# Compactness and blow up results for doubly perturbed Yamabe problems on
manifolds with non umbilic boundary

Marco G. Ghimenti  M. G. Ghimenti,
Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5, 56126
Pisa, Italy<EMAIL_ADDRESS>and  Anna Maria Micheletti  A. M.
Micheletti,
Dipartimento di Matematica Università di Pisa Largo B. Pontecorvo 5, 56126
Pisa, Italy<EMAIL_ADDRESS>
###### Abstract.

We study the stability of compactness of solutions for the Yamabe boundary
problem on a compact Riemannian manifold with non umbilic boundary. We prove
that the set of solutions of Yamabe boundary problem is a compact set when
perturbing the mean curvature of the boundary from below and the scalar
curvature with a function whose maximum is not too positive. In addition, we
prove the counterpart of the stability result: there exists a blowing up
sequence of solutions when we perturb the mean curvature from above or the
mean curvature from below and the scalar curvature with a function with a
large positive maximum.

###### Key words and phrases:

Non umbilic boundary, Yamabe problem, Compactness, Blow up analysis

###### 2000 Mathematics Subject Classification:

35J65, 53C21

> Dedicated to Norman Dancer on the occasion of his 75th birthday

##  1\. Introduction

Let $(M,g)$, a smooth, compact Riemannian manifold of dimension $n\geq 7$ with
non umbilic boundary. We recall that the boundary of $M$ is respectively
called umbilic if the trace-free second fundamental form of $\partial M$ is
non zero everywhere. Here we study the linearly perturbed problem

(1.1)
$\left\\{\begin{array}[]{cc}-\Delta_{g}u+\frac{n-2}{4(n-1)}R_{g}u+\varepsilon_{1}\alpha
u=0&\text{ in }M\\\ \frac{\partial
u}{\partial\nu}+\frac{n-2}{2}h_{g}u+\varepsilon_{2}\beta
u=(n-2)u^{\frac{n}{n-2}}&\text{ on }\partial M\end{array}\right..$

Where $\Delta_{g}$ is the Laplace-Beltrami operator and $\nu$ denotes the
outer normal. Also, $\varepsilon_{1},\varepsilon_{2}$ are positive parameters
and $\alpha,\beta:M\rightarrow\mathbb{R}$ are smooth functions. We can restate
Problem (1.1) in the more compact form

$\left\\{\begin{array}[]{cc}L_{g}u-\varepsilon_{1}\alpha u=0&\text{ in }M\\\
B_{g}u-\varepsilon_{2}\beta u+(n-2)u^{\frac{n}{n-2}}=0&\text{ on }\partial
M\end{array}\right.,$

where $L_{g}=\Delta_{g}-\frac{n-2}{4(n-1)}R_{g}$ and
$B_{g}=-\frac{\partial}{\partial\nu}-\frac{n-2}{2}h_{g}$.

Problem (1.1) is the perturbed version of the Yamabe boundary problem when the
target metric has zero scalar curvature, that is, given a compact Riemannian
manifold with boundary, finding a Riemannian metric, conformal to the original
one, with zero scalar curvature and constant boundary mean curvature. This
represents an extension of the Yamabe problem on manifold with boundary and,
since the target metric is conformally flat, also a generalization of the
Riemann mapping theorem to higher dimensions. Solving this problem is
equivalent to find a positive solution of the equation

(1.2)
$\left\\{\begin{array}[]{cc}-\Delta_{g}u+\frac{n-2}{4(n-1)}R_{g}u=0&\text{ in
}M\\\ \frac{\partial
u}{\partial\nu}+\frac{n-2}{2}h_{g}u=(n-2)u^{\frac{n}{n-2}}&\text{ on }\partial
M\end{array}\right.$

which is, as noticed before, the unperturbed version of (1.1). In this paper
we study if the perturbation term affect the property of solutions. In
particular we want to investigate if the compactness of the set of the
solution of the problem holds true for the perturbed problem. Our main results
are the following.

######  Theorem 1.

Let $(M,g)$ a smooth, $n$-dimensional Riemannian manifold of positive type
with regular boundary $\partial M$. Suppose that $n\geq 7$ and that $\pi(x)$,
the trace free second fundamental form of $\partial M$, is non zero
everywhere.

Let $\alpha,\beta:M\rightarrow\mathbb{R}$ smooth functions such that $\beta<0$
on $\partial M$ and ${\displaystyle\max_{q\in\partial
M}}\\{\alpha(q)-\frac{n-6}{4(n-1)(n-2)^{2}}\|\pi(q)\|^{2}\\}<0$. Then, there
exist two constants $C>0$ and $0<\bar{\varepsilon}<1$ such that, for any
$0\leq\varepsilon_{1},\varepsilon_{2}\leq\bar{\varepsilon}$ and for any $u>0$
solution of (1.1), it holds

$C^{-1}\leq u\leq C\text{ and }\|u\|_{C^{2,\eta}(M)}\leq C$

for some $0<\eta<1$. The constant $C$ does not depend on
$u,\varepsilon_{1},\varepsilon_{2}$.

######  Theorem 2.

Let $(M,g)$ a smooth, $n$-dimensional Riemannian manifold of positive type
with regular boundary $\partial M$. Suppose that $n\geq 7$ and that the trace
free second fundamental form of $\partial M$, is non zero everywhere. Let
$\alpha,\beta:M\rightarrow\mathbb{R}$ smooth functions.

  * •

If $\beta>0$ on $\partial M$ then for $\varepsilon_{1},\varepsilon_{2}>0$
small enough there exists a sequence of solutions
$u_{\varepsilon_{1},\varepsilon_{2}}$ of (1.1) which blows up at a suitable
point of $\partial M$ as $(\varepsilon_{1},\varepsilon_{2})\rightarrow(0,0)$.

  * •

If $\beta<0$ on $\partial M$, $\varepsilon_{1}=1$, $\alpha>0$ on $M$ and
${\displaystyle\inf_{q\in\partial M}\alpha(q)+\frac{1}{B}\varphi(q)>0}$, then
for $\varepsilon_{2}>0$ small enough there exists a sequence of solutions
$u_{\varepsilon_{2}}$ of (1.1) which blows up at a suitable point of $\partial
M$ as $\varepsilon_{2}\rightarrow 0$.

Here $B$ and $\varphi(q)$ are defined in Lemma 23.

We remark that in the above Theorem 2, $B$ is strictly positive, $\varphi(q)$
is strictly negative, and both are completely determined by $(M,g)$.

The result of Theorem 1 (and its counterpart Theorem 2) is somewhat
unexpected: in classical Yamabe problem [6, 7, 8] the compactness of solution
is guaranteed as soon as $\alpha$ is negative. In a forthcoming paper we prove
that also for boundary Yamabe problem on manifold with umbilic boundary
compactness is granted when $\alpha$ is negative while for $\alpha$ positive
everywhere there exists a blowing up sequence of solutions. So, this is an
example in which the strong analogy between classical Yamabe problem and
boundary Yamabe problem breaks down.

The boundary Yamabe problem was firstly introduced by Escobar in [9].
Existence results for (1.2) were proved by Escobar [9], Marques [20], Almaraz
[3], Brendle and Chen [5], Mayer and Ndiaye [22]. Solutions of (1.2) could be
found by minimization of the quotient

$Q(M,\partial M):=\inf_{u\in H^{1}\smallsetminus
0}\frac{\int\limits_{M}\left(|\nabla
u|^{2}+\frac{n-2}{4(n-1)}R_{g}u^{2}\right)dv_{g}+\int\limits_{\partial
M}\frac{n-2}{2}h_{g}u^{2}d\sigma_{g}}{\left(\int\limits_{\partial
M}|u|^{\frac{2(n-1)}{n-2}}d\sigma_{g}\right)^{\frac{n-2}{n-1}}}.$

In particular, the solution of is unique, up to symmetries, when
$-\infty<Q\leq 0$ while multiplicity results hold when $Q>0$. Manifolds for
which $Q>0$ are called of _positive type,_ and it is natural to ask, in that
case, when the full set of the solutions of (1.2) forms a $C^{2}$-compact set.
This is in complete analogy with classical Yamabe problem. In addition, the
celebrated strategy of Khuri, Marques and Schoen [18] to prove compactness of
solutions of Yamabe problem up to dimension $n=24$ can be succesfully adapted
to Problem (1.2). Indeed, with this method compactness has been proved firstly
in the case of locally flat manifolds not conformally equivalent to euclidean
balls in [11], then for manifold with non umbilic boundary in [1], and,
recently, for manifold with umbilic boundary on which the Weyl tensor does not
vanish, in [13, 15]. These results have been successively extended, but an
exhaustive list of reference of compactness results is beyond the scope of
this introduction. In [2] Almaraz proved that, for $n\geq 25$, it is possible
to construct umbilic boundary manifolds, not conformally equivalent to
euclidean balls, for which Problem (1.2) admits a non compact set of
solutions. It is conjectured that also for boundary Yamabe the critical
dimension is $n=24$, but compactness for dimension $n\leq 24$ is not yet
proved in all generality.

Another parallelism arises studying stability of Yamabe problem with respect
of small perturbations of curvatures. For classical Yamabe problem, Druet,
Hebey and Robert [6, 7, 8] proved that the set of solutions of
$-\Delta_{g}u+\frac{n-2}{4(n-1)}a(x)u=cu^{\frac{n+2}{n-2}}\text{ in }M$ is
still compact if $a(x)\leq R_{g}(x)$ on $M$. Thus they claim that the Yamabe
problem is _stable_ with respect of perturbation of scalar curvature from
below. On the other hand, they found counterexamples to compactness, and so
_instability,_ when $a(x)$ is greater than $R_{g}(x)$. In [14] the same
problem is studied in the case of boundary Yamabe equation by perturbing the
mean curvature term, and the same compactness versus blow up phenomenon
appeared. So, a first analogy between the role of scalar curvature in
classical case and mean curvature in boundary case is established. An analogy
between the role of scalar curvature in classical and boundary Yamabe problem
when the boundary is umbilic will be investigated by the authors in a
forthcoming paper.

As far as we know, Theorem 1 is the first case in which stability is possible
when pertrubing a curvature from above, and, therefore, in which the
parallelism between classical and boundary Yamabe problem is lost. The result
of Theorem 1 is strictly related to non umbilicity of the boundary. In fact,
the trace-free second fundamental form competes with the perturbation of the
scalar curvature. Thus, when the tensor does not vanish, it could compensate a
small positive perturbation. This is clearly observable in Proposition 16,
which is a key tool to prove the compactness result (and for the blow-up
counterpart, in Lemma 23).

The paper is organized as follows. Hereafter we recall basic definitions and
preliminary notions useful to achieve the result. Section 2 is devoted to the
proof of the compactness theorem, while in Section 3 we prove the non
compactness result.

###  1.1. Notations and preliminary definitions

######  Remark 3 (Notations).

We will use the indices $1\leq i,j,k,m,p,r,s\leq n-1$ and $1\leq a,b,c,d\leq
n$. Moreover we use the Einstein convention on repeated indices. We denote by
$g$ the Riemannian metric, by $R_{abcd}$ the full Riemannian curvature tensor,
by $R_{ab}$ the Ricci tensor and by $R_{g}$ and $h_{g}$ respectively the
scalar curvature of $(M,g)$ and the mean curvature of $\partial M$. The bar
over an object (e.g. $\bar{R}_{g}$) will means the restriction to this object
to the metric of $\partial M$

On the half space
$\mathbb{R}_{+}^{n}=\left\\{y=(y_{1},\dots,y_{n-1},y_{n})\in\mathbb{R}^{n},\\!\
y_{n}\geq 0\right\\}$ we set $B_{r}(y_{0})=\left\\{y\in\mathbb{R}^{n},\\!\
|y-y_{0}|\leq r\right\\}$ and
$B_{r}^{+}(y_{0})=B_{r}(y_{0})\cap\left\\{y_{n}>0\right\\}$. When $y_{0}=0$ we
will use simply $B_{r}=B_{r}(y_{0})$ and $B_{r}^{+}=B_{r}^{+}(y_{0})$. On the
half ball $B_{r}^{+}$ we set
$\partial^{\prime}B_{r}^{+}=B_{r}^{+}\cap\partial\mathbb{R}_{+}^{n}=B_{r}^{+}\cap\left\\{y_{n}=0\right\\}$
and $\partial^{+}B_{r}^{+}=\partial B_{r}^{+}\cap\left\\{y_{n}>0\right\\}$. On
$\mathbb{R}_{+}^{n}$ we will use the following decomposition of coordinates:
$(y_{1},\dots,y_{n-1},y_{n})=(\bar{y},y_{n})=(z,t)$ where
$\bar{y},z\in\mathbb{R}^{n-1}$ and $y_{n},t\geq 0$.

