}-B_{\mu,\xi}+\psi_{\varepsilon,\mu,\xi}\right)=0\
\ \ \ \Longleftrightarrow\ \ \ \
D_{\left(\mu,\xi\right)}\left[J_{\varepsilon}\left(u_{0}-B_{\mu,\xi}+\psi_{\varepsilon,\mu,\xi}\right)\right]=0.$
(4.14)

For every $\varepsilon,t>0$ and $\tau\in\mathbb{R}^{n}$, we define

$\mu_{\varepsilon}\left(t\right):=t\varepsilon\quad\text{and}\quad\xi_{\varepsilon}\left(\tau\right):=\exp_{\xi_{\varepsilon}}\left(\sqrt{\varepsilon}\tau\right).$

Since
$\varphi_{0}\left(\xi_{0}\right)=\left|\nabla\varphi_{0}\left(\xi_{0}\right)\right|=0$,
it then follows from (4.7) and (4.13) that

$\varepsilon^{-3}\left(J_{\varepsilon}\left(w_{\varepsilon}-B_{\mu_{\varepsilon}\left(t\right),\xi_{\varepsilon}\left(\tau\right)}+\psi_{\varepsilon,\mu_{\varepsilon}\left(t\right),\xi_{\varepsilon}\left(\tau\right)}\right)-\frac{1}{6}\int_{\mathbb{R}^{n}}\left|w_{\varepsilon}\right|^{3}\,dv_{g}-\frac{1}{6}\int_{\mathbb{R}^{n}}U_{0}^{3}dv_{\delta_{0}}\right)\\\
\longrightarrow
F\left(t,\tau\right):=\left(\frac{1}{4}D^{2}\varphi_{0}\left(\xi_{0}\right).\left(\tau,\tau\right)+\frac{1}{2}\left(h+2\left.\frac{dw_{\varepsilon}}{d\varepsilon}\right|_{\varepsilon=0}\right)\left(\xi_{0}\right)\right)t^{2}\int_{\mathbb{R}^{n}}U_{0}^{2}\,dv_{\delta_{0}}\\\
-\frac{256}{3}\sqrt{6}\,\omega_{5}u_{0}\left(\xi_{0}\right)^{3/2}t^{3}$ (4.15)

as $\varepsilon\to 0$, uniformly with respect to $\xi\in M$ and $t$ in compact
subsets of $\left(0,\infty\right)$. By using the assumptions (i) and (ii)
together with (4.2), we obtain that the function $F$ attains a strict maximum
at $\left(t_{0},0\right)$ for some number $t_{0}>0$. By using (4.15), we then
obtain that there exists a family of maximum points
$\left(\left(t_{\varepsilon},\tau_{\varepsilon}\right)\right)_{0<\varepsilon\ll
1}$ of the function $\left(t,\tau\right)\mapsto
J_{\varepsilon}\left(w_{\varepsilon}-B_{\mu_{\varepsilon}\left(t\right),\xi_{\varepsilon}\left(\tau\right)}+\psi_{\varepsilon,\mu_{\varepsilon}\left(t\right),\xi_{\varepsilon}\left(\tau\right)}\right)$
in $\left[t_{0}/2,2t_{0}\right]\times B\left(0,1\right)$ that converges to
$\left(t_{0},0\right)$ as $\varepsilon\to 0$. It then follows from (4.14) that
for small $\varepsilon>0$, the function
$u_{\varepsilon}:=w_{\varepsilon}-B_{\mu_{\varepsilon}\left(t_{\varepsilon}\right),\xi_{\varepsilon}\left(\tau_{\varepsilon}\right)}+\psi_{\varepsilon,\mu_{\varepsilon}\left(t_{\varepsilon}\right),\xi_{\varepsilon}\left(\tau_{\varepsilon}\right)}$
is a solution to (1.2). It is easy to verify that $u_{\varepsilon}$ satisfies
(1.4). This ends the proof of Theorem 1.3. ∎

###### Proof of Corollary 2.1.

Here again, the proof of Robert and Vétois [RV4]*Theorem 1.3 continues to
apply with straightforward adaptations in the case where $n\geq 7$. Therefore,
in what follows, we only consider the case where $n=6$. Remark that in this
case, we can rewrite the equation satisfied by $u_{0}$ as

$\Delta_{g}u_{0}+\left(h_{0}+2u_{0}\right)u_{0}=3u_{0}^{2}\quad\text{in }M.$

By applying [RV4]*Lemma 13.1, we obtain that there exist a neighborhood
$\Omega_{0}$ of $\xi_{0}$ and families of functions
$(\tilde{h}_{\varepsilon})_{0\leq\varepsilon\ll 1}$ in $C^{p}\left(M\right)$
and $\left(\tilde{u}_{\varepsilon}\right)_{0\leq\varepsilon\ll 1}$ in
$C^{2}\left(M\right)$ such that $\tilde{h}_{\varepsilon}\to h_{0}$ in
$C^{p}\left(M\right)$ and $\tilde{u}_{\varepsilon}\to u_{0}$ in
$C^{2}\left(M\right)$ as $\varepsilon\to 0$, and for every $\varepsilon>0$,
$\tilde{h}_{\varepsilon}=h_{0}$ and $\tilde{u}_{\varepsilon}=u_{0}$ in
$\Omega_{0}$, and $\tilde{u}_{\varepsilon}$ is a nondegenerate, positive
solution to the equation

$\Delta_{g}\tilde{u}_{\varepsilon}+\big{(}\tilde{h}_{\varepsilon}+2u_{0}\big{)}\tilde{u}_{\varepsilon}=3\tilde{u}_{\varepsilon}^{2}\quad\text{in
}M.$ (4.16)

For every $\varepsilon>0$, since $\tilde{h}_{\varepsilon}=h_{0}$ and
$\tilde{u}_{\varepsilon}=u_{0}$ in $\Omega_{0}$,
$\varphi_{0}\left(\xi_{0}\right)=0$ and $\xi_{0}$ is a local maximum point of
the function $\varphi_{0}$, a straightforward perturbation argument (together
with a regularization argument in case $p=1$) yields the existence of a family
of functions $(\tilde{h}_{\varepsilon,\alpha})_{0\leq\alpha\ll 1}$ in
$C^{\max\left(p,2\right)}\left(M\right)$ such that
$\tilde{h}_{\varepsilon,\alpha}\to\tilde{h}_{\varepsilon}$ in
$C^{p}\left(M\right)$ as $\alpha\to 0$ and for every $\alpha>0$, the function
$\widetilde{\varphi}_{\varepsilon,\alpha}:=\tilde{h}_{\varepsilon,\alpha}+2u_{0}-c_{n}\operatorname{Scal}_{g}$
satisfies
$\widetilde{\varphi}_{\varepsilon,\alpha}\left(\xi_{0}\right)=\left|\nabla\widetilde{\varphi}_{\varepsilon,\alpha}\left(\xi_{0}\right)\right|=0$
and $D^{2}\widetilde{\varphi}_{\varepsilon,\alpha}\left(\xi_{0}\right)<0$.
Since $\tilde{u}_{\varepsilon}$ is a nondegenerate, positive solution to
(4.16), we then obtain that there exists a family of positive solutions
$(\tilde{u}_{\varepsilon,\alpha})_{0\leq\alpha\ll 1}$ to the equations

$\Delta_{g}\tilde{u}_{\varepsilon,\alpha}+\big{(}\tilde{h}_{\varepsilon,\alpha}+2u_{0}\big{)}\tilde{u}_{\varepsilon,\alpha}=3\tilde{u}_{\varepsilon,\alpha}^{2}\quad\text{in
}M$ (4.17)

such that $\tilde{u}_{\varepsilon,\alpha}\to\tilde{u}_{\varepsilon}$ in
$C^{2}\left(M\right)$ as $\alpha\to 0$. Since $\tilde{h}_{\varepsilon}\to
h_{0}$ and $\tilde{h}_{\varepsilon,\alpha}\to\tilde{h}_{\varepsilon}$ in
$C^{p}\left(M\right)$ as $\varepsilon\to 0$ and $\alpha\to 0$, respectively,
by standard elliptic theory, we obtain that $\tilde{u}_{\varepsilon}\to u_{0}$
and $\tilde{u}_{\varepsilon,\alpha}\to\tilde{u}_{\varepsilon}$ in
$C^{p+1}\left(M\right)$ as $\varepsilon\to 0$ and $\alpha\to 0$, respectively.
As is easily seen, the equation (4.17) can be rewritten as

$\Delta_{g}\tilde{u}_{\varepsilon,\alpha}+\hat{h}_{\varepsilon,\alpha}\tilde{u}_{\varepsilon,\alpha}=\tilde{u}_{\varepsilon,\alpha}^{2}\quad\text{in
}M,\quad\text{where
}\hat{h}_{\varepsilon,\alpha}:=\tilde{h}_{\varepsilon,\alpha}+2u_{0}-2\tilde{u}_{\varepsilon,\alpha}\quad\text{in
}M.$

