\documentclass[reqno]{amsart}
\usepackage{hyperref}
\usepackage{mathrsfs, amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 289, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/289\hfil Singular limiting solutions]
{Singular limiting solutions to 4-dimensional elliptic problems involving
exponentially dominated nonlinearity and nonlinear terms}

\author[S. Baraket, I. Bazarbacha, M. Trabelsi \hfil EJDE-2015/289\hfilneg]
{Sami Baraket, Imen Bazarbacha, Maryem Trabelsi}

\address{Sami Baraket \newline
Department of Mathematics,
College of Science, King Saud University,
P.O. Box 2455, Riyadh 11451, Saudi Arabia}
\email{sbaraket@ksu.edu.sa}

\address{Imen Bazarbacha \newline
D\'epartement de Math\'ematiques,
Facult\'e des Sciences de Tunis,
Campus Universitaire, 2092 Tunis,
University Tunis El Manar, Tunisia}
\email{imen.bazarbacha@gmail.com}

\address{Maryem Trabelsi \newline
D\'epartement de Math\'ematiques,
Facult\'e des Sciences de Tunis,
Campus Universitaire, 2092 Tunis,
University Tunis El Manar, Tunisia}
\email{trabelsi.maryem@gmail.com}

\thanks{Submitted August 15, 2015. Published November 20, 2015.}
\subjclass[2010]{35J60, 53C21, 58J05}
\keywords{Biharmonic operator; nonlinear operator; singular limits;
\hfill\break\indent  Green's function; nonlinear domain decomposition method}

\begin{abstract}
 Let $\Omega \in \mathbb{R}^4$ be a bounded open regular set,
 $x_1, x_2, \dots, x_m \in \Omega$, $\lambda, \rho >0$ and
 $ \mathscr{Q}_\lambda$ be a non linear operator (which will be defined later).
 We prove that the problem
 \[
 \Delta^2u +\mathscr{Q}_\lambda(u)= \rho^4 e^u
 \]
 has a positive weak solution in $\Omega$ with 
 $ u = \Delta u=0$ on $\partial \Omega$, which is singular at each $x_i$
 as the parameters $\lambda$ and $\rho$ tends to $0$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction and statement of results}

Semilinear equations involving fourth order elliptic operator and
exponential nonlinearity appear naturally in conformal geometry
and in particular in the prescription of the so called
$Q$-curvature in four-dimensional Riemannian manifolds
\cite{Cha-1,Chang-Yang}
\[
Q_g = \frac{1}{12}  ( -  \Delta_g S_g + S^2_g - 3 |\operatorname{Ric}_g|^2),
\]
where $\operatorname{Ric}_g$ denotes the Ricci tensor and $S_g$ is the
scalar curvature of the metric $g$. Recall that the $Q$-curvature
changes under a conformal change of metric
\[
g_{w} = e^{2 w}  g,
\]
according to
\begin{equation}
P_{g}  w + 2  Q_g = 2  \tilde{Q}_{g_w}  e^{ 4  w},
\label{eq:1.5}
\end{equation}
where
\[
P_{g} : =  \Delta_g^2 +  \delta  ( \frac{2}{3} S_g  I -
2  \operatorname{Ric}_g ) \,d, \]
is the Panietz operator, which
is an elliptic $4$-th order partial differential operator
\cite{Chang-Yang} and which transforms according to
\[
e^{4 w}  P_{e^{2w} g}  =  P_g,
\]
under a conformal change of metric $ g_w  : = e^{2w}  g$.
In the special case where the manifold is the Euclidean space,
the Panietz operator is simply given by
\[
P_{g_{\rm eucl}} = \Delta^{2},
\]
in which case \eqref{eq:1.5} can be written as
\[
\Delta^{2}  w =  \tilde{Q}  e^{4  w},
\]
the solutions of which give rise to conformal metric $g_w = e^{2
 w}  g_{\rm eucl}$ whose $Q$-curvature is given by $\tilde Q$.
There is by now an extensive literature about this problem and we
refer to \cite{Chang-Yang} and \cite{Malchiodi-Djadli} for
references and recent developments.