Fixed a point $q\in\partial M$, we denote by $\psi_{q}:B_{r}^{+}\rightarrow M$
the Fermi coordinates centered at $q$. We denote by $B_{g}^{+}(q,r)$ the image
of $B_{r}^{+}$. When no ambiguity is possible, we will denote $B_{g}^{+}(q,r)$
simply by $B_{r}^{+}$, omitting the chart $\psi_{q}$.

We recall also that $\omega_{n-2}$ denotes the volume of the $n-1$ dimensional
unit sphere $\mathbb{S}^{n-1}$.

At last we introduce here the standard bubble ${\displaystyle
U(y):=\frac{1}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{\frac{n-2}{2}}}}$
which is the unique solution, up to translations and rescaling, of the
nonlinear critical problem .

(1.3) $\left\\{\begin{array}[]{ccc}-\Delta U=0&&\text{on
}\mathbb{R}_{+}^{n};\\\ \frac{\partial U}{\partial
y_{n}}=-(n-2)U^{\frac{n}{n-2}}&&\text{on
$\partial$}\mathbb{R}_{+}^{n}.\end{array}\right.$

Set

(1.4)
$j_{l}:=\partial_{l}U=-(n-2)\frac{y_{l}}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{\frac{n}{2}}}$
$\partial_{k}\partial_{l}U=(n-2)\left\\{\frac{ny_{l}y_{k}}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{\frac{n+2}{2}}}-\frac{\delta^{kl}}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{\frac{n}{2}}}\right\\}$
(1.5)
$j_{n}:=y^{b}\partial_{b}U+\frac{n-2}{2}U=-\frac{n-2}{2}\frac{|y|^{2}-1}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{\frac{n}{2}}},$

we recall that $j_{1},\dots,j_{n}$ are a base of the space of the $H^{1}$
solutions of the linearized problem

(1.6) $\left\\{\begin{array}[]{ccc}&-\Delta\phi=0&\text{on
}\mathbb{R}_{+}^{n},\\\ &\frac{\partial\phi}{\partial
t}+nU^{\frac{2}{n-2}}\phi=0&\text{on $\partial$}\mathbb{R}_{+}^{n},\\\
&\phi\in H^{1}(\mathbb{R}_{+}^{n}).\end{array}\right.$

Given a point $q\in\partial M$, we introduce now the function $\gamma_{q}$
which arises from the second order term of the expansion of the metric $g$ on
$M$ (see 1.14). The choice of this function plays a twofold role in this
paper. On the one hand, using the function $\gamma_{q}$ we are able to perform
the estimates of Lemmas 12, 13 and Proposition 14. On the other hand, it gives
the correct correction to the standard bubble in order to perform finite
dimensional reduction.

For the proof of the following Lemma we refer to [1, Prop 5.1] and [16,
Proposition 7]

######  Lemma 4.

Assume $n\geq 3$. Given a point $q\in\partial M$, there exists a unique
$\gamma_{q}:\mathbb{R}_{+}^{n}\rightarrow\mathbb{R}$ a solution of the linear
problem

(1.7)
$\left\\{\begin{array}[]{ccc}-\Delta\gamma=2h_{ij}(q)t\partial_{ij}^{2}U&&\text{on
}\mathbb{R}_{+}^{n};\\\ \frac{\partial\gamma}{\partial
t}+nU^{\frac{2}{n-2}}\gamma=0&&\text{on
$\partial$}\mathbb{R}_{+}^{n}.\end{array}\right.$

which is $L^{2}(\mathbb{R}_{+}^{n})$-orthogonal to the functions
$j_{1},\dots,j_{n}$ defined in (1.4) and (1.5).

In addition it holds

(1.8) $|\nabla^{\tau}\gamma_{q}(y)|\leq C(1+|y|)^{3-\tau-n}\text{ for
}\tau=0,1,2.$ (1.9) $\int_{\mathbb{R}_{+}^{n}}\gamma_{q}\Delta\gamma_{q}dy\leq
0,$

(1.10)
$\int_{\partial\mathbb{R}_{+}^{n}}U^{\frac{n}{n-2}}(t,z)\gamma_{q}(t,z)dz=0$
(1.11) $\gamma_{q}(0)=\frac{\partial\gamma_{q}}{\partial
y_{1}}(0)=\dots=\frac{\partial\gamma_{q}}{\partial y_{n-1}}(0)=0.$

Finally the map $q\mapsto\gamma_{q}$ is $C^{2}(\partial M)$.

###  1.2. Expansion of the metric

It is well known that there exists a metric $\tilde{g}$, conformal to $g$,
such that $h_{\tilde{g}}\equiv 0$ (see [9, Lemma 3.3]). So, up to a global
conformal change of coordinates Problem (1.1) becomes

(1.12)
$\left\\{\begin{array}[]{cc}-\Delta_{g}u+\frac{n-2}{4(n-1)}R_{g}u+\varepsilon_{1}\alpha
u=0&\text{ in }M\\\ \frac{\partial u}{\partial\nu}+\varepsilon_{2}\beta
u=(n-2)u^{\frac{n}{n-2}}&\text{ on }\partial M\end{array}\right..$

With this change of coordinates the expansion of the metric is

(1.13) $\displaystyle|g(y)|^{1/2}=$ $\displaystyle
1-\frac{1}{2}\left[\|\pi\|^{2}+\text{Ric}(0)\right]y_{n}^{2}-\frac{1}{6}\bar{R}_{ij}(0)y_{i}y_{j}+O(|y|^{3})$
$\displaystyle g^{ij}(y)=$
$\displaystyle\delta_{ij}+2h_{ij}(0)y_{n}+\frac{1}{3}\bar{R}_{ikjl}(0)y_{k}y_{l}+2\frac{\partial
h_{ij}}{\partial y_{k}}(0)ty_{k}$ (1.14)
$\displaystyle+\left[R_{injn}(0)+3h_{ik}(0)h_{kj}(0)\right]y_{n}^{2}+O(|y|^{3})$
(1.15) $\displaystyle g^{an}(y)=$ $\displaystyle\delta_{an}$

where $\pi$ is the second fundamental form and $h_{ij}(0)$ are its
coefficients, and $\text{Ric}(0)=R_{nini}(0)=R_{nn}(0)$ (see [9]).

##  2\. The compactness result

We start this section by recalling a Pohozaev type identity. This indentity
gives us a fundamental sign condition to rule out the possibility of blowing
up sequence, as shown in subsection 2.5. A recall of preliminary results on
blow up points is collected in subsection 2.2, while a careful analysis of
blow up sequences is performed in subsection 2.4. This allows us to conclude
the section with the proof of Theorem 1. Throughout this section we work in
$\tilde{g}$ metric. For the sake of readability we will omit the tilde symbol
in all this section.

###  2.1. A Pohozaev type identity

A Pohozaev type identity is often used in Yamabe boundary problem. Here we use
the same local version which is introduced in [1, 13].

######  Theorem 5 (Pohozaev Identity).

Let $u$ a $C^{2}$-solution of the following problem

$\left\\{\begin{array}[]{cc}-\Delta_{g}u+\frac{n-2}{4(n-1)}R_{g}u+\varepsilon_{1}\alpha
u=0=0&\text{ in }B_{r}^{+}\\\ \frac{\partial
u}{\partial\nu}+\varepsilon_{2}\beta u=(n-2)u^{\frac{n}{n-2}}&\text{ on
}\partial^{\prime}B_{r}^{+}\end{array}\right.$

for $B_{r}^{+}=\psi_{q}^{-1}(B_{g}^{+}(q,r))$ for $q\in\partial M$. Let us
define

$P(u,r):=\int\limits_{\partial^{+}B_{r}^{+}}\left(\frac{n-2}{2}u\frac{\partial
u}{\partial r}-\frac{r}{2}|\nabla u|^{2}+r\left|\frac{\partial u}{\partial
r}\right|^{2}\right)d\sigma_{r}+\frac{r(n-2)^{2}}{2(n-1)}\int\limits_{\partial(\partial^{\prime}B_{r}^{+})}u^{\frac{2(n-1)}{n-2}}d\bar{\sigma}_{g},$

and

$\hat{P}(u,r):=-\int\limits_{B_{r}^{+}}\left(y^{a}\partial_{a}u+\frac{n-2}{2}u\right)[(L_{g}-\Delta)u]dy+\\\
+\varepsilon_{1}\int\limits_{B_{r}^{+}}\left(y^{a}\partial_{a}u+\frac{n-2}{2}u\right)\alpha
udy\\\
+\frac{n-2}{2}\varepsilon_{2}\int\limits_{\partial^{\prime}B_{r}^{+}}\left(\bar{y}^{k}\partial_{k}u+\frac{n-2}{2}u\right)\beta
ud\bar{y}.$

Then $P(u,r)=\hat{P}(u,r)$.

Here $a=1,\dots,n$, $k=1,\dots,n-1$ and $y=(\bar{y},y_{n})$, where
$\bar{y}\in\mathbb{R}^{n-1}$ and $y_{n}\geq 0$.

###  2.2. Isolated and isolated simple blow up points

We collect here the definitions of some type of blow up points, and the basic
properties about the behavior of these blow up points (see [1, 11, 17, 21]).

Let $\left\\{u_{i}\right\\}_{i}$ be a sequence of positive solution to

(2.1) $\left\\{\begin{array}[]{cc}L_{g_{i}}u-\varepsilon_{1,i}\alpha
u=0&\text{ in }M\\\ B_{g_{i}}u+(n-2)u^{\frac{n}{n-2}}-\varepsilon_{2,i}\beta
u=0&\text{ on }\partial M\end{array}\right..$

where $g_{i}\rightarrow g_{0}$ in the $C_{\text{loc}}^{3}$ topology and
$0<\varepsilon_{1,i},\varepsilon_{2,i}<\bar{\varepsilon}$ for some
$0<\bar{\varepsilon}\leq 1$. As before, we suppose without loss of generality
that $h_{g_{0}}\equiv 0$ and $h_{g_{i}}\equiv 0$ for all $i$.

######  Definition 6.

1) We say that $x_{0}\in\partial M$ is a blow up point for the sequence
$u_{i}$ of solutions of (2.1) if there is a sequence $x_{i}\in\partial M$ of
local maxima of $\left.u_{i}\right|_{\partial M}$ such that $x_{i}\rightarrow
x_{0}$ and $u_{i}(x_{i})\rightarrow+\infty.$

Shortly we say that $x_{i}\rightarrow x_{0}$ is a blow up point for
$\left\\{u_{i}\right\\}_{i}$.

2) We say that $x_{i}\rightarrow x_{0}$ is an isolated blow up point for
$\left\\{u_{i}\right\\}_{i}$ if $x_{i}\rightarrow x_{0}$ is a blow up point
for $\left\\{u_{i}\right\\}_{i}$ and there exist two constants $\rho,C>0$ such
that

$u_{i}(x)\leq Cd_{\bar{g}}(x,x_{i})^{\frac{2-n}{2}}\text{ for all
}x\in\partial M\smallsetminus\left\\{x_{i}\right\\},\
d_{\bar{g}}(x,x_{i})<\rho.$

Given $x_{i}\rightarrow x_{0}$ an isolated blow up point for
$\left\\{u_{i}\right\\}_{i}$, and given $\psi_{i}:B_{\rho}^{+}(0)\rightarrow
M$ the Fermi coordinates centered at $x_{i}$, we define the spherical average
of $u_{i}$ as

$\bar{u}_{i}(r)=\frac{2}{\omega_{n-1}r^{n-1}}\int_{\partial^{+}B_{r}^{+}}u_{i}\circ\psi_{i}d\sigma_{r}$

and

$w_{i}(r):=r^{\frac{2-n}{2}}\bar{u}_{i}(r)$

for $0<r<\rho.$

3) We say that $x_{i}\rightarrow x_{0}$ is an isolated simple blow up point
for $\left\\{u_{i}\right\\}_{i}$ solutions of (2.1) if $x_{i}\rightarrow
x_{0}$ is an isolated blow up point for $\left\\{u_{i}\right\\}_{i}$ and there
exists $\rho$ such that $w_{i}$ has exactly one critical point in the interval
$(0,\rho)$.

Given $x_{i}\rightarrow x_{0}$ a blow up point for
$\left\\{u_{i}\right\\}_{i}$, we set

$M_{i}:=u_{i}(x_{i})\ \text{ and }\ \delta_{i}:=M_{i}^{\frac{2}{2-n}}.$

Obviously $M_{i}\rightarrow+\infty$ and $\delta_{i}\rightarrow 0$.

The proofs of the following propositions can be found in [3] and in [11].