Since $\tilde{h}_{\varepsilon,\alpha}\to\tilde{h}_{\varepsilon}$ and
$\tilde{u}_{\varepsilon,\alpha}\to\tilde{u}_{\varepsilon}$ in
$C^{p}\left(M\right)$ as $\alpha\to 0$, we obtain that
$\hat{h}_{\varepsilon,\alpha}\to\tilde{h}_{\varepsilon}$ in
$C^{p}\left(M\right)$. By applying again [RV4]*Lemma 13.1, we then obtain that
for every $\varepsilon,\alpha>0$, there exist a neighborhood
$\Omega_{\varepsilon,\alpha}$ of $\xi_{0}$ and families of functions
$(\hat{h}_{\varepsilon,\alpha,\beta})_{0\leq\beta\ll 1}$ in
$C^{\max\left(p,2\right)}\left(M\right)$ and
$\left(\hat{u}_{\varepsilon,\alpha,\beta}\right)_{0\leq\beta\ll 1}$ in
$C^{2}\left(M\right)$ such that
$\hat{h}_{\varepsilon,\alpha,\beta}\to\hat{h}_{\varepsilon,\alpha}$ in
$C^{p}\left(M\right)$ and
$\hat{u}_{\varepsilon,\alpha,\beta}\to\tilde{u}_{\varepsilon,\alpha}$ in
$C^{2}\left(M\right)$ as $\beta\to 0$, and for every $\beta>0$,
$\hat{h}_{\varepsilon,\alpha,\beta}=\hat{h}_{\varepsilon,\alpha}$ and
$\hat{u}_{\varepsilon,\alpha,\beta}=\tilde{u}_{\varepsilon,\alpha}$ in
$\Omega_{\varepsilon,\alpha}$, and $\hat{u}_{\varepsilon,\alpha,\beta}$ is a
nondegenerate, positive solution to the equation

$\Delta_{g}\hat{u}_{\varepsilon,\alpha,\beta}+\hat{h}_{\varepsilon,\alpha,\beta}\hat{u}_{\varepsilon,\alpha,\beta}=\hat{u}_{\varepsilon,\alpha,\beta}^{2}\quad\text{in
}M.$

Remark that

$\hat{h}_{\varepsilon,\alpha,\beta}+2\hat{u}_{\varepsilon,\alpha}-c_{n}\operatorname{Scal}_{g}=\hat{h}_{\varepsilon,\alpha}+2\tilde{u}_{\varepsilon,\alpha}-c_{n}\operatorname{Scal}_{g}=\widetilde{\varphi}_{\varepsilon,\alpha}\quad\text{in
}\Omega_{\varepsilon,\alpha}\cap\Omega_{0}.$

Recall that
$\widetilde{\varphi}_{\varepsilon,\alpha}\left(\xi_{0}\right)=\left|\nabla\widetilde{\varphi}_{\varepsilon,\alpha}\left(\xi_{0}\right)\right|=0$
and $D^{2}\widetilde{\varphi}_{\varepsilon,\alpha}\left(\xi_{0}\right)<0$. We
can now conclude by applying Theorem 1.3 together with a straightforward
diagonal argument. This ends the proof of Corollary 2.1. ∎

###### Proof of Corollary 2.2.

Remark that the equation

$\Delta_{g}u_{0}+h_{0}u_{0}=u_{0}^{2^{*}-1}\quad\text{in }M$

can be rewritten as

$\Delta_{g}u_{0}+\left(c_{n}\operatorname{Scal}_{g}+\varphi_{0}\right)u_{0}=\left\\{\begin{aligned}
&3u_{0}^{2}&&\text{if }n=6\\\ &u_{0}^{2^{*}-1}&&\text{if }n\geq
7\end{aligned}\right\\}\text{ in }M.$ (4.18)

The results by Aubin [A], Schoen [Sc1] and Trudinger [T] give that on the one
hand, there exists a positive solution $u_{0}$ to (4.18) provided the operator
$\Delta_{g}+c_{n}\operatorname{Scal}_{g}+\varphi_{0}$ is coercive in
$H^{1}\left(M\right)$ and

$\inf_{u\in
C^{\infty}\left(M\right)\backslash\left\\{0\right\\}}\frac{\displaystyle\int_{M}\left(\left|\nabla
u\right|_{g}^{2}+\left(c_{n}\operatorname{Scal}_{g}+\varphi_{0}\right)u^{2}\right)dv_{g}}{\displaystyle\left(\int_{M}\left|u\right|^{2^{*}}dv_{g}\right)^{2/2^{*}}}<K_{n}^{-2},$
(4.19)

where $K_{n}$ is the best constant for the embedding
$D^{1,2}\left(\mathbb{R}^{n}\right)\hookrightarrow
L^{2^{*}}\left(\mathbb{R}^{n}\right)$, and on the other hand, (4.19) is
satisfied when either $\varphi_{0}<0$ at some point in $M$ or $\varphi_{0}=0$
in $M$ and $\left(M,g\right)$ is not conformally diffeomorphic to the standard
sphere. In the latter case, by remarking that

$\int_{M}\varphi_{0}u^{2}dv_{g}\leq\left\|\max\left(\varphi_{0},0\right)\right\|_{L^{1}\left(M\right)}\left\|u\right\|_{C^{0}\left(M\right)}^{2},$

we obtain that (4.19) is satisfied when
$\left\|\max\left(\varphi_{0},0\right)\right\|_{L^{1}\left(M\right)}<\varepsilon_{0}$
for some constant $\varepsilon_{0}>0$ depending only on $\left(M,g\right)$.
Therefore, in both cases (i) and (ii), we obtain that there exist functions
$h_{0}\in C^{p}\left(M\right)$ and $u_{0}\in C^{2}\left(M\right)$ such that
$u_{0}$ is a positive solution to (1.1) and
$\varphi_{0}\left(\xi_{0}\right)=\left|\nabla\varphi_{0}\left(\xi_{0}\right)\right|=0$.
We can then conclude by applying Corollary 2.1. ∎

##  5\. Proofs of Theorems 1.4 and 2.3 and Corollaries 2.4 and 2.5

This section is devoted to our constructions of blowing-up solutions of type
(1.4) in the case of dimensions $n\in\left\\{3,4,5\right\\}$. We begin with
proving Theorem 1.4.

###### Proof of Theorem 1.4.

We begin with constructing a suitable ansatz. Since $u_{0}\not\equiv 0$ is a
nondegenerate solution to (1.1), we obtain the existence of a family of
solutions $\left(w_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ in
$C^{2}\left(M\right)$ to the equations

$\Delta_{g}w_{\varepsilon}+h_{\varepsilon}w_{\varepsilon}=\left|w_{\varepsilon}\right|^{2^{*}-2}w_{\varepsilon}\quad\text{in
}M$ (5.1)

such that $w_{\varepsilon}\to u_{0}$ in $C^{2}\left(M\right)$ as
$\varepsilon\to 0$ and $w_{\varepsilon}$ is differentiable with respect to
$\varepsilon$. Moreover, since $\left(M,g\right)$, $h_{0}$ and $u_{0}$ are
symmetric with respect to $\xi_{0}$, we can assume that for each
$\varepsilon$, the function $w_{\varepsilon}$ is also symmetric with respect
to $\xi_{0}$. By differentiating (5.1), we obtain

$\left.\frac{dw_{\varepsilon}}{d\varepsilon}\right|_{\varepsilon=0}=-\big{(}\Delta_{g}+\tilde{h}_{0}\big{)}^{-1}\left(hu_{0}\right),$
(5.2)

where $\tilde{h}_{0}=h_{0}-\left(2^{*}-1\right)\left|u_{0}\right|^{2^{*}-2}$.
We let $g_{\xi_{0}}:=\Lambda_{\xi_{0}}^{2^{*}-2}g$ be a conformal metric to
$g$ satisfying (3.14) and $\chi$ be a smooth, nonnegative cutoff function in
$\left[0,\infty\right)$ such that $\chi=1$ in $\left[0,r_{0}/2\right)$ and
$\chi=0$ in $\left[r_{0},\infty\right)$ for some number $r_{0}>0$ smaller than
the injectivity radius of the metric $g_{\xi_{0}}$. For every $\mu>0$ and
$x\in M$, we then define

$U_{\mu}\left(x\right):=\left\\{\begin{aligned}
&B_{\mu}\left(x\right)+k_{n}\mu^{\frac{n-2}{2}}\big{(}\chi\left(d_{g_{\xi_{0}}}\left(x,\xi_{0}\right)\right)H_{g,\tilde{h}_{0}}\left(x,\xi_{0}\right)&&\\\
&\quad+\left(1-\chi\left(d_{g_{\xi_{0}}}\left(x,\xi_{0}\right)\right)\right)G_{g,\tilde{h}_{0}}\left(x,\xi_{0}\right)\big{)}&&\text{in
Case 1}\\\ &B_{\mu}\left(x\right)&&\text{in Case 2},\end{aligned}\right.$

where $k_{n}$ is as in (3.18),

$\displaystyle\left\\{\begin{aligned} &\text{Case 1: }n=3\text{ or
}\left[n=4,5\text{ and
}h\left(\xi_{0}\right)=c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\right]\\\
&\text{Case 2: }n\in\left\\{4,5\right\\}\text{ and }h\left(\xi_{0}\right)\neq
c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right),\end{aligned}\right.\allowdisplaybreaks$
$\displaystyle
B_{\mu}\left(x\right):=\Lambda_{\xi_{0}}\left(x\right)\chi\left(d_{g_{\xi_{0}}}\left(x,\xi_{0}\right)\right)\left(\frac{\sqrt{n\left(n-2\right)}\mu}{\mu^{2}+d_{g_{\xi_{0}}}\left(x,\xi_{0}\right)^{2}}\right)^{\frac{n-2}{2}}$
(5.3)

and

$H_{g,\tilde{h}_{0}}\left(x,\xi_{0}\right):=G_{g,\tilde{h}_{0}}\left(x,\xi_{0}\right)-\left(\left(n-2\right)\omega_{n-1}\right)^{-1}\Lambda_{\xi_{0}}\left(x\right)d_{g_{\xi_{0}}}\left(x,\xi_{0}\right)^{2-n}.$