 Wei in \cite{wei}, have studied the behavior of
solutions to the following nonlinear  problem in $\mathbb{R}^4$. More
precisely, consider the  problem
\begin{equation}
\begin{gathered}\Delta^{2} u = \lambda\;f(u) \quad\text{in } \Omega\\
u =\Delta u = 0 \quad \text{on }\partial\Omega.
\end{gathered} \label{eq:weichap3}
\end{equation}
Before showing his result, we  introduce some notation. Let
$G(x,x')$ defined, over $ \Omega\times\Omega$, the Green function
associated to the  bi-laplacian operator with a Navier boundary
conditions, which is the solution of
\begin{equation} \label{Green}
\begin{gathered}
\Delta_{x}^{2} G(x,x') = 64\pi^{2} \delta_{x=x'} \quad\text{in } \Omega\\
G(x,x')=\Delta_{x}
G(x,x') = 0\quad  \text{on } \partial\Omega
\end{gathered}
\end{equation}
and denote by $H(x,x')=G(x,x')+8\log|x-x'|$ its smooth part.
Consider now the functional
\begin{equation}
E  : (x^{1},\dots,x^m) \in ({\mathbb{R}}^4)^m \mapsto
 {\sum_{j=1}^{m}}H(x^{j},x^{j}) + \sum_{j \neq l}
G(x^{j},x^{l})\label{eq:fr4}
\end{equation}
and $u^{*}$ the solution of
\begin{equation}
\begin{gathered}
\Delta^{2} u^{*}= 64\pi^{2} {\sum_{i=1}^m}\delta_{x^i} \quad
 \text{in } \Omega\\
u^{*} =\Delta u^{*} = 0 \quad  \text{on } \partial\Omega.
\end{gathered} \label{eq:u*}
\end{equation}
The author proved the following result.

\begin{theorem}[\cite{wei}] \label{thm1}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^4$ and $f$ a smooth
nonnegative increasing function such that
\[
e^{-u}f(u)\text{ and }\varepsilon^{-u}\int_0^{u} f(s)
ds\text{ tends to $1$ as } u\to+\infty.
\]
For $u_\lambda$ solution of \eqref{eq:weichap3}, denote by
$ {\Sigma_{\lambda} = \lambda
\int_{\Omega}f(u_\lambda) dx}$. Then, three cases occur:
\begin{enumerate}
\item $\Sigma_{\lambda} \to 0$ therefore,
$\|u_\lambda\|_{L^{\infty(\Omega)}} \to 0$ as $\lambda
 \to 0$.

\item $\Sigma_{\lambda} \to +\infty$ then
$u_\lambda \to +\infty$ as $\lambda
 \to 0$.

\item $\Sigma_{\lambda} \to 64 \pi^{2} m$, for some positive integer $m$.
Then the limiting Function $u^* = { \lim_{\lambda\to 0}
u_{\lambda}}$ has $m$ blow-up points, $\{x^{1}, \dots, x^{m}\}$,
where $u_\lambda(x^i)\to +\infty$ as $\lambda \to 0$. Moreover,
 $(x^{1}, \dots, x^{m})$ is a critical point of $E$.
\end{enumerate}
\end{theorem}

Now, we are interested in positive solutions of the problem
\begin{equation}
\begin{gathered}
\Delta^{2} u = \rho^4 e^{u} \quad \text{in } \Omega\\
u  = \Delta u=0 \quad \text{on } \partial\Omega
\end{gathered} \label{eq:1.1}
\end{equation}
when the parameter $\rho$ tends to $0$. Obviously, the application
of the implicit function theorem yields the existence of a smooth
one parameter family of solutions $(u_\rho)_\rho$ which converges
uniformly to $0$ as $\rho$ tends to $0$. This branch of solutions
is usually referred to as the branch of {\em minimal
solutions} which gives the converse of the case $(1)$ given in the
last Theorem.

First, let us mention that in \cite{beg}, Ben Ayed, El Mehdi and Grossi
considered a bi-harmonic equation with large exponent in the non linear term;
 that is $\Delta^{2} u = u^p$, under Navier boundary conditions.
The authors have studied the asymptotic behavior of positive solutions
 obtained by minimizing suitable functionals.