######  Proposition 7.

Let $x_{i}\rightarrow x_{0}$ is an isolated blow up point for
$\left\\{u_{i}\right\\}_{i}$ and $\rho$ as in Definition 6. We set

$v_{i}(y)=M_{i}^{-1}(u_{i}\circ\psi_{i})(M_{i}^{\frac{2}{2-n}}y),\text{ for
}y\in B_{\rho M_{i}^{\frac{n-2}{2}}}^{+}(0).$

Then, given $R_{i}\rightarrow\infty$ and $c_{i}\rightarrow 0$, up to
subsequences, we have

  1. (1)

$|v_{i}-U|_{C^{2}\left(B_{R_{i}}^{+}(0)\right)}<c_{i}$;

  2. (2)

${\displaystyle\lim_{i\rightarrow\infty}\frac{R_{i}}{\log M_{i}}=0}$.

######  Proposition 8.

Let $x_{i}\rightarrow x_{0}$ be an isolated simple blow-up point for
$\left\\{u_{i}\right\\}_{i}$. Let $\eta$ small. If $0<\bar{\varepsilon}\leq 1$
is small enough and $0<\varepsilon_{1},\varepsilon_{2}<\bar{\varepsilon}$,
then there exist $C,\rho>0$ such that

$M_{i}^{\lambda_{i}}|\nabla^{k}u_{i}(\psi_{i}(y))|\leq C|y|^{2-k-n+\eta}$

for $y\in B_{\rho}^{+}(0)\smallsetminus\left\\{0\right\\}$ and $k=0,1,2$. Here
$\lambda_{i}=\left(\frac{2}{n-2}\right)(n-2-\eta)-1$.

###  2.3. A splitting lemma

Here we summarize a result which proves that only a finite number of blow up
points may occur to a blowing up sequence of solution. For its proof we refer
to [19, Proposition 5.1], [24, Lemma 3.1], [17, Proposition 1.1], [1,
Propositions 4.2 and 8.2].

######  Proposition 9.

Given $K>0$ and $R>0$ there exist two constants $C_{0},C_{1}>0$ (depending on
$K$, $R$ and $(M,g)$) such that if $u$ is a solution of

(2.2) $\left\\{\begin{array}[]{cc}L_{g}u-\varepsilon_{1}\alpha=0&\text{ in
}M\\\ B_{g}u-\varepsilon_{2}\beta u+(n-2)u^{\frac{n}{n-2}}=0&\text{ on
}\partial M\end{array}\right.$

and $\max_{\partial M}u>C_{0}$, then there exist $q_{1},\dots,q_{N}\in\partial
M$, with $N=N(u)\geq 1$ with the following properties: for $j=1,\dots,N$

  1. (1)

set $r_{j}:=Ru(q_{j})^{1-p}$, then $\left\\{B_{r_{j}}\cap\partial
M\right\\}_{j}$ are a disjoint collection;

  2. (2)

we have
$\left|u(q_{j})^{-1}u(\psi_{j}(y))-U(u(q_{j})^{p-1}y)\right|_{C^{2}(B_{2r_{j}}^{+})}<K$
(here $\psi_{j}$ are the Fermi coordinates at point $q_{j}$;

  3. (3)

we have

$\displaystyle
u(x)d_{\bar{g}}\left(x,\left\\{q_{1},\dots,q_{n}\right\\}\right)^{\frac{1}{p-1}}\leq
C_{1}$ $\displaystyle\text{ for all }x\in\partial M$ $\displaystyle
u(q_{j})d_{\bar{g}}\left(q_{j},q_{k}\right)^{\frac{1}{p-1}}\geq C_{0}$
$\displaystyle\text{ for any }j\neq k.$

In addition, if $n\geq 7$ and $|\pi(x)|\neq 0$ for any $x\in\partial M$, there
exists $d=d(K,R)$ such that

$\min_{\begin{array}[]{c}i\neq j\\\ 1\leq i,j\leq
N(u)\end{array}}d_{\bar{g}}(q_{i}(u),q_{j}(u))\geq d.$

Here $\bar{g}$ is the geodesic distance on $\partial M$.

###  2.4. Blowup estimates

In this section we provide a fine estimate for the approximation of the
rescaled solution near an isolated simple blow up point.

######  Proposition 10.

Let $x_{i}\rightarrow x_{0}$ be an isolated simple blow-up point for
$\left\\{u_{i}\right\\}_{i}$ and $\beta<0$. Then $\varepsilon_{2,i}\rightarrow
0$.

###### Proof.

We compute the Pohozaev identity in a ball of radius $r$ and we set
$\frac{r}{\delta_{i}}=:R_{i}\rightarrow\infty$.

By Proposition 8 we have that

(2.3) $P(u_{i},r)\leq\delta_{i}^{\lambda_{i}(n-2)}.$

We estimate now $\hat{P}(u_{i},r)$. By comparing this term with $P(u_{i},r)$
we will get the proof.

$\displaystyle\hat{P}(u_{i},r):$
$\displaystyle=-\int\limits_{B_{r}^{+}}\left(y^{a}\partial_{a}u_{i}+\frac{n-2}{2}u_{i}\right)[(L_{g}-\Delta)u_{i}]dy+\varepsilon_{1,i}\int\limits_{B_{r}^{+}}\left(y^{a}\partial_{a}u_{i}+\frac{n-2}{2}u_{i}\right)\alpha
u_{i}dy$
$\displaystyle+\frac{n-2}{2}\varepsilon_{2,i}\int\limits_{\partial^{\prime}B_{r}^{+}}\left(\bar{y}^{k}\partial_{k}u_{i}+\frac{n-2}{2}u_{i}\right)\beta
u_{i}d\bar{y}=:I_{1}(u_{i},r)+I_{2}(u_{i},r)+I_{3}(u_{i},r).$

The terms $I_{3}$ has been estimated in [14, Proposition 8] and it holds

(2.4) $I_{3}(u_{i},r)=\varepsilon_{2,i}\delta_{i}(B+o(1)),$

where $B$ is a positive real constant.

For $I_{2}(u_{i},r)$ we have, by change of variables,

$I_{2}(u_{i},r)=\varepsilon_{1,i}\delta_{i}^{2}\frac{n-2}{2}\alpha(x_{i})\int\limits_{\mathbb{R}_{+}^{n}}\frac{1-|y|^{2}}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{n-1}}dy+\varepsilon_{1,i}\delta_{i}^{2}O(\delta_{i}^{2}).$

Now, set
$I_{m}^{\alpha}:=\int_{0}^{\infty}\frac{s^{\alpha}ds}{\left(1+s^{2}\right)^{m}}$
we have

$\int\limits_{\mathbb{R}_{+}^{n}}\frac{1-|y|^{2}}{\left[(1+y_{n})^{2}+|\bar{y}|^{2}\right]^{n-1}}dy\\\
=\omega_{n-2}\left[I_{n-1}^{n-2}\int_{0}^{\infty}\frac{1-t^{2}}{(1+t)^{n-2}}dt-
I_{n-1}^{n}\int_{0}^{\infty}\frac{1}{(1+t)^{n-2}}dt\right]\\\
=\omega_{n-2}\left[I_{n-1}^{n-2}\frac{n-5}{(n-3)(n-4)}-I_{n-1}^{n}\frac{1}{n-4}\right]$

using the identities
$\int_{0}^{\infty}\frac{t^{k}dt}{(1+t)^{m}}=\frac{k!}{(m-1)(m-2)\cdots(m-1-k)}$
and $\int_{0}^{\infty}\frac{dt}{(1+t)^{m}}=\frac{1}{m-1}$. At this point,
since $I_{m}^{\alpha}=\frac{2m}{2m-\alpha-1}I_{m+1}^{\alpha}$ and
$I_{m}^{\alpha}=\frac{2m-\alpha-3}{\alpha+1}I_{m}^{\alpha+2}$ (see [1, Lemma
9.4]) we have

$\frac{(n-5)I_{n-1}^{n-2}}{(n-3)(n-4)}-\frac{I_{n-1}^{n}}{n-4}=-\frac{4I_{n-1}^{n}}{(n-1)(n-4)}=-\frac{8I_{n}^{n}}{(n-3)(n-4)},$

thus

$\displaystyle\int\limits_{B_{r}^{+}}\left(y^{a}\partial_{a}u_{i}+\frac{n-2}{2}u_{i}\right)\varepsilon_{1,i}\alpha
u_{i}dy$
$\displaystyle=-\frac{4(n-2)I_{n}^{n}\omega_{n-2}}{(n-3)(n-4)}\varepsilon_{1,i}\delta_{i}^{2}\alpha(x_{i})+o(\delta_{i}^{2})$
(2.5) $\displaystyle=\varepsilon_{1,i}\delta_{i}^{2}(A+o(1))$

where $A$ is a real constant.

For the term $I_{1}(u_{i},r)$ we slightly improve the estimate provided by
Almaraz in [1]. By the expansion of the metric (1.13), (1.14) and (1.15) we
have

$I_{1}(u_{i},r)\leq-\delta_{i}\int\limits_{B_{r/\delta_{i}}^{+}}\left(y^{a}\partial_{a}v_{i}+\frac{n-2}{2}v_{i}\right)v_{i}h_{kl}(0)y_{n}\partial_{k}\partial_{l}v_{i}dy+O(\delta_{i}^{2})$

By simmetry reasons we have that

$\lim_{i\rightarrow\infty}\int\limits_{B_{r/\delta_{i}}^{+}}\left(y^{a}\partial_{a}v_{i}+\frac{n-2}{2}v_{i}\right)v_{i}h_{kl}(0)y_{n}\partial_{k}\partial_{l}v_{i}dy\\\
=\int_{\mathbb{R}_{+}^{n}}\left(y^{a}\partial_{a}U+\frac{n-2}{2}U\right)Uh_{kl}(0)y_{n}\partial_{k}\partial_{l}Udy\\\
=h_{g}(0)\int_{\mathbb{R}_{+}^{n}}\left(y^{a}\partial_{a}U+\frac{n-2}{2}U\right)Uy_{n}\partial_{1}\partial_{1}Udy=0$

since we choose a metric for which the mean curvature of the boundary is zero.
So

(2.6) $|I_{1}(u_{i},r)|\leq\delta_{i}o^{+}(1)$

where $o^{+}(1)$ is a nonnegative constant that vanishes when
$i\rightarrow\infty.$

Comparing $\hat{P}(u_{i},r)$ and $P(u_{i},r)$, by (2.3), (2.4), (2.5) and
(2.6) we get

$-c\delta_{i}o^{+}(1)+(A+o(1))\varepsilon_{1,i}\delta_{i}^{2}+(B+o(1))\varepsilon_{2,i}\delta_{i}\leq\delta_{i}^{\lambda_{i}(n-2)},$

so

$-co^{+}(1)+(A+o(1))\varepsilon_{1,i}\delta_{i}+(B+o(1))\varepsilon_{2,i}\leq\delta_{i}^{\lambda_{i}(n-2)-1}.$

Being $\varepsilon_{1,i}<\bar{\varepsilon}<1$, the above inequality holds only
if $\varepsilon_{2,i}\rightarrow 0$. ∎

Since $\varepsilon_{2,i}\rightarrow 0$, $\delta_{i}\rightarrow 0$ and
$\varepsilon_{1,i}<\bar{\varepsilon}<1$, the proof of the next proposition is
analogous to Prop. 4.3 of [1].

######  Proposition 11.

Let $x_{i}\rightarrow x_{0}$ be an isolated simple blow-up point for
$\left\\{u_{i}\right\\}_{i}$. Then there exist $C,\rho>0$ such that

  1. (1)

$M_{i}u_{i}(\psi_{i}(y))\leq C|y|^{2-n}$ for all $y\in
B_{\rho}^{+}(0)\smallsetminus\left\\{0\right\\}$;

  2. (2)

$M_{i}u_{i}(\psi_{i}(y))\geq C^{-1}G_{i}(y)$ for all $y\in
B_{\rho}^{+}(0)\smallsetminus B_{r_{i}}^{+}(0)$ where
$r_{i}:=R_{i}M_{i}^{\frac{2}{2-n}}$ and $G_{i}$ is the Green’s function which
solves

$\left\\{\begin{array}[]{ccc}L_{g_{i}}G_{i}=0&&\text{in
}B_{\rho}^{+}(0)\smallsetminus\left\\{0\right\\}\\\ G_{i}=0&&\text{on
}\partial^{+}B_{\rho}^{+}(0)\\\ B_{g_{i}}G_{i}=0&&\text{on
}\partial^{\prime}B_{\rho}^{+}(0)\smallsetminus\left\\{0\right\\}\end{array}\right.$

and $|y|^{n-2}G_{i}(y)\rightarrow 1$ as $|z|\rightarrow 0$.