By (1.7), the mass $m_{g,\tilde{h}_{0}}(\xi_{0})$ is equal to
$H_{g,\tilde{h}_{0}}(\xi_{0},\xi_{0})$. For every $\varepsilon>0$ and $u\in
H^{1}\left(M\right)$, we then define

$J_{\varepsilon}\left(u\right):=\frac{1}{2}\int_{M}\left(\left|\nabla
u\right|_{g}^{2}+h_{\varepsilon}u^{2}\right)dv_{g}-\frac{1}{2^{*}}\int_{M}\left|u\right|^{2^{*}}dv_{g}.$

To apply a Lyapunov–Schmidt reduction method based on the ansatz
$w_{\varepsilon}-U_{\mu}$ (see for example the article by Robert and Vétois
[RV2]*Theorem 1.1), we need to estimate the asymptotic behavior as
$\varepsilon,\mu\to 0$ of
$J_{\varepsilon}\left(w_{\varepsilon}-U_{\mu}\right)$ together with an error
term of the form

$R_{\varepsilon,\mu}:=\left\|w_{\varepsilon}-U_{\mu}-\left(\Delta_{g}+1\right)^{-1}\left(\left(\left|w_{\varepsilon}-U_{\mu}\right|^{2^{*}-2}+1-h_{\varepsilon}\right)\left(w_{\varepsilon}-U_{\mu}\right)\right)\right\|_{H^{1}\left(M\right)}.$
(5.4)

We begin with estimating
$J_{\varepsilon}\left(w_{\varepsilon}-U_{\mu}\right)$. By integrating by parts
and using (5.1) together with straightforward estimates, we obtain

$J_{\varepsilon}\left(w_{\varepsilon}-U_{\mu}\right)=\frac{1}{n}\int_{M}\left|w_{\varepsilon}\right|^{2^{*}}dv_{g}+\frac{1}{2}\int_{M}\left(\left|\nabla
U_{\mu}\right|_{g}^{2}+\tilde{h}_{\varepsilon}U_{\mu}^{2}\right)dv_{g}-\frac{1}{2^{*}}\int_{M}\left|U_{\mu}\right|^{2^{*}}dv_{g}\\\
+\int_{M}\left|U_{\mu}\right|^{2^{*}-2}U_{\mu}w_{\varepsilon}\,dv_{g}+\operatorname{O}\Bigg{(}\int_{M_{\varepsilon,\mu}}\left(\left|U_{\mu}\right|^{2^{*}-2}w_{\varepsilon}^{2}+\left|w_{\varepsilon}\right|^{2^{*}-2}U_{\mu}^{2}\right)\,dv_{g}\\\
+\int_{M\backslash
M_{\varepsilon,\mu}}\left(\left|U_{\mu}\right|^{2^{*}-1}\left|w_{\varepsilon}\right|+\left|w_{\varepsilon}\right|^{2^{*}-3}\left|U_{\mu}\right|^{3}\right)dv_{g}\Bigg{)}$
(5.5)

uniformly with respect to $\varepsilon,\mu>0$, where

$\tilde{h}_{\varepsilon}:=h_{\varepsilon}-\left(2^{*}-1\right)\left|w_{\varepsilon}\right|^{2^{*}-2}\quad\text{and}\quad
M_{\varepsilon,\mu}:=\left\\{x\in
M:\,\left|w_{\varepsilon}\left(x\right)\right|\leq\left|U_{\mu}\left(x\right)\right|\right\\}.$

Remark that

$U_{\mu}=\operatorname{O}\left(\mu^{\frac{n-2}{2}}\right)\,\text{and}\,\big{(}\Delta_{g}+\tilde{h}_{\varepsilon}\big{)}U_{\mu}=k_{n}\mu^{\frac{n-2}{2}}\big{(}\tilde{h}_{\varepsilon}-\tilde{h}_{0}\big{)}G_{g,\tilde{h}_{0}}\left(\cdot,\xi_{0}\right)=\operatorname{o}\left(\mu^{\frac{n-2}{2}}\right)$
(5.6)

as $\varepsilon,\mu\to 0$, uniformly in $M\backslash
B_{g_{\xi_{0}}}\left(\xi_{0},r_{0}\right)$, where
$B_{g_{\xi_{0}}}\left(\xi_{0},r_{0}\right)$ is the ball of center $\xi_{0}$
and radius $r_{0}$ with respect to the metric $g_{\xi_{0}}$. By identifying
$T_{\xi_{0}}M$ with $\mathbb{R}^{n}$ and using (5.6) together with the
conformal invariance property of the conformal Laplacian and the change of
variable $x=\exp_{\xi_{0}}\left(\mu y\right)$ for $y\in
B\left(0,r_{0}/\mu\right)$, where $\exp_{\xi_{0}}$ is the exponential map at
the point $\xi_{0}$ with respect to the metric $g_{\xi_{0}}$ and
$B\left(0,r_{0}/\mu\right)$ is the ball of center 0 and radius $r_{0}/\mu$
with respect to the Euclidean metric, we obtain

$\displaystyle\int_{M}\left(\left|\nabla
U_{\mu}\right|_{g}^{2}+\tilde{h}_{\varepsilon}U_{\mu}^{2}\right)dv_{g}=\int_{B\left(0,r_{0}/\mu\right)}\widehat{U}_{\mu}\big{(}\Delta_{\hat{g}_{\mu}}\widehat{U}_{\mu}+\mu^{2}\hat{h}_{\varepsilon,\mu}\widehat{U}_{\mu}\big{)}dv_{\hat{g}_{\mu}}+\operatorname{o}\left(\mu^{n-2}\right),\allowdisplaybreaks$
(5.7)
$\displaystyle\int_{M}\left|U_{\mu}\right|^{2^{*}}dv_{g}=\int_{B\left(0,r_{0}/\mu\right)}\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}}dv_{\hat{g}_{\mu}}+\operatorname{o}\left(\mu^{n-2}\right),\allowdisplaybreaks$
(5.8)
$\displaystyle\int_{M}\left|U_{\mu}\right|^{2^{*}-2}U_{\mu}w_{\varepsilon}\,dv_{g}=\int_{B\left(0,r_{0}/\mu\right)}\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}-2}\widehat{U}_{\mu}\widehat{w}_{\varepsilon,\mu}\,dv_{\hat{g}_{\mu}}+\operatorname{o}\left(\mu^{n-2}\right),\allowdisplaybreaks$
(5.9)
$\displaystyle\int_{M_{\varepsilon,\mu}}\left(\left|U_{\mu}\right|^{2^{*}-2}w_{\varepsilon}^{2}+\left|w_{\varepsilon}\right|^{2^{*}-2}U_{\mu}^{2}\right)\,dv_{g}$
$\displaystyle\quad=\int_{\Omega_{\varepsilon,\mu}}\left(\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}-2}\widehat{w}_{\varepsilon,\mu}^{2}+\left|\widehat{w}_{\varepsilon,\mu}\right|^{2^{*}-2}\widehat{U}_{\mu}^{2}\right)dv_{\hat{g}_{\mu}}+\operatorname{o}\left(\mu^{n-2}\right)$
(5.10)

and

$\int_{M\backslash
M_{\varepsilon,\mu}}\left(\left|U_{\mu}\right|^{2^{*}-1}\left|w_{\varepsilon}\right|+\left|w_{\varepsilon}\right|^{2^{*}-3}\left|U_{\mu}\right|^{3}\right)dv_{g}\\\
=\int_{B\left(0,r_{0}/\mu\right)\backslash\Omega_{\varepsilon,\mu}}\left(\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}-1}\left|\widehat{w}_{\varepsilon,\mu}\right|+\left|\widehat{w}_{\varepsilon,\mu}\right|^{2^{*}-3}\big{|}\widehat{U}_{\mu}\big{|}^{3}\right)dv_{\hat{g}_{\mu}}+\operatorname{o}\left(\mu^{n-2}\right)$
(5.11)

as $\mu\to 0$, uniformly with respect to $\varepsilon>0$, where

$\displaystyle\Omega_{\varepsilon,\mu}:=\left\\{y\in
B\left(0,r_{0}/\mu\right):\,\left|\hat{w}_{\varepsilon,\mu}\left(y\right)\right|\leq\big{|}\widehat{U}_{\mu}\left(y\right)\big{|}\right\\},\allowdisplaybreaks$
$\displaystyle\hat{g}_{\mu}\left(y\right):=\exp_{\xi_{0}}^{*}g_{\xi_{0}}\left(\mu
y\right),\quad\widehat{w}_{\varepsilon,\mu}:=\mu^{\frac{n-2}{2}}w_{\varepsilon}\big{(}\exp_{\xi_{0}}\left(\mu
y\right)\big{)},\allowdisplaybreaks$
$\displaystyle\hat{h}_{\varepsilon,\mu}:=\left(c_{n}\operatorname{Scal}_{g_{\xi_{0}}}+\Lambda_{\xi_{0}}^{2-2^{*}}\big{(}\tilde{h}_{\varepsilon}-c_{n}\operatorname{Scal}_{g}\big{)}\right)\big{(}\exp_{\xi_{0}}\left(\mu
y\right)\big{)}$