In \cite{Cla-mun-mus}, the authors  studied existence and qualitative properties
 of positives solutions to the boundary-value problem
\begin{equation}
\begin{gathered}
\Delta^{2} u = \rho^4\; k ( x ) e^{u} \quad\text{in } \Omega\\
u = \Delta u =0 \quad \text{on }\partial\Omega
\end{gathered} \label{eq:1.k}
\end{equation}
where $k \in C^2 ( \Omega )$, is a non-negative, not identically zero function,
$\Omega$ a bounded open regular domain in $ \mathbb{R}^4$ and $\rho > 0$
is a small, positive parameter which tends to $0$.

 Recently, the existence of other branches of solutions as $\rho$ tends to
$0$ is studied in \cite{Bdop}. The authors construct a non-minimal
solutions with singular limit as the parameter $\rho$ tends to
$0$. Their results which give the converse of the case $(3)$
given in the last Theorem, can be stated as follows.

\begin{theorem}[\cite{Bdop}] \label{thm2}
 Let $\Omega$ be a smooth open subset of $\mathbb{R}^4$.
Assume  $(x^1,\ldots ,x^m)$ is a  nondegenerate critical
point of $E$. Then there exist $\rho_0 > 0$ and a one parameter family
$(u_{\rho})_{\rho  \in (0 , \rho_0) }$ of solutions of \eqref{eq:1.1},
such that
\[
\lim_{\rho\to 0} u_{\rho} =
u^{*},\quad\text{in }{\mathcal{C}}_{\rm loc}^{4,\alpha}(\Omega -
\{x^1 , \ldots ,x^m \}  ).
\]
\end{theorem}

To prove Theorem \ref{thm2}, the authors present, for the first time,  a
rather efficient method to solve such singularly perturbed
problems in the context of partial differential equations. This
method based on some nonlinear domain decomposition has already
been used successfully in geometric context (constant mean
curvature surfaces, constant scalar curvature metrics, extremal
K\"ahler metrics, \ldots). In this article, we adopt this method in
the study of the following problem.

Let $\Omega \subset {\mathbb{R}}^4$ be a regular bounded open
domain in ${\mathbb{R}}^4$. We are interested in positive
solutions of
\begin{equation}
\Delta^{2} u+ \mathscr{Q}_\lambda(u) =
\rho^4 e^{u} \text{ in}~ \Omega  \label{eq:11.1}
\end{equation}
satisfying $u=\Delta u =0$ on $\partial\Omega$
and $\mathscr{Q}_\lambda$ is the nonlinear operator given by
\begin{equation}
\begin{aligned}
\mathscr{Q}_\lambda(u)
&:=\lambda \big[ (\Delta u)^2+\Delta(|\nabla u|^2)
 +2 \nabla u \cdot \nabla( \Delta u) \big] \\
&\quad +2\lambda^2 \big[ \Delta u|\nabla u|^{2}+ \nabla
u\cdot\nabla (|\nabla u|^2)\big]+\lambda^3|\nabla u|^4.
\end{aligned}
\end{equation}
 Using the transformation
\begin{equation}
w := (\lambda \rho^4 e^u)^\lambda,
\end{equation}
if $u$ is a solution of \eqref{eq:11.1} then $w$ solves the  equation
\begin{equation}
\Delta^2 w ={ { w^ \frac{\lambda+1}{\lambda}}} \quad\text{in } \Omega .
 \label{eq:eq}
\end{equation}
Remark that the exponent $ {q=\frac{\lambda + 1}{\lambda}}$ tends to
$\infty$ as $\lambda$ tends to $0$.

We denote by $\varepsilon$ the smallest positive parameter
satisfying
\begin{equation}
\rho^4=\frac{384 \varepsilon^4}{(1+\varepsilon^{2})^4}\label{eq:12}.
\end{equation}
We remark that $\rho\sim \varepsilon$ as $\varepsilon\to 0$. We will suppose in the following that
\begin{itemize}
\item[(A1)]
$\lambda^{1+\delta/2}\varepsilon^{-\delta}= \mathcal{O}(1)$
as $\varepsilon \to 0$ for any $\delta\in(0,1)$.
\end{itemize}
In particular, if we take $\lambda = \mathcal{O}(\varepsilon^{2/3})$, then condition
 (A1) is satisfied. Under the assumption (A1), we can treat equation
\eqref{eq:11.1} as a perturbation of the  equation
 $$
\Delta^2 u = \rho^4 e^{u} \quad \text{in }\Omega\subset\mathbb{R}^4 .
$$
Our question is: Does there exist $u_{\varepsilon}$ a sequence of solutions which
converges to some singular function as the parameters $\varepsilon$ tend to $0$?