By Proposition 7 and Proposition 11 we have that, if $x_{i}\rightarrow x_{0}$
is an isolated simple blow-up point for $\left\\{u_{i}\right\\}_{i}$, then it
holds

$v_{i}\leq CU\text{ in }B_{\rho M_{i}^{\frac{2}{2-n}}}^{+}(0).$

Which follows is the core of the compacntess claim: we provide the estimates
of the blowup profile of an isolated simple blow up point $x_{i}\rightarrow
x_{0}$ for a sequence $\left\\{u_{i}\right\\}_{i}$ of solutions of (2.1). The
strategy to achieve these results is similar to the one contained in [1, Lemma
6.1] and in [14, Section 5], so we will give only the general scheme and
emphasize the main differences, while we refer to the cited papers for
detailed proofs. Set

(2.7) $\delta_{i}:=u_{i}^{\frac{2}{2-n}}(x_{i})=M_{i}^{\frac{2}{2-n}}\ \ \
v_{i}(y):=\delta_{i}^{\frac{n-2}{2}}u_{i}(\delta_{i}y)\text{ for }y\in
B_{\frac{R}{\delta_{i}}}^{+}(0),$

we have that $v_{i}$ satisfies

(2.8)
$\left\\{\begin{array}[]{cc}L_{\hat{g}_{i}}v_{i}-\varepsilon_{1,i}\alpha(\delta_{i}y)v_{i}=0&\text{
in }B_{\frac{R}{\delta_{i}}}^{+}(0)\\\
B_{\hat{g}_{i}}v_{i}+(n-2)v_{i}^{\frac{n}{n-2}}-\varepsilon_{2,i}\beta(\delta_{i}y)v_{i}=0&\text{
on }\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}(0)\end{array}\right.$

where $\hat{g}_{i}:=g_{i}(\delta_{i}y)$.

######  Lemma 12.

Assume $n\geq 7$. Let $\gamma_{x_{i}}$ be defined in (1.7). There exist
$R,C>0$ such that

$|v_{i}(y)-U(y)-\delta_{i}\gamma_{x_{i}}(y)|\leq
C\left(\delta_{i}^{2}+\varepsilon_{1,i}\delta_{i}^{2}+\varepsilon_{2,i}\delta_{i}\right)$

for $|y|\leq R/\delta_{i}$.

###### Proof.

Let $y_{i}$ such that

$\mu_{i}:=\max_{|y|\leq
R/\delta_{i}}|v_{i}(y)-U(y)-\delta_{i}\gamma_{x_{i}}(y)|=|v_{i}(y_{i})-U(y_{i})-\delta_{i}\gamma_{x_{i}}(y_{i})|.$

We can assume, without loss of generality, that
$|y_{i}|\leq\frac{R}{2\delta_{i}}.$ This will be useful in the next.

By contradiction, suppose that

(2.9)
$\max\left\\{\mu_{i}^{-1}\delta_{i}^{2},\mu_{i}^{-1}\varepsilon_{1,i}\delta_{i}^{2},\mu_{i}^{-1}\varepsilon_{2,i}\delta_{i}\right\\}\rightarrow
0\text{ when }i\rightarrow\infty.$

Defined

$w_{i}(y):=\mu_{i}^{-1}\left(v_{i}(y)-U(y)-\delta_{i}\gamma_{x_{i}}(y)\right)\text{
for }|y|\leq R/\delta_{i},$

we have, by direct computation, that

(2.10) $\left\\{\begin{array}[]{cc}L_{\hat{g}_{i}}w_{i}=A_{i}&\text{ in
}B_{\frac{R}{\delta_{i}}}^{+}(0)\\\
B_{\hat{g}_{i}}w_{i}+b_{i}w_{i}=F_{i}&\text{ on
}\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}(0)\end{array}\right.$

where

$\displaystyle b_{i}=$
$\displaystyle(n-2)\frac{v_{i}^{\frac{n}{n-2}}-(U+\delta_{i}\gamma_{x_{i}})^{\frac{n}{n-2}}}{v_{i}-U-\delta_{i}\gamma_{x_{i}}},$
$\displaystyle Q_{i}=$
$\displaystyle-\frac{1}{\mu_{i}}\left\\{\left(L_{\hat{g}_{i}}-\Delta\right)(U+\delta_{i}\gamma_{x_{i}})+\delta_{i}\Delta\gamma_{x_{i}}\right\\},$
$\displaystyle A_{i}=$ $\displaystyle
Q_{i}+\frac{\varepsilon_{1,i}\delta_{i}^{2}}{\mu_{i}}\alpha_{i}(\delta_{i}y)v_{i}(y),$
$\displaystyle\bar{Q}_{i}=$
$\displaystyle-\frac{1}{\mu_{i}}\left\\{(n-2)(U+\delta_{i}\gamma_{x_{i}})^{\frac{n}{n-2}}-(n-2)U^{\frac{n}{n-2}}-n\delta_{i}U^{\frac{2}{n-2}}\gamma_{x_{i}}\right\\},$
$\displaystyle F_{i}=$
$\displaystyle\bar{Q}_{i}+\frac{\varepsilon_{2,i}\delta_{i}}{\mu_{i}}\beta_{i}(\delta_{i}y)v_{i}(y).$

Since $v_{i}\rightarrow U$ in $C_{\text{loc}}^{2}(\mathbb{R}_{+}^{n})$ we
have, at once,

(2.11) $\displaystyle b_{i}$ $\displaystyle\rightarrow
nU^{\frac{2}{n-2}}\text{ in }C_{\text{loc}}^{2}(\mathbb{R}_{+}^{n})$ (2.12)
$\displaystyle|b_{i}(y)|$ $\displaystyle\leq(1+|y|)^{-2}\text{ for }|y|\leq
R/\delta_{i}.$

We proceed now by estimating $Q_{i}$ and $\bar{Q}_{i}$. As in [1, Lemma 6.1],
using the expansion of the metric and the decays properties of $U$ and
$\gamma_{x_{i}}$ we obtain

(2.13) $Q_{i}=O(\mu_{i}^{-1}\delta_{i}^{2}\left(1+|y|\right)^{2-n})\text{ and
}\bar{Q}_{i}=O(\mu_{i}^{-1}\delta_{i}^{2}\left(1+|y|\right)^{3-n})$

from which we get

(2.14) $\displaystyle A_{i}$
$\displaystyle=O(\mu_{i}^{-1}\delta_{i}^{2}\left(1+|y|\right)^{2-n})+O(\mu_{i}^{-1}\varepsilon_{1,i}\delta_{i}^{2}\left(1+|y|\right)^{2-n})$
$\displaystyle F_{i}$
$\displaystyle=O(\mu_{i}^{-1}\delta_{i}^{2}\left(1+|y|\right)^{3-n})+O(\mu_{i}^{-1}\varepsilon_{2,i}\delta_{i}\left(1+|y|\right)^{2-n}).$

In light of (2.9) we also have $A_{i}\in L^{p}(B_{R/\delta_{i}}^{+})$ and
$F_{i}\in L^{p}(\partial^{\prime}B_{R/\delta_{i}}^{+})$ for all $p\geq 2$.
Since $|w_{i}(y)|\leq 1$, by (2.9) (2.11), (2.12), (2.14) and by standard
elliptic estimates we conclude that $\left\\{w_{i}\right\\}_{i}$, up to
subesequences, converges in $C_{\text{loc}}^{2}(\mathbb{R}_{+}^{n})$ to some
$w$ solution of

(2.15) $\left\\{\begin{array}[]{cc}\Delta w=0&\text{ in }\mathbb{R}_{+}^{n}\\\
\frac{\partial}{\partial\nu}w+nU^{\frac{n}{n-2}}w=0&\text{ on
}\partial\mathbb{R}_{+}^{n}\end{array}\right..$

Now we prove that $|w(y)|\leq C(1+|y|^{-1})$ for $y\in\mathbb{R}_{+}^{n}$.
Consider $G_{i}$ the Green function for the conformal Laplacian
$L_{\hat{g}_{i}}$ defined on $B_{r/\delta_{i}}^{+}$ with boundary conditions
$B_{\hat{g}_{i}}G_{i}=0$ on $\partial^{\prime}B_{r/\delta_{i}}^{+}$ and
$G_{i}=0$ on $\partial^{+}B_{r/\delta_{i}}^{+}$. It is well known that
$G_{i}=O(|\xi-y|^{2-n})$. By the Green formula and by (2.14) we have

$\displaystyle w_{i}(y)=$
$\displaystyle-\int_{B_{\frac{R}{\delta_{i}}}^{+}}G_{i}(\xi,y)A_{i}(\xi)d\mu_{\hat{g}_{i}}(\xi)-\int_{\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}}\frac{\partial
G_{i}}{\partial\nu}(\xi,y)w_{i}(\xi)d\sigma_{\hat{g}_{i}}(\xi)$
$\displaystyle+\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}G_{i}(\xi,y)\left(b_{i}(\xi)w_{i}(\xi)-F_{i}(\xi)\right)d\sigma_{\hat{g}_{i}}(\xi),$

so

$\displaystyle|w_{i}(y)|$
$\displaystyle\leq\frac{\delta_{i}^{2}}{\mu_{i}}\int_{B_{\frac{R}{\delta_{i}}}^{+}}|\xi-y|^{2-n}(1+|\xi|)^{2-n}d\xi+\frac{\varepsilon_{1,i}\delta_{i}^{2}}{\mu_{i}}\int_{B_{\frac{R}{\delta_{i}}}^{+}}|\xi-y|^{2-n}(1+|\xi|)^{2-n}d\xi$
$\displaystyle+\int_{\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}}|\xi-y|^{1-n}w_{i}(\xi)d\sigma(\xi)$
$\displaystyle+\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}|\bar{\xi}-y|^{2-n}\left((1+|\bar{\xi}|)^{-2}+\frac{\delta_{i}^{2}}{\mu_{i}}(1+|\bar{\xi}|)^{3-n}+\frac{\varepsilon_{2,i}\delta_{i}}{\mu_{i}}(1+|\bar{\xi}|)^{2-n}\right)d\bar{\xi},$

Notice that in the third integral we used that $|y|\leq\frac{R}{2\delta_{i}}$
to estimate $|\xi-y|\geq|\xi|-|y|\geq\frac{R}{2\delta_{i}}$ on
$\partial^{+}B_{R/\delta_{i}}^{+}$. Moreover, since $v_{i}(\xi)\leq CU(\xi)$,
we get $|w_{i}(\xi)|\leq C\frac{\delta_{i}^{n-2}}{\mu_{i}}$ on
$\partial^{+}B_{R/\delta_{i}}^{+}$. Hence

(2.16)
$\int_{\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}}|\xi-y|^{1-n}w_{i}(\xi)d\sigma(\xi)\leq
C\int_{\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}}\frac{\delta_{i}^{2n-3}}{\mu_{i}}d\sigma_{\hat{g}_{i}}(\xi)\leq
C\frac{\delta_{i}^{n-2}}{\mu_{i}}.$

For the other terms we use the formula

(2.17) $\int_{\mathbb{R}^{m}}|\xi-y|^{l-m}(1+|\xi|)^{-\eta}d\xi\leq
C(1+|y|)^{l-\eta}$

where $y\in\mathbb{R}^{m+k}\supseteq\mathbb{R}^{m}$, $\eta,l\in\mathbb{N}$,
$0<l<\eta<m$ (see [1, Lemma 9.2] and [4, 12]), obtaining at last

$|w_{i}(y)|\leq
C\left((1+|y|)^{-1}+\frac{\delta_{i}^{2}}{\mu_{i}}(1+|y|)^{4-n}+\frac{\varepsilon_{1,i}\delta_{i}^{2}}{\mu_{i}}(1+|y|)^{4-n}+\frac{\varepsilon_{2,i}\delta_{i}}{\mu_{i}}(1+|y|)^{3-n}\right)$

for $|y|\leq\frac{R}{2\delta_{i}}$. By assumption (2.9) we prove

(2.18) $|w(y)|\leq C(1+|y|)^{-1}\text{ for }y\in\mathbb{R}_{+}^{n}$

as claimed.

Now we can derive a contradiction. Notice that, since $v_{i}\rightarrow U$
near $0$, and by (1.11) we have $w_{i}(0)\rightarrow 0$ as well as
$\frac{\partial w_{i}}{\partial y_{j}}(0)\rightarrow 0$ for $j=1,\dots,n-1$.
This implies that

(2.19) $w(0)=\frac{\partial w}{\partial y_{1}}(0)=\dots=\frac{\partial
w}{\partial y_{n-1}}(0)=0.$

It is known (see [1, Lemma 2]) that any solution of (2.15) that decays as
(2.18) is a linear combination of $\frac{\partial U}{\partial
y_{1}},\dots,\frac{\partial U}{\partial
y_{n-1}},\frac{n-2}{2}U+y^{b}\frac{\partial U}{\partial y_{b}}$. This,
combined with (2.19), implies that $w\equiv 0$.