and

$\widehat{U}_{\mu}\left(y\right):=\\\ \left\\{\begin{aligned}
&\chi\left(\mu\left|y\right|\right)U_{0}\left(y\right)+k_{n}\mu^{n-2}\Lambda_{\xi_{0}}\left(\exp_{\xi_{0}}\left(\mu
y\right)\right)^{-1}\Big{(}\chi\left(\mu\left|y\right|\right)H_{g,\tilde{h}_{0}}\left(\exp_{\xi_{0}}\left(\mu
y\right),\xi_{0}\right)\\\
&\qquad+\left(1-\chi\left(\mu\left|y\right|\right)\right)G_{g,\tilde{h}_{0}}\left(\exp_{\xi_{0}}\left(\mu
y\right),\xi_{0}\right)\Big{)}\hskip 104.0pt\text{in Case 1}\\\
&\chi\left(\mu\left|y\right|\right)U_{0}\left(y\right)\hskip 235.0pt\text{in
Case 2}\end{aligned}\right.$

for all points $y\in B\left(0,r_{0}/\mu\right)$, where
$\left|y\right|:=\left|y\right|_{\delta_{0}}$ and $U_{0}$ is as in (3.30).
Since $w_{\varepsilon}\to u_{0}$ in $C^{2}\left(M\right)$ as $\varepsilon\to
0$, $u_{0}\left(\xi_{0}\right)=0$ and $w_{\varepsilon}$ is symmetric with
respect to $\xi_{0}$, a Taylor expansion gives

$\widehat{w}_{\varepsilon,\mu}\left(y\right)=\frac{d}{d\varepsilon}\left[w_{\varepsilon}\left(\xi_{0}\right)\right]_{\varepsilon=0}\varepsilon\mu^{\frac{n-2}{2}}+\operatorname{O}\left(\varepsilon^{2}\mu^{\frac{n-2}{2}}+\mu^{\frac{n+2}{2}}\left|y\right|^{2}\right)$
(5.12)

uniformly with respect to $\varepsilon,\mu>0$ and $y\in
B\left(0,r_{0}/\mu\right)$. By using (1.6) and (1.7) (together with (3.14) and
a result from the book of Habermann [Ha]*Lemma 5.2.10 in the case where
$n=3$), we obtain

$\widehat{U}_{\mu}\left(y\right)=\left\\{\begin{aligned}
&U_{0}\left(y\right)+k_{n}m_{g,\tilde{h}_{0}}\left(\xi_{0}\right)\mu^{n-2}+\operatorname{O}\left(\mu^{n-2}\eta\left(\mu
y\right)\right)&&\text{in Case 1}\\\
&U_{0}\left(y\right)+\operatorname{O}\left(\mu^{n-2}\right)&&\text{in Case
2}\end{aligned}\right.$ (5.13)

uniformly with respect to $\mu>0$ and $y\in B\left(0,r_{0}/\mu\right)$, for
some function $\eta\in C^{0}\left(\mathbb{R}^{n}\right)$ such that
$\eta\left(0\right)=0$. To estimate the right-hand side of (5.7), we remark
that in Case 1, for every $\mu>0$, the function $\widehat{U}_{\mu}$ can be
rewritten as

$\widehat{U}_{\mu}=W_{\mu}\left(y\right)+k_{n}\mu^{n-2}\Lambda_{\xi_{0}}\left(\exp_{\xi_{0}}\left(\mu
y\right)\right)^{-1}G_{g,\tilde{h}_{0}}\left(\exp_{\xi_{0}}\left(\mu
y\right),\xi_{0}\right)$ (5.14)

for all points $y\in B\left(0,r_{0}/\mu\right)\backslash\left\\{0\right\\}$,
where

$W_{\mu}\left(y\right):=\chi\left(\mu\left|y\right|\right)\left(U_{0}\left(y\right)-\left(n\left(n-2\right)\right)^{\frac{n-2}{4}}\left|y\right|^{2-n}\right).$

Recall moreover that $\Delta_{\delta_{0}}U_{0}=U_{0}^{2^{*}-1}$ in
$\mathbb{R}^{n}$ and $\Delta_{\delta_{0}}\left|\cdot\right|^{2-n}=0$ in
$\mathbb{R}^{n}\backslash\left\\{0\right\\}$. Since the functions
$\chi\left(\mu\,\cdot\right)U_{0}$ and $W_{\mu}$ are radially symmetric about
the origin, by using (3.14) and (5.14) together with straightforward
estimates, we then obtain

$\displaystyle\Delta_{\hat{g}_{\mu}}W_{\mu}\left(y\right)$
$\displaystyle=\Delta_{\delta_{0}}W_{\mu}\left(y\right)+\operatorname{O}\big{(}\mu^{N}\left|y\right|^{N-1}\left|\nabla
W_{\mu}\left(y\right)\right|\big{)}$
$\displaystyle=U_{0}\left(y\right)^{2^{*}-1}+\operatorname{O}\big{(}\mu^{N}\left|y\right|^{N-n-2}\big{)}\quad\text{in
Case 1}$ (5.15)

and

$\displaystyle\Delta_{\hat{g}_{\mu}}\widehat{U}_{\mu}\left(y\right)$
$\displaystyle=\Delta_{\delta_{0}}\widehat{U}_{\mu}\left(y\right)+\operatorname{O}\big{(}\mu^{N}\left|y\right|^{N-1}\big{|}\nabla\widehat{U}_{\mu}\left(y\right)\big{|}\big{)}$
$\displaystyle=U_{0}\left(y\right)^{2^{*}-1}+\operatorname{O}\big{(}\mu^{N}\left|y\right|^{N-n}\big{)}\quad\text{in
Case 2}$ (5.16)

uniformly with respect to $\varepsilon,\mu>0$ and $y\in
B\left(0,r_{0}/\mu\right)\backslash\left\\{0\right\\}$. On the other hand, in
Case 1, the conformal invariance property of the conformal Laplacian gives

$\big{(}\Delta_{\hat{g}_{\mu}}+\mu^{2}\hat{h}_{0,\mu}\big{)}\big{(}\widehat{U}_{\mu}-W_{\mu}\big{)}\left(y\right)=\mu^{n}\left[\Lambda_{\xi_{0}}^{1-2^{*}}\big{(}\Delta_{g}+\tilde{h}_{0}\big{)}G_{g,\tilde{h}_{0}}\left(\cdot,\xi_{0}\right)\right]\left(\exp_{\xi_{0}}\left(\mu
y\right)\right)=0$ (5.17)

for all $y\in B\left(0,r_{0}/\mu\right)\backslash\left\\{0\right\\}$.
Moreover, by using (3.14) and (3.15) together with Taylor expansions, we
obtain

$\displaystyle\hat{h}_{\varepsilon,\mu}\left(y\right)W_{\mu}\left(y\right)+\big{(}\hat{h}_{\varepsilon,\mu}-\hat{h}_{0,\mu}\big{)}\left(y\right)\big{(}\widehat{U}_{\mu}-W_{\mu}\big{)}\left(y\right)$
$\displaystyle=\operatorname{O}\left(\left\\{\begin{aligned}
&\left|y\right|^{-1}\big{(}\left(1+\left|y\right|\right)^{-2}+\varepsilon\big{)}&&\text{if
}n=3\\\
&\left|y\right|^{2-n}\big{(}\mu^{2}\left|y\right|^{2}\left(1+\left|y\right|\right)^{-2}+\varepsilon\big{)}&&\text{if
}n=4,5\text{ and
}h_{0}\left(\xi_{0}\right)=c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\end{aligned}\right\\}\right)$
(5.18)

and

$\hat{h}_{\varepsilon,\mu}\left(y\right)\widehat{U}_{\mu}\left(y\right)=\big{(}h_{0}\left(\xi_{0}\right)-c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)+\operatorname{O}\big{(}\mu^{2}\left|y\right|^{2}+\varepsilon\big{)}\big{)}U_{0}\left(y\right)+\operatorname{O}\left(\mu^{n-2}\right)\,\,\,\,\text{in
Case 2}$ (5.19)

uniformly with respect to $\varepsilon,\mu>0$ and $y\in
B\left(0,r_{0}/\mu\right)\backslash\left\\{0\right\\}$. It follows from
(5)–(5.19) that

$\big{(}\Delta_{\hat{g}_{\mu}}\widehat{U}_{\mu}+\mu^{2}\hat{h}_{\varepsilon,\mu}\widehat{U}_{\mu}\big{)}\left(y\right)=U_{0}\left(y\right)^{2^{*}-1}\\\
+\left\\{\begin{aligned}
&\operatorname{O}\big{(}\mu^{N}\left|y\right|^{N-5}+\mu^{2}\left|y\right|^{-1}\big{(}\left(1+\left|y\right|\right)^{-2}+\varepsilon\big{)}\big{)}\hskip
113.0pt\text{if }n=3\\\
&\operatorname{O}\big{(}\mu^{N}\left|y\right|^{N-n-2}+\mu^{2}\left|y\right|^{2-n}\big{(}\mu^{2}+\varepsilon\big{)}\big{)}\hskip
8.0pt\text{if }n=4,5\text{ and
}h_{0}\left(\xi_{0}\right)=c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\\\
&\big{(}h_{0}\left(\xi_{0}\right)-c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)+\operatorname{O}\big{(}\mu^{2}\left|y\right|^{2}+\varepsilon\big{)}\big{)}\mu^{2}U_{0}\left(y\right)+\operatorname{O}\big{(}\mu^{N}\left|y\right|^{N-n}\big{)}\hskip
1.0pt\text{in Case 2}\end{aligned}\right.$ (5.20)

uniformly with respect to $\varepsilon,\mu>0$ and $y\in
B\left(0,r_{0}/\mu\right)$. For every $\varepsilon,t>0$, we define