 Our main result reads as follows.

\begin{theorem} \label{our gamma}
Given $\alpha \in (0,1)$. Let
 $\Omega$ be an open smooth bounded set of $\mathbb{R}^4$,
$\lambda>0$ satisfy condition {\rm (A1)},
and $S=\{x_1,\ldots ,x_m\}\subset\Omega$ be a non empty set.
 Assume that $(x_1,\ldots ,x_m)$ is a  nondegenerate
critical point of the function
$$
\mathscr{F}(x_1,\dots,x_m)
= {\sum_{j=1}^{m}}H(x_{j},x_{j}) + \sum_{i \neq j}
G(x_{i},x_{j})\quad in\;(\Omega)^m,
$$
then there exist $\rho_0 > 0$, $\lambda_0 >0$ and a family
$\{u_{\rho, \lambda}\}$ with $0 < \rho < \rho_0, 0  < \lambda < \lambda_0$
of solutions of \eqref{eq:11.1}, such that
\[
\lim_{\rho \to 0,\, \lambda\to 0}
 u_{\rho,\lambda} = \sum_{j=1}^m G(x_j,\cdot) \quad \text{in }
\mathcal{C}_{\rm loc}^{4,\alpha}(\Omega - \{x_1 , \ldots ,x_m \}).
\]
\end{theorem}

\section{Construction of the approximate solution}

 We first describe the rotationally symmetric approximate solutions of
\begin{equation}
\Delta^{2}u - \rho^4 e^{u} = 0 ,\label{eq:epbpdo}
\end{equation}
in $\mathbb{R}^4$, which will be crucial in the construction of the approximate
solution. Given $\varepsilon>0$, we define
\[
u_{\varepsilon,\tau } (x) :=  4  \log(1 + \varepsilon^{2})
+4 \log \tau- 4  \log(\varepsilon^{2} + (\tau|x|)^2).
\]
which is clearly a solution of \eqref{eq:epbpdo} when
\[
\rho^4 = \frac{384\varepsilon^4} {(1+\varepsilon^{2})^4}.
\]

For $\tau > 0$, we remark that  equation \eqref{eq:epbpdo} is
invariant under some dilation in the following sense: If $u$ is solution
of \eqref{eq:epbpdo}, then
\[
\tau\mapsto u(\tau \cdot ) + 4  \log  \tau.
\]
is also a solution of \eqref{eq:epbpdo}. So, for  $\varepsilon>0$ and $\tau > 0$ we
denote by $ u_{\varepsilon,\tau}$ the element of this new family of radial
solutions of \eqref{eq:epbpdo}.

For $\varepsilon=\tau=1$ and we denote by
$u_1 = u_{1,1}$ this particular solution. We also define the
following linear fourth order elliptic operator
\[
\mathscr{L} : = \Delta^{2} - \frac{384}{(1 + |x|^{2})^4},
\]
which corresponds to the linearization of \eqref{eq:epbpdo} about
the solution $u_1$.

\subsection{Radial solution  on $\mathbb{R}^4$}
For all $\varepsilon , \tau, \lambda > 0 $, we set
\begin{equation}
R_{\varepsilon,\lambda}: = \tau r_{\varepsilon,\lambda}/\varepsilon , \quad
\text{where }
r_{\varepsilon,\lambda}:=\max(\sqrt{\varepsilon},\sqrt{\lambda})
\label{eq:small bool}.
\end{equation}


The classification of bounded solutions of $\mathbb{L}  w =0$
in $\mathbb{R}^4$ is well known.  Some solutions are easy to
find. For example, we can define
\[
\phi_0(x) : = r  \partial_{r}u_1(x) + 4 = 4  \frac{1- r^{2}}{1+r^{2}},
\]
where $r=|x|$. Clearly $\mathbb{L}  \phi_0 =0$ and this
reflects the fact that \eqref{eq:epbpdo} is invariant under the
group of dilations $ \tau \mapsto u(\tau  \cdot ) + 4  \log \tau $.
We also define, for $i=1, \ldots, 4$
\[
\phi_i(x) : = -{\partial_{x_i}} u_1(x) =\frac{8x_i}{1+|x|^{2}},
\]
which are also solutions of $\mathbb{L}   \phi_i =0$ since
these solutions correspond to the invariance of the equation under
the group of translations $a \mapsto u( \cdot +a)$.
Then, we have the  following  classification.