Now, on one hand $|y_{i}|\leq\frac{R}{2\delta_{i}}$, so estimate (2.18) holds;
on the other hand, since $w_{i}(y_{i})=1$ and $w\equiv 0$, we get
$|y_{i}|\rightarrow\infty$, obtaining

$1=w_{i}(y_{i})\leq C(1+|y_{i}|)^{-1}\rightarrow 0$

which gives us the contradiction. ∎

######  Lemma 13.

Assume $n\geq 7$ and $\beta<0$. There exists $R,C>0$ such that

$\varepsilon_{2,i}\leq C\delta_{i}$

for $|y|\leq R/\delta_{i}$.

###### Proof.

We proceed by contradiction, supposing that

(2.20)
$\varepsilon_{2,i}^{-1}\delta_{i}=\left(\varepsilon_{2,i}\delta_{i}\right)^{-1}\delta_{i}^{2}\rightarrow
0\text{ when }i\rightarrow\infty.$

Thus, by Lemma 12, we have

$|v_{i}(y)-U(y)-\delta_{i}\gamma_{x_{i}}(y)|\leq
C\varepsilon_{2,i}\delta_{i}\text{ for }|y|\leq R/\delta_{i}.$

We define, similarly to Lemma 12,

$w_{i}(y):=\frac{1}{\varepsilon_{2,i}\delta_{i}}\left(v_{i}(y)-U(y)-\delta_{i}\gamma_{x_{i}}(y)\right)\text{
for }|y|\leq R/\delta_{i},$

and we have that $w_{i}$ satisfies (2.10), where $b_{i}$ is as in Lemma 12,
and

$\displaystyle Q_{i}=$
$\displaystyle-\frac{1}{\varepsilon_{2,i}\delta_{i}}\left\\{\left(L_{\hat{g}_{i}}-\Delta\right)(U+\delta_{i}\gamma_{x_{i}})+\delta_{i}\Delta\gamma_{x_{i}}\right\\},$
$\displaystyle A_{i}=$ $\displaystyle
Q_{i}+\frac{\varepsilon_{1,i}\delta_{i}^{2}}{\varepsilon_{2,i}\delta_{i}}\alpha_{i}(\delta_{i}y)v_{i}(y),$
$\displaystyle\bar{Q}_{i}=$
$\displaystyle-\frac{1}{\varepsilon_{2,i}\delta_{i}}\left\\{(n-2)(U+\delta_{i}\gamma_{x_{i}})^{\frac{n}{n-2}}-(n-2)U^{\frac{n}{n-2}}-n\delta_{i}U^{\frac{2}{n-2}}\gamma_{x_{i}}\right\\},$
$\displaystyle F_{i}=$
$\displaystyle\bar{Q}_{i}+\beta_{i}(\delta_{i}y)v_{i}(y).$

As before, $b_{i}$ satisfies inequality (2.12) while

(2.21) $\displaystyle A_{i}$
$\displaystyle=O\left(\frac{\delta_{i}^{2}}{\varepsilon_{2,i}\delta_{i}}\left(1+|y|\right)^{2-n}\right)$
$\displaystyle F_{i}$
$\displaystyle=O\left(\frac{\delta_{i}^{2}}{\varepsilon_{2,i}\delta_{i}}\left(1+|y|\right)^{3-n}\right)+O\left(\left(1+|y|\right)^{2-n}\right),$

so by classic elliptic estimates we can prove that the sequence $w_{i}$
converges in $C_{\text{loc}}^{2}(\mathbb{R}_{+}^{n})$ to some $w$.

We proceed as in Lemma 12 to deduce that, by (2.20)

$\displaystyle|w_{i}(y)|$ $\displaystyle\leq
C\left((1+|y|)^{-1}+\frac{\delta_{i}^{2}(1+|y|)^{4-n}}{\varepsilon_{2,i}\delta_{i}}+\left(1+|y|\right)^{3-n}\right)$
(2.22) $\displaystyle\leq C\left((1+|y|)^{-1}\right)\text{ for
}|y|\leq\frac{R}{2\delta_{i}}.$

Now let $j_{n}$ be defined as in (1.5). Similarly to [14, Lemma 12], since
$w_{i}$ satisfies (2.10), integrating by parts we obtain

(2.23)
$\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}j_{n}F_{i}d\sigma_{\hat{g}_{i}}=\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}j_{n}\left[B_{\hat{g}_{i}}w_{i}+b_{i}w_{i}\right]d\sigma_{\hat{g}_{i}}\\\
=\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}w_{i}\left[B_{\hat{g}_{i}}j_{n}+b_{i}j_{n}\right]d\sigma_{\hat{g}_{i}}+\int_{\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}}\left[\frac{\partial
j_{n}}{\partial\eta_{i}}w_{i}-\frac{\partial
w_{i}}{\partial\eta_{i}}j_{n}\right]d\sigma_{\hat{g}_{i}}\\\
+\int_{B_{\frac{R}{\delta_{i}}}^{+}}\left[w_{i}L_{\hat{g}_{i}}j_{n}-j_{n}L_{\hat{g}_{i}}w_{i}\right]d\mu_{\hat{g}_{i}}$

where $\eta_{i}$ is the inward unit normal vector to
$\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}$. One can check easily that

$\lim_{i\rightarrow+\infty}\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}j_{n}\bar{Q}_{i}d\sigma_{\hat{g}_{i}}=0.$

Also, since $\beta<0$, by Proposition 7, we have

$\beta(\delta_{i}y)v_{i}(y)\rightarrow\beta(x_{0})U(y)\text{ for
}i\rightarrow+\infty.$

and thus

(2.24)
$\lim_{i\rightarrow+\infty}\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}\beta(\delta_{i}y)v_{i}(y)j_{n}(y)=\frac{n-2}{2}\beta(x_{0})\int_{\mathbb{R}^{n-1}}\frac{1-|\bar{y}|^{2}}{\left(1+|\bar{y}|^{2}\right)^{n-1}}=:B>0$

so

(2.25)
$\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}j_{n}F_{i}d\sigma_{\hat{g}_{i}}=B+o(1).$

By (2.23) and (2.25) we derive a contradiction. Indeed, by the decay of
$j_{n}$ and by the decay of $w_{i}$, given by (2.22) and by (2.20), we have

(2.26)
$\lim_{i\rightarrow+\infty}\int_{\partial^{+}B_{\frac{R}{\delta_{i}}}^{+}}\left[\frac{\partial
j_{n}}{\partial\eta_{i}}w_{i}-\frac{\partial
w_{i}}{\partial\eta_{i}}j_{n}\right]d\sigma_{\hat{g}_{i}}=0$

Since $\Delta j_{n}=0$, one can check that

(2.27)
$\lim_{i\rightarrow+\infty}\int_{B_{\frac{R}{\delta_{i}}}^{+}}w_{i}L_{\hat{g}_{i}}j_{n}d\mu_{\hat{g}_{i}}=0.$

Also, we can prove that

(2.28)
$\lim_{i\rightarrow+\infty}\int_{B_{\frac{R}{\delta_{i}}}^{+}}j_{n}Q_{i}d\mu_{\hat{g}_{i}}=0.$

Finally

(2.29)
$\displaystyle\lim_{i\rightarrow+\infty}\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}w_{i}\left[B_{\hat{g}_{i}}j_{n}+b_{i}j_{n}\right]d\sigma_{\hat{g}_{i}}$
$\displaystyle=\int_{\partial\mathbb{R}_{+}^{n}}w\left[\frac{\partial
j_{n}}{\partial y_{n}}+nU^{\frac{2}{n-2}}j_{n}\right]d\sigma_{\hat{g}_{i}}=0$

since $\frac{\partial j_{n}}{\partial y_{n}}+nU^{\frac{2}{n-2}}j_{n}=0$ when
$y_{n}=0$.

In light of (2.26) (2.28) and (2.27) we infer, by (2.23), that

(2.30)
$\int_{\partial^{\prime}B_{\frac{R}{\delta_{i}}}^{+}}j_{n}F_{i}d\sigma_{\hat{g}_{i}}=-\int_{B_{\frac{R}{\delta_{i}}}^{+}}\left[j_{n}A_{i}w_{i}\right]d\mu_{\hat{g}_{i}}+o(1).$

Again we have $\alpha(\delta_{i}y)v_{i}(y)\rightarrow\alpha(x_{0})U(y)\text{
for }i\rightarrow+\infty$, so,

(2.31)
$\lim_{i\rightarrow\infty}\int_{B_{\frac{R}{\delta_{i}}}^{+}}j_{n}(y)\alpha(\delta_{i}y)v_{i}(y)d\mu_{\hat{g}_{i}}=\alpha(x_{0})\lim_{i\rightarrow\infty}\int\limits_{\mathbb{R}_{+}^{n}}\left(s^{a}\partial_{a}v_{i}+\frac{n-2}{2}v_{i}\right)v_{i}ds=:A\in\mathbb{R}.$

Thus

(2.32)
$\int_{B_{\frac{R}{\delta_{i}}}^{+}}\left[j_{n}A_{i}w_{i}\right]=-\frac{\delta_{i}^{2}}{\varepsilon_{2,i}\delta_{i}}(A+o(1))=o(1)$

by (2.20). At this point, by (2.25), (2.30) and (2.32), we get

(2.33) $B+o(1)=o(1).$

which gives us a contradiction since $B>0$. ∎

The following proposition is the main result of this section. The proof can be
obtained with a first estimate in the spirit of the previous Lemmas, which is
iterated until we get the final result. In fact, once one have the result of
Lemma 13, the proof of the Proposition is very similar to the one of [1,
Proposition 6.1]. For the sake of brevity we prefer to omit it.

######  Proposition 14.

Assume $n\geq 7$ and $\beta<0$. Let $\gamma_{x_{i}}$ be defined in (1.7).
There exist $R,C>0$ such that

$\displaystyle\left|\nabla_{\bar{y}}^{\tau}\left(v_{i}(y)-U(y)-\delta_{i}\gamma_{x_{i}}(y)\right)\right|$
$\displaystyle\leq C\delta_{i}^{2}(1+|y|)^{4-\tau-n}$
$\displaystyle\left|y_{n}\frac{\partial}{\partial_{n}}\left(v_{i}(y)-U(y)-\delta_{i}\gamma_{x_{i}}(y)\right)\right|$
$\displaystyle\leq C\delta_{i}^{2}(1+|y|)^{4-n}$

for $|y|\leq\frac{R}{2\delta_{i}}$. Here $\tau=0,1,2$ and
$\nabla_{\bar{y}}^{\tau}$ is the differential operator of order $\tau$ with
respect the first $n-1$ variables.

###  2.5. Sign estimates of Pohozaev identity terms

We estimate now $P(u_{i},r)$, where $\left\\{u_{i}\right\\}_{i}$ is a family
of solutions of (2.1) which has an isolated simple blow up point
$x_{i}\rightarrow x_{0}$. This estimate, contained in Proposition 16, is a
crucial point when proving the vanishing of the second fundamental form at an
isolated simple blow up point.

The leading term of $P(u_{i},r)$ will be
$-\int_{B_{r/\delta_{i}}^{+}}\left(y^{b}\partial_{b}u+\frac{n-2}{2}u\right)\left[(L_{\hat{g}_{i}}-\Delta)v\right]dy$,
so we set

(2.34)
$R(u,v)=-\int_{B_{r/\delta_{i}}^{+}}\left(y^{b}\partial_{b}u+\frac{n-2}{2}u\right)\left[(L_{\hat{g}_{i}}-\Delta)v\right]dy,$

and we recall the following result by Almaraz [1, Propositions 5.2 and 7.1].

######  Lemma 15.

For $n\geq 7$ we have

$\displaystyle R(U+\delta^{2}\gamma_{q},U+\delta^{2}\gamma_{q})=$
$\displaystyle\delta^{2}\frac{(n-6)\omega_{n-2}I_{n}^{n}}{(n-1)(n-2)(n-3)(n-4)}\left[\|\pi\|^{2}\right]$
$\displaystyle-\frac{1}{2}\delta^{2}\int_{\mathbb{R}_{+}^{n}}\gamma_{q}\Delta\gamma_{q}dy+o(\delta^{2})$

where
$I_{n}^{n}:=:=\int_{0}^{\infty}\frac{s^{n}ds}{\left(1+s^{2}\right)^{n}}$.