$\mu_{\varepsilon}\left(t\right):=\left\\{\begin{aligned}
&t\varepsilon^{\frac{2}{n-2}}&&\text{in Case 1}\\\
&t\nu_{\varepsilon}&&\text{in Case 2.1}\\\ &t\varepsilon^{2}&&\text{in Case
2.2},\end{aligned}\right.$ (5.21)

where Case 2.1 and Case 2.2 are subcases of Case 2 defined as

$\left\\{\begin{aligned} &\text{Case 2.1: }n=4\text{ and
}h\left(\xi_{0}\right)\neq c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\\\
&\text{Case 2.2: }n=5\text{ and }h\left(\xi_{0}\right)\neq
c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\end{aligned}\right.$

and $\nu_{\varepsilon}>0$ is such that
$\nu_{\varepsilon}\ln\left(1/\nu_{\varepsilon}\right)=\varepsilon$ and
$\nu_{\varepsilon}\to 0$ as $\varepsilon\to 0$. It follows from (3.14) and
(5.7)–(5.21) together with standard integral estimates (see for instance the
article by Esposito, Pistoia and Vétois [EPV]*Section 4) that

$\displaystyle\int_{M}\left(\left|\nabla
U_{\mu_{\varepsilon}\left(t\right)}\right|_{g}^{2}+\tilde{h}_{\varepsilon}U_{\mu_{\varepsilon}\left(t\right)}^{2}\right)dv_{g}=\int_{\mathbb{R}^{n}}U_{0}^{2^{*}}dv_{\delta_{0}}$
$\displaystyle\qquad+\left\\{\begin{aligned}
&k_{n}m_{g,\tilde{h}_{0}}\left(\xi_{0}\right)t^{n-2}\varepsilon^{2}\int_{\mathbb{R}^{n}}U_{0}^{2^{*}-1}dv_{\delta_{0}}+\operatorname{o}\left(\varepsilon^{2}\right)&&\text{in
Case 1}\\\
&8\,\omega_{3}\left(h_{0}\left(\xi_{0}\right)-c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)+\operatorname{o}\left(1\right)\right)t^{2}\varepsilon\nu_{\varepsilon}&&\text{in
Case 2.1}\\\
&\left(h_{0}\left(\xi_{0}\right)-c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)+\operatorname{o}\left(1\right)\right)t^{2}\varepsilon^{4}\int_{\mathbb{R}^{5}}U_{0}^{2}\,dv_{\delta_{0}}&&\text{in
Case 2.2},\end{aligned}\right.\allowdisplaybreaks$ (5.22)
$\displaystyle\int_{M}\left|U_{\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}}dv_{g}=\int_{\mathbb{R}^{n}}U_{0}^{2^{*}}dv_{\delta_{0}}$
$\displaystyle\qquad+\left\\{\begin{aligned}
&2^{*}k_{n}m_{g,\tilde{h}_{0}}\left(\xi\right)t^{n-2}\varepsilon^{2}\int_{\mathbb{R}^{n}}U_{0}^{2^{*}-1}dv_{\delta_{0}}+\operatorname{o}\left(\varepsilon^{2}\right)&&\text{in
Case 1}\\\
&\operatorname{o}\left(\varepsilon\nu_{\varepsilon}\right)&&\text{in Case
2.1}\\\ &\operatorname{o}\left(\varepsilon^{4}\right)&&\text{in Case
2.2}\end{aligned}\right.$ (5.23)

and

$\int_{M}\left|U_{\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}-2}U_{\mu_{\varepsilon}\left(t\right)}w_{\varepsilon}\,dv_{g}\\\
=\left\\{\begin{aligned}
&\frac{d}{d\varepsilon}\left[w_{\varepsilon}\left(\xi_{0}\right)\right]_{\varepsilon=0}t^{\frac{n-2}{2}}\varepsilon^{2}\int_{\mathbb{R}^{n}}U_{0}^{2^{*}-1}dv_{\delta_{0}}+\operatorname{o}\left(\varepsilon^{2}\right)&&\text{in
Case 1}\\\
&\frac{d}{d\varepsilon}\left[w_{\varepsilon}\left(\xi_{0}\right)\right]_{\varepsilon=0}t\varepsilon\nu_{\varepsilon}\int_{\mathbb{R}^{n}}U_{0}^{3}\,dv_{\delta_{0}}+\operatorname{o}\left(\varepsilon\nu_{\varepsilon}\right)&&\text{in
Case 2.1}\\\
&\frac{d}{d\varepsilon}\left[w_{\varepsilon}\left(\xi_{0}\right)\right]_{\varepsilon=0}t^{3/2}\varepsilon^{4}\int_{\mathbb{R}^{n}}U_{0}^{7/3}\,dv_{\delta_{0}}+\operatorname{o}\left(\varepsilon^{4}\right)&&\text{in
Case 2.2}\end{aligned}\right.$ (5.24)

as $\varepsilon\to 0$, uniformly with respect to $t$ in compact subsets of
$\left(0,\infty\right)$. Moreover, by using (5.12) and (5.21), we obtain that
for every compact set $A\subset\left(0,\infty\right)$, there exists a constant
$r_{A}>0$ such that for every $\varepsilon>0$ and $t\in A$,

$B\left(0,r_{A}\varrho_{\varepsilon}\right)\subset\Omega_{\varepsilon,\mu_{\varepsilon}\left(t\right)},\quad\text{where
}\varrho_{\varepsilon}:=\left\\{\begin{aligned}
&\varepsilon^{-\frac{n+2}{n\left(n-2\right)}}&&\text{in Case 1}\\\
&\nu_{\varepsilon}^{-3/4}&&\text{in Case 2.1}\\\ &\varepsilon^{-4/3}&&\text{in
Case 2.2.}\end{aligned}\right.$ (5.25)

It follows from (5.12), (5.13) and (5.25) that

$\int_{\Omega_{\varepsilon,\mu_{\varepsilon}\left(t\right)}}\left(\big{|}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}\big{|}^{2^{*}-2}\widehat{w}_{\varepsilon,\mu_{\varepsilon}\left(t\right)}^{2}+\left|\widehat{w}_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}-2}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}^{2}\right)dv_{\hat{g}_{\mu}}=\operatorname{O}\Bigg{(}\int_{B\left(0,r_{A}\varrho_{\varepsilon}\right)}\\\
\bigg{(}\left(\varepsilon\mu_{\varepsilon}\left(t\right)^{\frac{n-2}{2}}+\mu_{\varepsilon}\left(t\right)^{\frac{n+2}{2}}\left|\cdot\right|^{2}\right)^{2}U_{0}^{2^{*}-2}+\left(\varepsilon\mu_{\varepsilon}\left(t\right)^{\frac{n-2}{2}}+\mu_{\varepsilon}\left(t\right)^{\frac{n+2}{2}}\left|\cdot\right|^{2}\right)^{2^{*}-2}U_{0}^{2}\bigg{)}dv_{\delta_{0}}\\\
+\int_{B\left(0,r_{0}/\mu_{\varepsilon}\left(t\right)\right)\backslash
B\left(0,r_{A}\varrho_{\varepsilon}\right)}U_{0}^{2^{*}}dv_{\delta_{0}}\Bigg{)}$
(5.26)

and

$\int_{B\left(0,r_{0}/\mu_{\varepsilon}\left(t\right)\right)\backslash\Omega_{\varepsilon,\mu_{\varepsilon}\left(t\right)}}\Big{(}\big{|}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}\big{|}^{2^{*}-1}\left|\widehat{w}_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right|+\left|\widehat{w}_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}-3}\big{|}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}\big{|}^{3}\Big{)}dv_{\hat{g}_{\mu}}\\\
=\operatorname{O}\bigg{(}\int_{B\left(0,r_{0}/\mu_{\varepsilon}\left(t\right)\right)\backslash
B\left(0,r_{A}\varrho_{\varepsilon}\right)}\bigg{(}\left(\varepsilon\mu_{\varepsilon}\left(t\right)^{\frac{n-2}{2}}+\mu_{\varepsilon}\left(t\right)^{\frac{n+2}{2}}\left|\cdot\right|^{2}\right)U_{0}^{2^{*}-1}\\\
+\left(\varepsilon\mu_{\varepsilon}\left(t\right)^{\frac{n-2}{2}}+\mu_{\varepsilon}\left(t\right)^{\frac{n+2}{2}}\left|\cdot\right|^{2}\right)^{2^{*}-3}U_{0}^{3}\bigg{)}dv_{\delta_{0}}\bigg{)}$
(5.27)

uniformly with respect to $\varepsilon>0$ and $t\in A$. By using (5.10),
(5.11), (5.21), (5.26) and (5.27) together with standard integral estimates,
we then obtain