######  Proposition 16.

Let $x_{i}\rightarrow x_{0}$ be an isolated simple blow-up point for $u_{i}$
solutions of (2.1). Let $\beta<0$ and $n\geq 7$. Fixed $r$, we have, for $i$
large

$\displaystyle\hat{P}(u_{i},r)\geq$
$\displaystyle\delta_{i}^{2}\frac{(n-6)\omega_{n-2}I_{n}^{n}}{(n-1)(n-2)(n-3)(n-4)}\left[\|\pi\|^{2}\right]$
$\displaystyle-\varepsilon_{1,i}\delta_{i}^{2}\frac{4(n-2)I_{n}^{n}\omega_{n-2}}{(n-3)(n-4)}\alpha(x_{i})+o(\delta_{i}^{2}).$

###### Proof.

We remind that the definition of $\hat{P}$ is given in Theorem 5 and we take
$v_{i}(y)$ as in (2.7). By Proposition 14 and by (1.8) of Lemma 4, for
$|y|<R/\delta_{i}$ we have

$\left|v_{i}(y)-U(y)\right|=O(\delta_{i}^{2}(1+|y|^{4-n})+O(\delta_{i}(1+|y|^{3-n})=O(\delta_{i}(1+|y|^{3-n})$
$\left|y_{a}\partial_{a}v_{i}(y)-y_{a}\partial_{a}U(y)\right|=O(\delta_{i}^{2}(1+|y|^{4-n})+O(\delta_{i}(1+|y|^{3-n})=O(\delta_{i}(1+|y|^{3-n}),$

so, recalling (2.5) we have

$\int\limits_{B_{r}^{+}}\left(y^{a}\partial_{a}u_{i}+\frac{n-2}{2}u_{i}\right)\varepsilon_{1,i}\alpha_{i}u_{i}dy=-\frac{4(n-2)I_{n}^{n}\omega_{n-2}}{(n-3)(n-4)}\varepsilon_{1,i}\delta_{i}^{2}\alpha(x_{i})+o(\delta_{i}^{2}).$

Analogously we obtain

$\int\limits_{\partial^{\prime}B_{r}^{+}}\left(\bar{y}^{k}\partial_{k}u_{i}+\frac{n-2}{2}u_{i}\right)\varepsilon_{2,i}\beta_{i}u_{i}d\bar{y}\\\
=\varepsilon_{2,i}\delta_{i}\frac{n-2}{2}\beta(x_{i})\int\limits_{\mathbb{R}^{n-1}}\frac{1-|\bar{y}|^{2}}{\left[1+|\bar{y}|^{2}\right]^{n-1}}d\bar{y}+\varepsilon_{2,i}\delta_{i}O(\delta_{i}^{2})>0.$

So, for $i$ sufficiently large we obtain

$\displaystyle\hat{P}(u_{i},r)\geq$
$\displaystyle-\int_{B_{r/\delta_{i}}^{+}}\left(y^{b}\partial_{b}v_{i}+\frac{n-2}{2}v_{i}\right)\left[(L_{\hat{g}_{i}}-\Delta)v_{i}\right]dy$
$\displaystyle-\frac{4(n-2)I_{n}^{n}\omega_{n-2}}{(n-3)(n-4)}\varepsilon_{1,i}\delta_{i}^{2}\alpha(x_{i})+o(\delta_{i}^{2}).$

Then, by the estimates on $v_{i}$ obtained in the previous section, using
Lemma 15, and recalling that, by inequality (1.9),
$\int_{\mathbb{R}_{+}^{n}}\gamma_{q}\Delta\gamma_{q}dy\leq 0$, we get

$\displaystyle\hat{P}(u_{i},r)\geq$ $\displaystyle
R(U+\delta^{2}\gamma_{q},U+\delta^{2}\gamma_{q})$
$\displaystyle-\frac{4(n-2)I_{n}^{n}\omega_{n-2}}{(n-3)(n-4)}\varepsilon_{1,i}\delta_{i}^{2}\alpha(x_{i})+o(\delta_{i}^{2})$
$\displaystyle\geq$
$\displaystyle\delta_{i}^{2}\frac{(n-6)\omega_{n-2}I_{n}^{n}}{(n-1)(n-2)(n-3)(n-4)}\left[|h_{kl}(x_{i})|^{2}\right]$
$\displaystyle-\frac{4(n-2)I_{n}^{n}\omega_{n-2}}{(n-3)(n-4)}\varepsilon_{1,i}\delta_{i}^{2}\alpha(x_{i})+o(\delta_{i}^{2})$

which gives the proof. ∎

######  Proposition 17.

Assume $n\geq 7$,
$0\leq\varepsilon_{1,i},\varepsilon_{2,1}\leq\bar{\varepsilon}<1$, $\beta<0$
and

${\displaystyle\max_{q\in\partial
M}\left\\{\alpha(q)-\frac{n-6}{4(n-1)(n-2)^{2}}\|\pi(q)\|^{2}\right\\}<0}.$

Let $x_{i}\rightarrow x_{0}$ be an isolated simple blow-up point for $u_{i}$
solutions of (2.1). Then

  1. (1)

For $i$ large,
$\hat{P}(u_{i},r)\geq\delta_{i}^{2}C_{1}\left[\|\pi(x_{i})\|^{2}\right]+o(\delta_{i}^{2})$
for some $C_{1}>0$;

  2. (2)

$\|\pi(x_{0})\|=0.$

###### Proof.

By Proposition 10 and Proposition 8 we have

$P(u_{i},r)\leq C\delta_{i}^{n-2}.$

On the other hand recalling Proposition 16, Theorem 5, the assumption on
$\alpha$, and that $\varepsilon_{1,i}<1$, we have

$P(u_{i},r)=\hat{P}(u_{i},r)\geq\delta_{i}^{2}C_{1}\left[\|\pi(x_{i})\|^{2}\right]+o(\delta_{i}^{2}),$

with $C_{1}>0$. In addition, we get $\|\pi(x_{i})\|^{2}\leq\delta_{i}^{n-4}$,
which gives the proof. ∎

Once we have the result of Proposition 16, with strategy similar to 17, we can
prove the following Proposition. For a detailed proof we refer to [1,
Proposition 8.1].

######  Proposition 18.

Let $x_{i}\rightarrow x_{0}$ be an isolated blow up point for $u_{i}$
solutions of (2.1). Assume $n\geq 7$,
$0\leq\varepsilon_{1,i},\varepsilon_{2,1}\leq\bar{\varepsilon}<1$, $\beta<0$,
${\displaystyle\max_{q\in\partial
M}\left\\{\alpha(q)-\frac{n-6}{4(n-1)(n-2)^{2}}\|\pi(q)\|^{2}\right\\}<0}$ and
$\|\pi(x_{0})\|\neq 0$. Then $x_{0}$ is isolated simple.

###  2.6. Proof of Theorem 1

Using what we have obtained throughout this section, we can now prove the
compactness result.

###### Proof of Theorem 1.

. By contradiction, suppose that $x_{i}\rightarrow x_{0}$ is a blowup point
for $u_{i}$ solutions of (1.1). Let $q_{1}^{i},\dots q_{N(u_{i})}^{i}$ the
sequence of points given by Proposition 9. By Claim 3 of Proposition 9 there
exists a sequence of indices $k_{i}\in 1,\dots N$ such that
$d_{\bar{g}}\left(x_{i},q_{k_{i}}^{i}\right)\rightarrow 0$. Up to relabeling,
we say $k_{i}=1$ for all $i$. Then also $q_{1}^{i}\rightarrow x_{0}$ is a blow
up point for $u_{i}$. By Proposition 9 and Proposition 18 we have that
$q_{1}^{i}\rightarrow x_{0}$ is an isolated simple blow up point for $u_{i}$.
Then by Proposition 17 we deduce that $\|\pi(x_{0})\|=0$, contradicting the
assumption of the theorem. This concludes the proof. ∎

##  3\. The non compactness result

In this section we perform the Ljapunov-Schmidt finite dimensional reduction,
which relies on three steps. First, we start finding a solution of the
infinite dimensional problem (3.7) with a ansatz $u=W_{\delta,q}+\delta
V_{\delta,q}+\phi$ where $W_{\delta,q}+\delta V_{\delta,q}$ is a model
solution and $\phi=\phi_{\delta,q}$ is a small remainder. Then, we study a
finite dimensional reduced problem which depends only on $\delta,q$. Finally,
we give the proof of Theorem 2.

In the Ljapunov-Schmidt procedure, it will be necessary that
$-L_{g}+\varepsilon_{1}\alpha$ is a positive definite operator. Since $-L_{g}$
is positive definite, in the case $\alpha<0$, we choose $\varepsilon_{1}$
small enough in order to ensure the positivity of
$-L_{g}+\varepsilon_{1}\alpha$ .

###  3.1. The finite dimensional reduction

Since $-L_{g}+\varepsilon_{1}\alpha$ is a positive definite operator, we
define an equivalent scalar product on $H^{1}$ as

(3.1) $\left\langle\left\langle
u,v\right\rangle\right\rangle_{g}=\int_{M}(\nabla_{g}u\nabla_{g}v+\frac{n-2}{4(n-1)}R_{g}uv+\varepsilon_{1}\alpha
uv)d\mu_{g}$

which leads to the norm $\|\cdot\|_{g}$ equivalent to the usual one.

Given $1\leq t\leq\frac{2(n-1)}{n-2}$ we have the well known embedding

$i:H^{1}(M)\rightarrow L^{t}(\partial M),$

and we define, by the scalar product
$\left\langle\left\langle\cdot,\cdot\right\rangle\right\rangle_{g}$,

$i_{\alpha}^{*}:L^{t^{\prime}}(\partial M)\rightarrow H^{1}(M)$

in the following sense: given $f\in L^{\frac{2(n-1)}{n-2}}(\partial M)$ there
exists a unique $v\in H^{1}(M)$ such that

(3.2) $\displaystyle v=i_{\alpha}^{*}(f)$
$\displaystyle\iff\left\langle\left\langle
v,\varphi\right\rangle\right\rangle_{g}=\int_{\partial M}f\varphi
d\sigma\text{ for all }\varphi$ (3.5)
$\displaystyle\iff\left\\{\begin{array}[]{ccc}-\Delta_{g}v+\frac{n-2}{4(n-1)}R_{g}v+\varepsilon_{1}\alpha=0&&\text{on
}M;\\\ \frac{\partial v}{\partial\nu}=f&&\text{on
$\partial$}M.\end{array}\right.$

At this point Problem (1.1) is equivalent to find $v\in H^{1}(M)$ such that

$v=i_{\alpha}^{*}(f(v)-\varepsilon_{2}\beta v)$

where

$f(v)=(n-2)\left(v^{+}\right)^{\frac{n}{n-2}}.$

Notice that, if $v\in H_{g}^{1}$, then $f(v)\in L^{\frac{2(n-1)}{n}}(\partial
M)$.

Also, problem (1.1) has a variational structure and a positive solution for
(1.1) is a critical point for the following functional defined on $H^{1}(M)$

$\displaystyle J_{\varepsilon_{1},\varepsilon_{2},g}(v)=J_{g}(v):$
$\displaystyle=\frac{1}{2}\int_{M}|\nabla_{g}v|^{2}+\frac{n-2}{4(n-1)}R_{g}v^{2}+\varepsilon_{1}\alpha
v^{2}d\mu_{g}$ $\displaystyle+\frac{1}{2}\int_{\partial M}\varepsilon_{2}\beta
v^{2}d\sigma_{g}-\frac{(n-2)^{2}}{2(n-1)}\int_{\partial
M}\left(v^{+}\right)^{\frac{2(n-1)}{n-2}}d\sigma_{g}.$

We define a model solution of (1.1) by means of the standard bubble $U$ and of
the function $\gamma_{q}$ introduced in Lemma 4

Given $q\in\partial M$ and $\psi_{q}^{\partial}:\mathbb{R}_{+}^{n}\rightarrow
M$ the Fermi coordinates in a neighborhood of $q$, we define

$\displaystyle W_{\delta,q}(\xi)$
$\displaystyle=U_{\delta}\left(\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right)\chi\left(\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right)=$
$\displaystyle=\frac{1}{\delta^{\frac{n-2}{2}}}U\left(\frac{y}{\delta}\right)\chi(y)=\frac{1}{\delta^{\frac{n-2}{2}}}U\left(x\right)\chi(\delta
x)$

where $y=(z,t)$, with $z\in\mathbb{R}^{n-1}$ and $t\geq 0$, $\delta
x=y=\left(\psi_{q}^{\partial}\right)^{-1}(\xi)$ and $\chi$ is a radial cut off
function, with support in ball centered in $0$, of radius $R$. In an analogous
way, we define

$V_{\delta,q}(\xi)=\frac{1}{\delta^{\frac{n-2}{2}}}\gamma_{q}\left(\frac{1}{\delta}\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right)\chi\left(\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right).$

Finally, given $j_{a}$ defined in (1.4) and (1.5) we define

$Z_{\delta,q}^{b}(\xi)=\frac{1}{\delta^{\frac{n-2}{2}}}j_{b}\left(\frac{1}{\delta}\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right)\chi\left(\left(\psi_{q}^{\partial}\right)^{-1}(\xi)\right).$

By means of
$\left\langle\left\langle\cdot,\cdot\right\rangle\right\rangle_{g}$ it is
possible to decompose $H^{1}$ in the direct sum of the following two subspaces

$\displaystyle K_{\delta,q}$ $\displaystyle=\text{Span}\left\langle
Z_{\delta,q}^{1},\dots,Z_{\delta,q}^{n}\right\rangle$ $\displaystyle
K_{\delta,q}^{\bot}$ $\displaystyle=\left\\{\varphi\in H^{1}(M)\ :\
\left\langle\left\langle\varphi,Z_{\delta,q}^{b}\right\rangle\right\rangle_{g}=0,\
b=1,\dots,n\right\\}$

and to define the projections

$\Pi=H^{1}(M)\rightarrow K_{\delta,q}\text{ and
}\Pi^{\bot}=H^{1}(M)\rightarrow K_{\delta,q}^{\bot}.$

As claimed before, we look for a solution $u_{q}$ of (1.1) having the form

$u_{q}=W_{\delta,q}+\delta V_{\delta,q}+\phi$

where $\phi\in K_{\delta,q}^{\bot}$. Using $i_{\alpha}^{*}$, (1.1) is
equivalent to the following pair of equations

(3.6) $\displaystyle\Pi\left\\{W_{\delta,q}+\delta V_{\delta,q}+\phi-
i_{\alpha}^{*}\left[f(W_{\delta,q}+\delta
V_{\delta,q}+\phi)-\varepsilon_{2}\beta(W_{\delta,q}+\delta
V_{\delta,q}+\phi)\right]\right\\}$ $\displaystyle=0$ (3.7)
$\displaystyle\Pi^{\bot}\left\\{W_{\delta,q}+\delta V_{\delta,q}+\phi-
i_{\alpha}^{*}\left[f(W_{\delta,q}+\delta
V_{\delta,q}+\phi)-\varepsilon_{2}\beta(W_{\delta,q}+\delta
V_{\delta,q}+\phi)\right]\right\\}$ $\displaystyle=0.$

Let us define the linear operator $L:K_{\delta,q}^{\bot}\rightarrow
K_{\delta,q}^{\bot}$ as

(3.8) $L(\phi)=\Pi^{\bot}\left\\{\phi-
i_{\alpha}^{*}\left(f^{\prime}(W_{\delta,q}+\delta
V_{\delta,q})[\phi]\right)\right\\},$

and a nonlinear term $N(\phi)$ and a remainder term R as

(3.9) $\displaystyle N(\phi)=$
$\displaystyle\Pi^{\bot}\left\\{i_{\alpha}^{*}\left(f(W_{\delta,q}+\delta
V_{\delta,q}+\phi)-f(W_{\delta,q}+\delta
V_{\delta,q})-f^{\prime}(W_{\delta,q}+\delta
V_{\delta,q})[\phi]\right)\right\\}$ (3.10) $\displaystyle R=$
$\displaystyle\Pi^{\bot}\left\\{i_{\alpha}^{*}\left(f(W_{\delta,q}+\delta
V_{\delta,q})\right)-W_{\delta,q}-\delta V_{\delta,q}\right\\},$

With these operators, the infinite dimensional equation (3.7) becomes

$L(\phi)=N(\phi)+R-\Pi^{\bot}\left\\{i_{\alpha}^{*}\left(\varepsilon_{2}\beta(W_{\delta,q}+\delta
V_{\delta,q}+\phi)\right)\right\\}.$

In this subsection we find, for any $\delta,q$ given, a function $\phi$ which
solves equation (3.7). Many of the estimates which follow are contained in
[16], which we refer to for further details. Here we describe only the main
steps of each proof.

######  Lemma 19.

Assume $n\geq 7$. It holds

$\|R\|_{g}=O(\delta^{2})$

###### Proof.

Take the unique $\Gamma$ such that

$\Gamma=i_{\alpha}^{*}\left(f(W_{\delta,q}+\delta V_{\delta,q})\right),$

that is the function solving

$\left\\{\begin{array}[]{ccc}-\Delta_{g}\Gamma+\frac{n-2}{4(n-1)}R_{g}\Gamma+\varepsilon_{1}\alpha\Gamma=0&&\text{on
}M;\\\ \frac{\partial\Gamma}{\partial\nu}=(n-2)\left((W_{\delta,q}+\delta
V_{\delta,q})^{+}\right)^{\frac{n}{n-2}}&&\text{on
$\partial$}M.\end{array}\right.$

We have, by (3.1) that

$\displaystyle\|R\|_{g}^{2}$
$\displaystyle\leq\left\|i_{g}^{*}\left(f(W_{\delta,q}+\delta
V_{\delta,q}\right)-W_{\delta,q}-\delta V_{\delta,q}\right\|_{g}^{2}=\|\Gamma-
W_{\delta,q}-\delta V_{\delta,q}\|_{g}^{2}$
$\displaystyle=\int_{M}\left[\Delta_{g}(W_{\delta,q}+\delta
V_{\delta,q})-\frac{n-2}{4(n-1)}R_{g}(W_{\delta,q}+\delta
V_{\delta,q})\right](\Gamma-W_{\delta,q}-\delta V_{\delta,q})d\mu_{g}$
$\displaystyle-\int_{M}\varepsilon_{1}\alpha(W_{\delta,q}+\delta
V_{\delta,q})(\Gamma-W_{\delta,q}-\delta V_{\delta,q})d\mu_{g}$
$\displaystyle+\int_{\partial M}\left[f(W_{\delta,q}+\delta
V_{\delta,q})-\frac{\partial}{\partial\nu}(W_{\delta,q}+\delta
V_{\delta,q})\right](\Gamma-W_{\delta,q}-\delta V_{\delta,q})d\sigma_{g}$
$\displaystyle=:I_{1}+I_{2}+I_{3}.$

For $I_{1}$ we have

$I_{1}\leq\left|\Delta_{g}(W_{\delta,q}+\delta
V_{\delta,q})-\frac{n-2}{4(n-1)}R_{g}(W_{\delta,q}+\delta
V_{\delta,q})\right|_{L_{g}^{\frac{2n}{n+2}}(M)}\|R\|_{g}$

and direct computation and by the expansions of the metric (1.13) (1.14) we
have (see [16, Lemma 9])

$\left|W_{\delta,q}+\delta^{2}V_{\delta,q}\right|_{L_{\tilde{g}}^{\frac{2n}{n+2}}(M)}=O(\delta^{2}),$
$\left|\Delta_{\tilde{g}}(W_{\delta,q}+\delta^{2}V_{\delta,q})\right|_{L_{\tilde{g}}^{\frac{2n}{n+2}}(M)}=O(\delta^{2}).$

Similarly

$I_{2}\leq\varepsilon_{1}O(\delta^{2})\left\|R\right\|_{g}=O(\delta^{2})\left\|R\right\|_{g}.$

The proof of estimate for $I_{3}$ is more delicate, and uses in a crucial way
that $\gamma_{q}$ solves (1.7). As shown in [16, Lemma 9] we have indeed

$I_{3}\leq O(\delta^{2})\left\|R\right\|_{g}$

which completes the proof. ∎

######  Lemma 20.

Given $(\varepsilon_{1},\varepsilon_{2})$, for any pair $(\delta,q)$ there
exists a positive constant $C=C(\delta,q)$ such that for any $\varphi\in
K_{\delta,q}^{\bot}$ it holds

$\|L(\varphi)\|_{g}\geq C\|\varphi\|_{g}.$

This lemma is a standard tool in finite dimensional reduction, so we refer to
[10, 23] for the proof.

Proving that $N$ is a contraction it is also standard. In fact there exists
$\eta<1$ such that, for any $\varphi_{1},\varphi_{2}\in K_{\delta,q}^{\bot}$
it holds

(3.11) $\|N(\varphi)\|_{g}\leq\eta\|\varphi\|_{g}\text{ and
}\|N(\varphi_{1})-N(\varphi_{2})\|_{g}\leq\eta\|\varphi_{1}-\varphi_{2}\|_{g}$

By Lemma 19, Lemma 20, and by (3.11) we get the last result of this
subsection.

######  Proposition 21.

Given $(\varepsilon_{1},\varepsilon_{2})$, for any pair $(\delta,q)$ there
exists a unique $\phi=\phi_{\delta,q}\in K_{\delta,q}^{\bot}$ which solves
(3.7) such that

$\|\phi\|_{g}=O(\delta^{2}+\varepsilon_{2}\delta).$

In addition the map $q\mapsto\phi$ is $C^{1}$.

###### Proof.

Lemma 20 and (3.11) and by the properties of $i_{\alpha}$, there exists $C>0$
such that

$\left\|L^{-1}\left(N(\phi)+R-\Pi^{\bot}\left\\{i_{\alpha}^{*}\left(\varepsilon_{2}\beta(W_{\delta,q}+\delta
V_{\delta,q}+\phi)\right)\right\\}\right)\right\|_{g}\\\ \leq
C\left((\eta\|\phi\|_{g}+\|R\|_{g}+\left\|i_{\alpha}^{*}\left(\varepsilon_{2}\beta(W_{\delta,q}+\delta
V_{\delta,q}+\phi)\right)\right\|_{g}\right).$

Now, it is easy to estimate that

$\displaystyle\left\|i_{\alpha}^{*}\left(\varepsilon_{2}\beta(W_{\delta,q}+\delta
V_{\delta,q}+\phi)\right)\right\|_{g}$
$\displaystyle\leq\varepsilon_{2}\left(\left\|W_{\delta,q}+\delta
V_{\delta,q}\right\|_{L_{g}^{\frac{2(n-1)}{n}}(\partial
M)}+\left\|\phi\right\|_{g}\right)$ (3.12) $\displaystyle\leq
C\left(\varepsilon_{2}\delta+\varepsilon_{2}\left\|\phi\right\|_{g}\right).$

By Lemma 19 and by the previous estimates, for the map

$T(\tilde{\phi}):=L^{-1}\left(N(\tilde{\phi})+R-\Pi^{\bot}\left\\{i_{\alpha}^{*}\left(\varepsilon_{2}\beta(W_{\delta,q}+\delta
V_{\delta,q}+\phi)\right)\right\\}\right)$

it holds

$\|T(\phi)\|_{g}\leq
C\left((\eta+\varepsilon_{2})\|\phi\|_{g}+\varepsilon_{2}\delta+\delta^{2}\right).$

So, it is possible to choose $\rho>0$ such that $T$ is a contraction from the
ball $\|\phi\|_{g}\leq\rho(\varepsilon_{2}\delta+\delta^{2})$ in itself.
Hence, by the fixed point Theorem, we have the first claim. The second claim
is proved by the implicit function Theorem. ∎

###  3.2. The reduced functional

Once a solution of Problem (3.7) is found, it is possible to look for a
critical point of $J_{g}\left(W_{\delta,q}+\delta V_{\delta,q}+\phi\right)$,
solving a finite dimensional problem which depends only on $(\delta,q)$.

######  Lemma 22.

Assume $n\geq 7$. It holds

$\left|J_{g}\left(W_{\delta,q}+\delta
V_{\delta,q}+\phi\right)-J_{g}\left(W_{\delta,q}+\delta
V_{\delta,q}\right)\right|=o(1)\left\|\phi\right\|_{g}$

$C^{0}$-uniformly for $q\in\partial M$.

###### Proof.

We have, for some $\theta\in(0,1)$

(3.13) $J_{g}(W_{\delta,q}+\delta V_{\delta,q}+\phi)-J_{g}(W_{\delta,q}+\delta
V_{\delta,q})=J_{g}^{\prime}(W_{\delta,q}+\delta V_{\delta,q})[\phi]\\\
+\frac{1}{2}J_{g}^{\prime\prime}(W_{\delta,q}+\delta
V_{\delta,q}+\theta\phi)[\phi,\phi]\\\
=\int_{M}\left(\nabla_{g}W_{\delta,q}+\delta\nabla_{g}V_{\delta,q}\right)\nabla_{g}\phi+\left(\frac{n-2}{4(n-1)}R_{g}+\varepsilon_{1}\alpha\right)\left(W_{\delta,q}+\delta
V_{\delta,q}\right)\phi d\mu_{g}\\\ -(n-2)\int_{\partial
M}\left(\left(W_{\delta,q}+\delta
V_{\delta,q}\right)^{+}\right)^{\frac{n}{n-2}}\phi d\sigma_{g}\\\
+\int_{\partial M}\varepsilon_{2}\beta\left(W_{\delta,q}+\delta
V_{\delta,q}\right)\phi d\sigma_{g}+\frac{1}{2}\|\phi\|_{g}^{2}\\\
-\frac{n}{2}\int_{\partial M}\left(\left(W_{\delta,q}+\delta
V_{\delta,q}+\theta\phi_{\delta,q}\right)^{+}\right)^{\frac{2}{n-2}}\phi_{\delta,q}^{2}d\sigma_{g}+\frac{1}{2}\int_{\partial
M}\varepsilon_{2}\beta|\phi|^{2}d\sigma_{g}.$

All the terms but $\int_{M}\varepsilon_{1}\alpha\left(W_{\delta,q}+\delta
V_{\delta,q}\right)\phi d\mu_{g}$ have been estimated in [16, Lemma 12], so we
summarize only the key steps. As in Lemma 19, the most delicate term is the
nonlinear term on the boundary. In particular we have that

$\int_{\partial M}\left[(n-2)\left(\left(W_{\delta,q}+\delta
V_{\delta,q}\right)^{+}\right)^{\frac{n}{n-2}}-\frac{\partial}{\partial\nu}W_{\delta,q}\right]\phi
d\sigma_{g}\\\ =\left|(n-2)\left(\left(W_{\delta,q}+\delta
V_{\delta,q}\right)^{+}\right)^{\frac{n}{n-2}}-\frac{\partial}{\partial\nu}W_{\delta,q}\right|_{L^{\frac{2(n-1)}{n}}(\partial
M)}\|\phi\|_{g}\\\ =o(|\delta\log\delta|)\|\phi\|_{g}=o(1)\|\phi\|_{g}.$

The other terms in (3.13) are easier to estimate and lead to higher order
terms.