$\int_{M_{\varepsilon,\mu_{\varepsilon}\left(t\right)}}\left(\left|U_{\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}-2}w_{\varepsilon}^{2}+\left|w_{\varepsilon}\right|^{2^{*}-2}U_{\mu_{\varepsilon}\left(t\right)}^{2}\right)dv_{g}+\int_{M\backslash
M_{\varepsilon,\mu_{\varepsilon}\left(t\right)}}\Big{(}\left|U_{\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}-1}\left|w_{\varepsilon}\right|\\\
+\left|w_{\varepsilon}\right|^{2^{*}-3}\left|U_{\mu_{\varepsilon}\left(t\right)}\right|^{3}\Big{)}dv_{g}=\operatorname{o}\left(\left\\{\begin{aligned}
&\varepsilon^{2}&&\text{in Case 1}\\\ &\varepsilon\nu_{\varepsilon}&&\text{in
Case 2.1}\\\ &\varepsilon^{4}&&\text{in Case
2.2}\end{aligned}\right\\}\right)$ (5.28)

as $\varepsilon\to 0$, uniformly with respect to $t$ in compact subsets of
$\left(0,\infty\right)$. Note that in Case 2.1, we used that
$\nu_{\varepsilon}=\operatorname{o}\left(\varepsilon\right)$ by (5.21). It
follows from (5.5), (5.22)–(5.24) and (5.28) that

$J_{\varepsilon}\left(w_{\varepsilon}-U_{\mu_{\varepsilon}\left(t\right)}\right)=\frac{1}{n}\int_{M}\left|w_{\varepsilon}\right|^{2^{*}}dv_{g}+\frac{1}{n}\int_{\mathbb{R}^{n}}U_{0}^{2^{*}}dv_{\delta_{0}}\\\
+\left\\{\begin{aligned}
&F\left(t\right)\varepsilon^{2}+\operatorname{o}\left(\varepsilon^{2}\right)&&\text{in
Case 1}\\\
&F\left(t\right)\varepsilon\nu_{\varepsilon}+\operatorname{o}\left(\varepsilon\nu_{\varepsilon}\right)&&\text{in
Case 2.1}\\\
&F\left(t\right)\varepsilon^{4}+\operatorname{o}\left(\varepsilon^{4}\right)&&\text{in
Case 2.2}\end{aligned}\right.$ (5.29)

as $\varepsilon\to 0$, uniformly with respect to $t$ in compact subsets of
$\left(0,\infty\right)$,where

$F\left(t\right):=\frac{d}{d\varepsilon}\left[w_{\varepsilon}\left(\xi_{0}\right)\right]_{\varepsilon=0}t^{\frac{n-2}{2}}\int_{\mathbb{R}^{n}}U_{0}^{2^{*}-1}dv_{\delta_{0}}\\\
+\left\\{\begin{aligned}
&-\frac{k_{n}}{2}\,m_{g,\tilde{h}_{0}}\left(\xi_{0}\right)t^{n-2}\int_{\mathbb{R}^{n}}U_{0}^{2^{*}-1}dv_{\delta_{0}}&&\text{in
Case 1}\\\
&4\,\omega_{3}\left(h_{0}\left(\xi_{0}\right)-c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\right)t^{2}&&\text{in
Case 2.1}\\\
&\frac{1}{2}\left(h_{0}\left(\xi_{0}\right)-c_{n}\operatorname{Scal}_{g}\left(\xi_{0}\right)\right)t^{2}\int_{\mathbb{R}^{5}}U_{0}^{2}\,dv_{\delta_{0}}&&\text{in
Case 2.2.}\end{aligned}\right.$

We now consider the error term $R_{\varepsilon,\mu}$ defined in (5.4). By
continuity of the embedding $H^{1}\left(M\right)\hookrightarrow
L^{2^{*}}\left(M\right)$, we obtain

$R_{\varepsilon,\mu}=\operatorname{O}\left(\left\|\left(\Delta_{g}+h_{\varepsilon}-\left|w_{\varepsilon}-U_{\mu}\right|^{2^{*}-2}\right)\left(w_{\varepsilon}-U_{\mu}\right)\right\|_{L^{\frac{2n}{n+2}}\left(M\right)}\right)$
(5.30)

uniformly with respect to $\varepsilon,\mu>0$. By using (5.1) and (5.30)
together with similar estimate as in (5.5)–(5.11), we obtain

$\displaystyle R_{\varepsilon,\mu}$
$\displaystyle=\operatorname{O}\bigg{(}\left\|\Delta_{g}U_{\mu}+\left(h_{\varepsilon}-\left(2^{*}-1\right)\left|w_{\varepsilon}\right|^{2^{*}-2}\right)U_{\mu}-\left|U_{\mu}\right|^{2^{*}-2}U_{\mu}\right\|_{L^{\frac{2n}{n+2}}\left(M\right)}\allowdisplaybreaks$
$\displaystyle\qquad+\left\|\left|U_{\mu}\right|^{2^{*}-2}\left|w_{\varepsilon}\right|+\left|w_{\varepsilon}\right|^{2^{*}-2}\left|U_{\mu}\right|\right\|_{L^{\frac{2n}{n+2}}\left(M_{\varepsilon,\mu}\right)}\allowdisplaybreaks$
$\displaystyle\qquad+\left\|\left|U_{\mu}\right|^{2^{*}-1}+\left|w_{\varepsilon}\right|^{2^{*}-3}U_{\mu}^{2}\right\|_{L^{\frac{2n}{n+2}}\left(M\backslash
M_{\varepsilon,\mu}\right)}\bigg{)}\allowdisplaybreaks$
$\displaystyle=\operatorname{O}\bigg{(}\Big{\|}\Delta_{\hat{g}_{\mu}}\widehat{U}_{\mu}+\mu^{2}\hat{h}_{\varepsilon,\mu}\widehat{U}_{\mu}-\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}-2}U_{\mu}\Big{\|}_{L^{\frac{2n}{n+2}}\left(B\left(0,r_{0}/\mu\right)\right)}\allowdisplaybreaks$
$\displaystyle\qquad+\Big{\|}\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}-2}\left|\widehat{w}_{\varepsilon,\mu}\right|+\left|\widehat{w}_{\varepsilon,\mu}\right|^{2^{*}-2}\big{|}\widehat{U}_{\mu}\big{|}\Big{\|}_{L^{\frac{2n}{n+2}}\left(\Omega_{\varepsilon,\mu}\right)}$
$\displaystyle\qquad+\Big{\|}\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}-1}+\left|\widehat{w}_{\varepsilon,\mu}\right|^{2^{*}-3}\widehat{U}_{\mu}^{2}\Big{\|}_{L^{\frac{2n}{n+2}}\left(B\left(0,r_{0}/\mu\right)\backslash\Omega_{\varepsilon,\mu}\right)}\bigg{)}+\operatorname{o}\left(\mu^{\frac{n-2}{2}}\right)$
(5.31)

as $\varepsilon,\mu\to 0$. By using (5.13) and (5.20), we obtain

$\Big{\|}\Delta_{\hat{g}_{\mu}}\widehat{U}_{\mu}+\mu^{2}\hat{h}_{\varepsilon,\mu}\widehat{U}_{\mu}-\big{|}\widehat{U}_{\mu}\big{|}^{2^{*}-2}U_{\mu}\Big{\|}_{L^{\frac{2n}{n+2}}\left(B\left(0,r_{0}/\mu\right)\right)}=\left\\{\begin{aligned}
&\operatorname{o}\left(\mu^{\frac{n-2}{2}}\right)&&\text{in Case 1}\\\
&\operatorname{O}\left(\mu^{\frac{n-2}{2}}\right)&&\text{in Case
2}\end{aligned}\right.$ (5.32)

as $\varepsilon,\mu\to 0$. By proceeding in a similar way as in (5.26)–(5.28),
by using (5.12), (5.13), (5.21) and (5.25), we obtain

$\Big{\|}\big{|}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}\big{|}^{2^{*}-2}\left|\widehat{w}_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right|+\left|\widehat{w}_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}-2}\big{|}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}\big{|}\Big{\|}_{L^{\frac{2n}{n+2}}\left(\Omega_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right)}+\Big{\|}\big{|}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}\big{|}^{2^{*}-1}\\\
+\left|\widehat{w}_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right|^{2^{*}-3}\widehat{U}_{\mu_{\varepsilon}\left(t\right)}^{2}\Big{\|}_{L^{\frac{2n}{n+2}}\left(B\left(0,r_{0}/\mu_{\varepsilon}\left(t\right)\right)\backslash\Omega_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right)}=\operatorname{o}\left(\mu_{\varepsilon}\left(t\right)^{\frac{n-2}{2}}\right)$
(5.33)

as $\varepsilon\to 0$, uniformly with respect to $t$ in compact subsets of
$\left(0,\infty\right)$. It follows from (5.21) and (5)–(5.33) that