At last, by Holder inequality we have

$\displaystyle\left|\int_{M}W_{\delta,q}\phi d\mu_{g}\right|$
$\displaystyle\leq
C|W_{\delta,q}|_{L_{g}^{\frac{2n}{n+2}}}|\phi|_{L_{g}^{\frac{2n}{n-2}}}\leq
C\delta^{2}\|\phi\|_{g}$ $\displaystyle\delta\left|\int_{M}V_{\delta,q}\phi
d\mu_{g}\right|$ $\displaystyle\leq
C\delta|V_{\delta,q}|_{L_{g}^{2}}|\phi|_{L_{g}^{2}}\leq C\delta\|\phi\|_{g}.$

so

$\int_{M}\varepsilon_{1}\alpha\left(W_{\delta,q}+\delta
V_{\delta,q}\right)\phi d\mu_{g}=O(\delta)\|\phi\|_{g}$

and we are in position to prove the result. ∎

######  Lemma 23.

Let $n\geq 7$. It holds

$J_{g}(W_{\delta,q}+\delta
V_{\delta,q})=A+\varepsilon_{1}\delta^{2}\alpha(q)B+\varepsilon_{2}\delta\beta(q)C+\delta^{2}\varphi(q)+o(\varepsilon_{1}\delta^{2})+o(\varepsilon_{2}\delta)+o(\delta^{2})$

where

$\displaystyle A=$
$\displaystyle\frac{(n-2)(n-3)}{2(n-1)^{2}}\omega_{n-2}I_{n-1}^{n}>0;$
$\displaystyle B=$
$\displaystyle\frac{n-2}{(n-1)(n-4)}\omega_{n-2}I_{n-1}^{n}>0;$ $\displaystyle
C=$ $\displaystyle\frac{n-2}{n-1}\omega_{n-2}I_{n-1}^{n}>0;$
$\displaystyle\varphi(q)$
$\displaystyle=\frac{1}{2}\int_{\mathbb{R}_{+}^{n}}\gamma_{q}\Delta\gamma_{q}dy-\frac{(n-6)(n-2)\omega_{n-1}I_{n-1}^{n}}{4(n-1)^{2}(n-4)}\|\pi(q)\|^{2}\leq
0.$

Here $I_{n-1}^{n}:=\int_{0}^{\infty}\frac{s^{n}}{(1+s^{2})^{n-1}}ds$ and
$\pi(q)$ is the trace free tensor of the second fundamental form.

###### Proof.

The main estimates of this proof are proved in [16, Proposition 13], which we
refer for to for a detailed proof. Here, we limit ourselves to estimate the
perturbation terms. We have

$\displaystyle J_{g}(W_{\delta,q}+\delta V_{\delta,q})=$
$\displaystyle\frac{1}{2}\int_{M}|\nabla_{g}(W_{\delta,q}+\delta
V_{\delta,q})|^{2}d\mu_{g}+\frac{n-2}{8(n-1)}\int_{M}R_{g}(W_{\delta,q}+\delta
V_{\delta,q})^{2}d\mu_{g}$
$\displaystyle+\frac{1}{2}\varepsilon_{1}\int_{M}\alpha(W_{\delta,q}+\delta
V_{\delta,q})^{2}d\mu_{g}$
$\displaystyle+\frac{1}{2}\varepsilon_{2}\int_{\partial
M}\beta(W_{\delta,q}+\delta V_{\delta,q})^{2}d\sigma_{g}$
$\displaystyle-\frac{(n-2)^{2}}{2(n-1)}\int_{\partial
M}\left(W_{\delta,q}+\delta V_{\delta,q}\right)^{\frac{2(n-1)}{n-2}}.$

We easily compute the terms involving $\varepsilon_{1}$ and $\varepsilon_{2}$
taking in account the expansion of the volume form (1.13), getting

$\frac{1}{2}\varepsilon_{1}\int_{M}\alpha(W_{\delta,q}+\delta
V_{\delta,q})^{2}d\mu_{g}=\frac{1}{2}\varepsilon_{1}\delta^{2}\alpha(q)\int_{\mathbb{R}^{n}}U(y)^{2}dy+o(\varepsilon_{1}\delta^{2})$

and

$\frac{1}{2}\varepsilon_{2}\int_{\partial M}\beta(W_{\delta,q}+\delta
V_{\delta,q})^{2}d\mu_{g}=\frac{1}{2}\varepsilon_{2}\delta\beta(q)\int_{\mathbb{R}^{n-1}}U(\bar{y},0)^{2}d\bar{y}+o(\varepsilon_{2}\delta).$

By direct computation, and by Remark 18 in [16] (see also [16, page 1332]) we
have that
$\int_{\mathbb{R}^{n}}U(y)^{2}dy=\frac{2(n-2)}{(n-1)(n-4)}\omega_{n-2}I_{n-1}^{n}$
and
$\int_{\mathbb{R}^{n-1}}U(\bar{y},0)^{2}d\bar{y}=\frac{2(n-2)}{n-1}\omega_{n-2}I_{n-1}^{n}$,
getting the value of the positive constants $B$ and $C$.

For the remaining terms we refer to [16, Proposition 13] in which is proved
that

$\frac{1}{2}\int_{M}|\nabla_{\tilde{g}_{q}}(W_{\delta,q}+\delta
V_{\delta,q})|^{2}d\mu_{g}+\frac{n-2}{8(n-1)}\int_{M}R_{g}(W_{\delta,q}+\delta
V_{\delta,q})^{2}d\mu_{g}\\\ -\frac{(n-2)^{2}}{2(n-1)}\int_{\partial
M}\left(W_{\delta,q}+\delta
V_{\delta,q}\right)^{\frac{2(n-1)}{n-2}}d\sigma_{g_{q}}=A+\delta^{2}\varphi(q)+o(\delta^{2}).$

We conclude by noticing that $\varphi(q)\leq 0$ by (1.9). ∎

###  3.3. Proof of Theorem 2

At first we recall that, in the hypotheses of Theorem 2, we have that the
function $\varphi$ defined in Lemma 23 is strictly negative on $\partial M$.
Infact $\|\pi(q)\|^{2}$ is non zero by assumption, and
$\int_{\mathbb{R}_{+}^{n}}\gamma_{q}\Delta\gamma_{q}$ is non positive by
(1.9). With this in mind, we are in position to prove the result.

###### Proof of Theorem 2.

. We start with the first case, $\beta>0$. We choose

$\displaystyle\varepsilon_{1}$ $\displaystyle=o(1)$ $\displaystyle\delta$
$\displaystyle=\lambda\varepsilon_{2}$

where $\lambda\in\mathbb{R}^{+}$. With this choice, by Lemma 22 and
Proposition 21 we have that

$\left|J_{g}\left(W_{\lambda\varepsilon_{2},q}+\lambda\varepsilon_{2}V_{\lambda\varepsilon_{2},q}+\phi\right)-J_{g}\left(W_{\lambda\varepsilon_{2},q}+\lambda\varepsilon_{2}V_{\lambda\varepsilon_{2},q}\right)\right|=o(\varepsilon_{2}^{2})$

and that, by Lemma 23,

$J_{g}\left(W_{\lambda\varepsilon_{2},q}+\lambda\varepsilon_{2}V_{\lambda\varepsilon_{2},q}\right)=A+\varepsilon_{2}^{2}\left(\lambda\beta(q)C+\lambda^{2}\varphi(q)\right)+o(\varepsilon_{2}^{2}).$

We recall a result which is a key tool in Ljapunov-Schmidt procedure, and
which is proved, for instance, in [16, Lemma 15]) and which relies on the
estimates of Lemma 22.

######  Remark.

Given $(\varepsilon_{1},\varepsilon_{2}),$ if
$(\bar{\lambda},\bar{q})\in(0,+\infty)\times\partial M$ is a critical point
for the reduced functional
$I_{\varepsilon_{1},\varepsilon_{2}}(\lambda,q):=J_{g}\left(W_{\lambda\varepsilon_{2},q}+\lambda\varepsilon_{2}V_{\lambda\varepsilon_{2},q}+\phi\right)$,
then the function
$W_{\bar{\lambda}\varepsilon_{2},\bar{q}}+\bar{\lambda}\varepsilon_{2}V_{\bar{\lambda}\varepsilon_{2},\bar{q}}+\phi$
is a solution of (1.1).

To conclude the proof it lasts to find a pair $(\bar{\lambda},\bar{q})$ which
is a critical point for $I_{\varepsilon_{1},\varepsilon_{2}}(\lambda,q)$.

In this first case we consider
$G(\lambda,q):=\lambda\beta(q)C+\lambda^{2}\varphi(q)$. We have that
$\beta(q)C$ is strictly positive on $\partial M$, by our assumptions, while,
as recalled before, $\varphi$ is strictly negative on $\partial M$. At this
point there exists a compact set $[a,b]\subset\mathbb{R}^{+}$ such that the
function $G$ admits an absolute maximum in $(a,b)\times\partial M$, which also
is the absolute maximum value of $G$ on $\mathbb{R}^{+}\times\partial M$. This
maximum is also $C^{0}$-stable, in the sense that, if $(\lambda_{0},q_{0})$ is
the maximum point for $G$, for any function $f\in C^{1}([a,b]\times\partial
M)$ with $\|f\|_{C^{0}}$ sufficiently small, then the function $G+f$ on
$[a,b]\times\partial M$ admits a maximum point $(\bar{\lambda},\bar{q})$ close
to $(\lambda_{0},q_{0})$. By the $C_{0}$ stability of this maximum point
$(\lambda_{0},q_{0})$, and by Lemma 23, given $\varepsilon_{2}$ sufficiently
small (and $\varepsilon_{1}=o(1)$), there exists a pair
$\left(\lambda_{\varepsilon_{1},\varepsilon_{2}},q_{\varepsilon_{1},\varepsilon_{2}}\right)$
which is a maximum point for $I_{\varepsilon_{1},\varepsilon_{2}}(\lambda,q)$.
This implies, in light of the above Remark, that there exists a pair
$\left(\bar{\lambda}_{\varepsilon_{1},\varepsilon_{2}},\bar{q}_{\varepsilon_{1},\varepsilon_{2}}\right)$
such that
$W_{\bar{\lambda}_{\varepsilon_{1},\varepsilon_{2}}\varepsilon_{2},\bar{q}_{\varepsilon_{1},\varepsilon_{2}}}+\bar{\lambda}_{\varepsilon_{1},\varepsilon_{2}}\varepsilon_{2}V_{\bar{\lambda}_{\varepsilon_{1},\varepsilon_{2}}\varepsilon_{2},\bar{q}_{\varepsilon_{1},\varepsilon_{2}}}+\phi$
is a solution of (1.1), and the proof for the case $\beta>0$ is complete.

The proof in the second case is similar. In this case, by assumption, we have
that $B\alpha(q)+\varphi(q)>0$ on $\partial M$. Then we choose

$\displaystyle\varepsilon_{1}$ $\displaystyle=1$ $\displaystyle\delta$
$\displaystyle=\lambda\varepsilon_{2}$

and we obtain that

$I_{\varepsilon_{1},\varepsilon_{2}}(\lambda,q)=A+\varepsilon_{2}^{2}\left[\lambda\beta(q)C+\lambda^{2}\left(\alpha(q)B+\varphi(q)\right)\right]+o(\varepsilon_{2}^{2}).$

In this case we define the function $G(\lambda,q)$ as

$G(\lambda,q):=\lambda\beta(q)C+\lambda^{2}\left(\alpha(q)B+\varphi(q)\right).$

and, by our assumptions, the coefficient of $\lambda$ is strictly negative on
$\partial M$ while the coefficient of $\lambda^{2}$ is strictly positive on
$\partial M$, so we can conclude the proof follows in a similar way. ∎

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