$R_{\varepsilon,\mu_{\varepsilon}\left(t\right)}=\left\\{\begin{aligned}
&\operatorname{o}\left(\varepsilon\right)&&\text{in Case 1}\\\
&\operatorname{O}\left(\nu_{\varepsilon}\right)&&\text{in Case 2.1}\\\
&\operatorname{O}\left(\varepsilon^{3}\right)&&\text{in Case
2.2}\end{aligned}\right.$ (5.34)

as $\varepsilon\to 0$, uniformly with respect to $t$ in compact subsets of
$\left(0,\infty\right)$. By using (5.29) and (5.34) and performing a
Lyapunov–Schmidt reduction in the subspace of $H^{1}\left(M\right)$ which
consists of symmetric functions with respect to $\xi_{0}$ (see the article by
Morabito, Pistoia and Vaira [MPV]; see also the article by Robert and Vétois
[RV2]*Theorem 1.1, where the proof of the main result can easily be adapted to
this case), we obtain that for small $\varepsilon,\mu>0$, there exists a
function $\psi_{\varepsilon,\mu}\in H^{1}\left(M\right)$ such that
$\psi_{\varepsilon,\mu}\to 0$ in $H^{1}\left(M\right)$ as $\varepsilon,\mu\to
0$,

$J_{\varepsilon}\left(w_{\varepsilon}-U_{\mu_{\varepsilon}\left(t\right)}+\psi_{\varepsilon,\mu_{\varepsilon}\left(t\right)}\right)=\frac{1}{n}\int_{M}\left|w_{\varepsilon}\right|^{2^{*}}dv_{g}+\frac{1}{n}\int_{\mathbb{R}^{n}}U_{0}^{2^{*}}dv_{\delta_{0}}\\\
+\left\\{\begin{aligned}
&F\left(t\right)\varepsilon^{2}+\operatorname{o}\left(\varepsilon^{2}\right)&&\text{in
Case 1}\\\
&F\left(t\right)\varepsilon\nu_{\varepsilon}+\operatorname{o}\left(\varepsilon\nu_{\varepsilon}\right)&&\text{in
Case 2.1}\\\
&F\left(t\right)\varepsilon^{4}+\operatorname{o}\left(\varepsilon^{4}\right)&&\text{in
Case 2.2}\end{aligned}\right.$ (5.35)

as $\varepsilon\to 0$, uniformly with respect to $t$ in compact subsets of
$\left(0,\infty\right)$, and

$D_{u}J_{\varepsilon}\left(w_{\varepsilon}-U_{\mu}+\psi_{\varepsilon,\mu}\right)=0\quad\Longleftrightarrow\quad\frac{d}{d\mu}\left[J_{\varepsilon}\left(u_{0}-U_{\mu}+\psi_{\varepsilon,\mu}\right)\right]=0.$
(5.36)

By using (1.8) and (5.2), we obtain that the function $F$ has a strict maximum
or minimum point in $\left(0,\infty\right)$, that we call $t_{0}$. By using
(5.35), we then obtain that for large $C>0$, there exists a family of maximum
or minimum points $\left(t_{\varepsilon}\right)_{0<\varepsilon\ll 1}$ of the
function
$J_{\varepsilon}\left(w_{\varepsilon}-U_{\mu_{\varepsilon}\left(\cdot\right)}+\psi_{\varepsilon,\mu_{\varepsilon}\left(\cdot\right)}\right)$
in $\left(1/C,C\right)$ that converges to $t_{0}$ as $\varepsilon\to 0$. It
then follows from (5.36) that for small $\varepsilon>0$, the function
$u_{\varepsilon}:=w_{\varepsilon}-U_{\mu_{\varepsilon}\left(t_{\varepsilon}\right)}+\psi_{\varepsilon,\mu_{\varepsilon}\left(t_{\varepsilon}\right)}$
is a solution to (1.2). It is easy to verify that $u_{\varepsilon}$ satisfies
(1.4). This ends the proof of Theorem 1.4. ∎

To prove Theorem 2.3, we also need the following:

######  Proposition 5.1.

Let $\left(M,g\right)$ be a smooth, closed Riemannian manifold of dimension
$n\geq 3$. Assume that $\left(M,g\right)$ is $2$-symmetric with respect to a
point $\xi_{0}\in M$. Let $\sigma_{0}$, $\sigma_{1}$, $\sigma_{2}$,
$\Gamma_{\sigma_{0}}$, $\Gamma_{\sigma_{1}}$ and $\Gamma_{\sigma_{2}}$ be as
in the definition of $2$-symmetric manifolds. Then the following assertions
hold true:

  1. (i)

For every point $\xi\in\Gamma_{\sigma_{1}}$ or $\xi\in\Gamma_{\sigma_{2}}$,
the connected component of $\Gamma_{\sigma_{1}}$ or $\Gamma_{\sigma_{2}}$
which contains $\xi$ is a totally geodesic submanifold of dimension $n-1$ of
$M$.

  2. (ii)

For every point $\xi\in\Gamma_{\sigma_{1}}\cap\Gamma_{\sigma_{2}}$, in
geodesic normal coordinates, the maps $d\sigma_{1}\left(\xi\right)$ and
$d\sigma_{2}\left(\xi\right)$ are two reflections about orthogonal hyperplanes
$H_{1}$ and $H_{2}$ of $T_{\xi}M$. If moreover $\xi=\xi_{0}$, then in geodesic
normal coordinates, the map $d\sigma_{0}\left(\xi_{0}\right)$ is the
reflection about the plane which passes through the origin and is spanned by
the normal vectors to the hyperplanes $H_{1}$ and $H_{2}$.

In particular, it follows from the point (ii) that
$\Omega\cap\Gamma_{\sigma_{0}}\neq\emptyset$.

###### Proof of Proposition 5.1.

We begin with proving the point (i). We let $\xi$ be a point in
$\Gamma_{\sigma_{1}}\backslash\Gamma_{\sigma_{2}}$ (the case where
$\xi\in\Gamma_{\sigma_{1}}\cap\Gamma_{\sigma_{2}}$ is treated in the point
(ii) and the case where
$\xi\in\Gamma_{\sigma_{2}}\backslash\Gamma_{\sigma_{1}}$ is similar to the
case where $\xi\in\Gamma_{\sigma_{1}}\backslash\Gamma_{\sigma_{2}}$). We let
$C_{\xi}$ be the connected component of $\Gamma_{\sigma_{1}}$ which contains
the point $\xi$. Since $\sigma_{1}$ is an isometry of $M$, we obtain that
$C_{\xi}$ is a totally geodesic submanifold of $M$. Let $n_{1}$ be the
dimension of $C_{\xi}$. Since $M$ is connected and
$M\backslash\Gamma_{\sigma_{1}}\neq\emptyset$, we obtain that $n_{1}<n$. We
now assume by contradiction that $n_{1}<n-1$. We then obtain that the set
$B_{g}\left(\xi,r\right)\backslash\Gamma_{\sigma_{1}}$ is connected for small
$r>0$, where $B_{g}\left(\xi,r\right)$ is the ball of center $\xi$ and radius
$r$ with respect to the metric $g$. Since
$\xi\in\Gamma_{\sigma_{1}}\backslash\Gamma_{\sigma_{2}}$,
$M=\Omega\sqcup\sigma_{1}\left(\Omega\right)\sqcup\sigma_{2}\left(\Omega\right)\sqcup\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\sqcup\left(\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}\right)$,
the sets $B_{g}\left(\xi,r\right)\backslash\Gamma_{\sigma_{1}}$, $\Omega$,
$\sigma_{1}\left(\Omega\right)$, $\sigma_{2}\left(\Omega\right)$ and
$\sigma_{1}\circ\sigma_{2}\left(\Omega\right)$ are open and the set
$\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}$ is closed, it follows that for
small $r>0$, the set $B_{g}\left(\xi,r\right)\backslash\Gamma_{\sigma_{1}}$ is
included in only one set
$S\in\left\\{\Omega,\sigma_{1}\left(\Omega\right),\sigma_{2}\left(\Omega\right),\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\right\\}$.
On the other hand , for every sequence of points
$\left(x_{k}\right)_{k\in\mathbb{N}}$ in $S$ such that $x_{k}\to\xi$ as
$k\to\infty$, by continuity of $\sigma_{1}$, we obtain that
$\sigma_{1}\left(x_{k}\right)\to\sigma_{1}\left(\xi\right)=\xi$ as
$k\to\infty$, which implies that
$\sigma_{1}\left(S\right)\cap\left(S\cup\Gamma_{\sigma_{1}}\right)\neq\emptyset$.
This clearly contradicts the definitions of $\sigma_{1}$, $\sigma_{2}$ and
$\Omega$, thus proving that $n_{1}=n-1$, which ends the proof of the point
(i).

We now prove the point (ii). We let $\xi$ be a point in
$\Gamma_{\sigma_{1}}\cap\Gamma_{\sigma_{2}}$. By similar considerations as in
the proof of the point (i), we obtain that the connected components of
$\Gamma_{\sigma_{1}}$ and $\Gamma_{\sigma_{2}}$ which contain the point $\xi$
are totally geodesic submanifold of $M$ of dimensions $n_{1}$ and $n_{2}$,
respectively, such that $n_{1}<n$ and $n_{2}<n$. We assume by contradiction
that $n_{1}<n-1$ or $n_{2}<n-1$. We then obtain that for small $r>0$, the set
$B_{g}\left(\xi,r\right)\backslash\left(\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}\right)$
has at most two distinct connected components. Similarly as in the proof of
the point (i), it follows that for small $r>0$,
$B_{g}\left(\xi,r\right)\backslash\left(\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}\right)\subset
S_{1}\cup S_{2}$ for some sets
$S_{1},S_{2}\in\left\\{\Omega,\sigma_{1}\left(\Omega\right),\sigma_{2}\left(\Omega\right),\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\right\\}$.
On the other hand , for every sequence of points
$\left(x_{k}\right)_{k\in\mathbb{N}}$ in $S_{1}\cup S_{2}$ such that
$x_{k}\to\xi$ as $k\to\infty$, by continuity of $\sigma_{1}$ and $\sigma_{2}$,
we obtain that
$\sigma_{1}\left(x_{k}\right)\to\sigma_{1}\left(\xi\right)=\xi$,
$\sigma_{2}\left(x_{k}\right)\to\sigma_{2}\left(\xi\right)=\xi$ and
$\sigma_{1}\circ\sigma_{2}\left(x_{k}\right)\to\sigma_{1}\circ\sigma_{2}\left(\xi\right)=\xi$
as $k\to\infty$. This together with the properties of $\sigma_{1}$ and
$\sigma_{2}$ implies that
$\xi\in\partial\Omega\cap\partial\left(\sigma_{1}\left(\Omega\right)\right)\cap\partial\left(\sigma_{2}\left(\Omega\right)\right)\cap\partial\left(\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\right)$,
which is in contradiction with
$B_{g}\left(\xi,r\right)\backslash\left(\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}\right)\subset
S_{1}\cup S_{2}$. This proves that $n_{1}=n_{2}=n-1$. Since the reflections
about hyperplanes are the only Euclidean isometries which fixed point sets are
hyperplane, it follows that in geodesic normal coordinates, the maps
$d\sigma_{1}\left(\xi\right)$ and $d\sigma_{2}\left(\xi\right)$ are two
reflections about hyperplanes $H_{1}$ and $H_{2}$ of $T_{\xi}M$. Moreover,
since $\sigma_{1}\circ\sigma_{2}=\sigma_{2}\circ\sigma_{1}$ in $M$, we obtain
that $d\sigma_{1}\left(\xi\right)\circ
d\sigma_{2}\left(\xi\right)=d\sigma_{2}\left(\xi\right)\circ
d\sigma_{1}\left(\xi\right)$, which implies that either $H_{1}=H_{2}$ or
$H_{1}$ is orthogonal to $H_{2}$. If $H_{1}=H_{2}$, then we obtain that
$\Gamma_{\sigma_{1}\circ\sigma_{2}}$ contains a neighborhood of the point
$\xi$, which is in contradiction with
$M=\Omega\sqcup\sigma_{1}\left(\Omega\right)\sqcup\sigma_{2}\left(\Omega\right)\sqcup\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\sqcup\left(\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}\right)$.
Therefore, we obtain that $H_{1}$ is orthogonal to $H_{2}$. Finally, in the
case where $\xi=\xi_{0}$, for every vector $v\in T_{\xi_{0}}M$, since
$d\left(\sigma_{0}\circ\sigma_{1}\circ\sigma_{2}\right)_{\xi_{0}}\left(v\right)=-v$,
$\sigma_{0}=\left(\sigma_{0}\circ\sigma_{1}\circ\sigma_{2}\right)\circ\sigma_{2}\circ\sigma_{1}$
and
$\xi_{0}\in\Gamma_{\sigma_{0}}\cap\Gamma_{\sigma_{1}}\cap\Gamma_{\sigma_{2}}$,
we obtain that
$d\left(\sigma_{0}\right)_{\xi_{0}}\left(v\right)=-d\left(\sigma_{2}\right)_{\xi_{0}}\circ
d\left(\sigma_{1}\right)_{\xi_{0}}\left(v\right)$. It follows that in geodesic
normal coordinates, the map $d\sigma_{0}\left(\xi_{0}\right)$ is the
reflection about the plane which passes through the origin and is spanned by
the normal vectors to the hyperplanes $H_{1}$ and $H_{2}$. This ends the proof
of the point (ii). ∎

We can now use Theorem 1.4 and Proposition 5.1 to prove Theorem 2.3.

###### Proof of Theorem 2.3.

######  Step 1.

The function $u_{0}$ can be extended as a solution to (1.1) such that $u_{0}$
is symmetric with respect to $\xi_{0}$, $u_{0}>0$ in $\Omega$ and
$u_{0}\circ\sigma_{1}=u_{0}\circ\sigma_{2}=-u_{0}$ in $M$.

###### Proof of Step 1.

Since
$M=\Omega\sqcup\sigma_{1}\left(\Omega\right)\sqcup\sigma_{2}\left(\Omega\right)\sqcup\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\sqcup\left(\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}\right)$,
we can extend $u_{0}$ to $M$ by letting

$\left\\{\begin{aligned} &u_{0}:=-u_{0}\circ\sigma_{1}\text{ in
}\sigma_{1}\left(\Omega\right),\quad u_{0}:=-u_{0}\circ\sigma_{2}\text{ in
}\sigma_{2}\left(\Omega\right),\\\
&u_{0}:=u_{0}\circ\sigma_{1}\circ\sigma_{2}\text{ in
}\sigma_{1}\circ\sigma_{2}\left(\Omega\right)\quad\text{and}\quad
u_{0}:=0\text{ in
}\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}.\end{aligned}\right.$

By using the composition properties of $\sigma_{0}$, $\sigma_{1}$ and
$\sigma_{2}$ and since $u_{0}\circ\sigma_{0}=u_{0}$ in $\Omega$, we obtain
that $u_{0}$ is symmetric with respect to $\xi_{0}$ for
$\sigma:=\sigma_{0}\circ\sigma_{1}\circ\sigma_{2}$, $u_{0}\in
H^{1}\left(M\right)$ and $u_{0}\circ\sigma_{1}=u_{0}\circ\sigma_{2}=-u_{0}$ in
$M$. To prove that $u_{0}$ is a solution to (1.1), for every function $v\in
C^{\infty}\left(M\right)$, we define

$v_{sym}:=v-v\circ\sigma_{1}-v\circ\sigma_{2}+v\circ\sigma_{1}\circ\sigma_{2}.$

Since $\partial\Omega\subset\Gamma_{\sigma_{1}}\cup\Gamma_{\sigma_{2}}$, we
obtain that $v_{sym}=0$ on $\partial\Omega$. By using (2.1) together with a
straightforward change of variable and the composition properties of
$\sigma_{0}$, $\sigma_{1}$ and $\sigma_{2}$, we then obtain

$\displaystyle\int_{M}\left(\left<\nabla u_{0},\nabla
v\right>_{g}+h_{0}u_{0}v\right)dv_{g}$
$\displaystyle=\int_{\Omega}\left(\left<\nabla u_{0},\nabla
v_{sym}\right>_{g}+h_{0}u_{0}v_{sym}\right)dv_{g}$
$\displaystyle=\int_{\Omega}u_{0}^{2^{*}-1}v_{sym}\,dv_{g}=\int_{M}\left|u_{0}\right|^{2^{*}-2}u_{0}v\,dv_{g}.$
(5.37)

Since (5) holds true for all functions $v\in C^{\infty}\left(M\right)$, we
obtain that $u_{0}$ is a solution to (1.1). Since $h_{0}\in
C^{1}\left(M\right)$, by standard elliptic regularity, we then obtain that
$u_{0}\in C^{2}\left(M\right)$. This ends the proof of Step 1. ∎

######  Step 2.

There exist families of functions $\left(h_{\gamma}\right)_{0<\gamma\ll 1}$ in
$C^{p}\left(M\right)$ and $\left(u_{\gamma}\right)_{0<\gamma\ll 1}$ in
$C^{2}\left(M\right)$ such that $h_{\gamma}\to h_{0}$ in $C^{p}\left(M\right)$
and $u_{\gamma}\to u_{0}$ in $C^{2}\left(M\right)$ as $\gamma\to 0$ and for
each $\gamma$, $h_{\gamma}$ is $2$-symmetric with respect to $\xi_{0}$,
$u_{\gamma}$ is symmetric with respect to $\xi_{0}$,
$u_{\gamma}\circ\sigma_{1}=u_{\gamma}\circ\sigma_{2}=-u_{\gamma}$ in $M$ and
$u_{\gamma}$ is a nondegenerate solution to the equation

$\Delta_{g}u_{\gamma}+h_{\gamma}u_{\gamma}=\left|u_{\gamma}\right|^{2^{*}-2}u_{\gamma}\quad\text{in
}M.$ (5.38)

###### Proof of Step 2.

We use a similar idea as in the article by Robert and Vétois [RV4]*Lemma 13.1,
but the fact that $u_{0}$ vanishes requires to make some nontrivial
modifications. Since $u_{0}\in C^{2}\left(M\right)$, a standard density
argument gives that there exists a sequence
$\left(v_{\beta}\right)_{\beta\in\left(0,1\right)}$ in
$C^{\infty}\left(M\right)$ such that $v_{\beta}\to u_{0}$ in
$C^{2}\left(M\right)$ as $\beta\to 0$. Moreover, since $u_{0}$ is symmetric
with respect to $\xi_{0}$ and
$u_{0}\circ\sigma_{1}=u_{0}\circ\sigma_{2}=-u_{0}$ in $M$, we can assume that
for every $\beta\in\left(0,1\right)$, $v_{\beta}$ is also symmetric with
respect to $\xi_{0}$ and
$v_{\beta}\circ\sigma_{1}=v_{\beta}\circ\sigma_{2}=-v_{\beta}$ in $M$. For