\begin{lemma}[\cite{Bdop}] \label{le:3.1}
Any bounded solution of $\mathscr{L}  w = 0$ defined in
$\mathbb{R}^4$ is a linear combination of $\phi_i$ for $i=0, 1,
\ldots ,4$. 
\end{lemma}

Let $B_r$ denote the ball of radius $r$ centered at the origin in
$\mathbb{R}^4$.

\begin{definition} \label{de:3.1} \rm
Given $k \in {\mathbb N}$, $\alpha \in (0,1)$ and $\mu \in
\mathbb{R}$, we introduce the H\"older weighted spaces
$\mathcal{C}^{k,\alpha}_\mu(\mathbb{R}^4)$ as the space of functions
 $w \in \mathcal{C}^{k,\alpha}_{\rm loc}(\mathbb{R}^4)$ for which the
 norm
\[
\| w \|_{\mathcal{C}^{k,\alpha}_\mu(\mathbb{R}^4)} : = \|
w\|_{\mathcal{C}^{k, \alpha} (\bar B_1)} +
 {\sup_{r\geqslant 1}}  \big( (1+r^2)^{-\delta/2}  \|w
(r\cdot) \|_{\mathcal{C}^{k,\alpha}_\mu(\bar B_1-B_{1/2})}\big),
\]
is finite. 
\end{definition}

Also, we define
\[
\mathcal{C}^{k,\alpha}_{{\rm rad}, \mu}(\mathbb{R}^4)
=\{f\in \mathcal{C}^{k,\alpha}_{\mu}(\mathbb{R}^4)text{ such that }
f(x)=f(|x|), \forall x\in \mathbb{R}^4\}.
\]


As a consequence of  Lemma \ref{le:3.1}, we recall
the surjectivity result of $\mathscr{L}$.

\begin{proposition}[\cite{Bdop}] \label{pr:surj-1}
\begin{enumerate}
\item Assume that $\mu > 1$ and $\mu \not\in \mathbb{Z}$, then
the operator $L_\mu: \mathcal{C}^{4,\alpha}_\mu(\mathbb{R}^4)
\to \mathcal{C}^{0,\alpha}_{\mu-4}(\mathbb{R}^4)$
defined by $L_\mu(w) = \mathscr{L} w$
is surjective.

\item Assume that $\delta > 0$ and $\delta\not\in \mathbb{Z}$, then
the operator $L_\delta : \mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4))
\to \mathcal{C}^{0,\alpha}_{{\rm rad},\delta-4}(\mathbb{R}^4)$ defined by
$L_\delta(w)= \mathscr{L}  w$ is surjective.
\end{enumerate}
\end{proposition}

We set $\bar{B}_1^* = \bar{B}_1 - \{0\}$.

\begin{definition} \label{de:5.1} \rm
Given $k\in {\mathbb N}$, $\alpha \in (0,1)$ and $\mu \in
\mathbb{R}$, we introduce the H\"older weighted spaces
$\mathcal{C}^{k,\alpha}_\mu(\bar B_1^*)$ as the space of functions in
$\mathcal{C}^{k,\alpha}_{\rm loc}(\bar B_1^*)$ for which the
norm
\[
\|u \|_{\mathcal{C}^{k,\alpha}_\mu(\bar B_1^*)} =
 {\sup_{r\leqslant 1/2}}  ( r^{-\mu}  \| u(r
\cdot) \|_{\mathcal{C}^{k,\alpha}(\bar B_2-B_1)} ),
\]
is finite. 
\end{definition}

Then, we define the subspace of radial functions in
$\mathcal{C}^{k,\alpha}_{\delta}(\bar B_1^*)$ by
\[
\mathcal{C}^{k,\alpha}_{{\rm rad}, \delta}(\bar B_1^*)=\{f\in
\mathcal{C}^{k,\alpha}_{\delta}(\mathbb{R}^4);\;such \;that\;
f(x)=f(|x|), \forall x\in \bar B_1^*\}.
\]
Our aim now is the construction of a radial solution $u$
of
\begin{equation}
\Delta^2 u + \mathscr{Q}_\lambda (u)
-\rho^4e^{u} =0\quad \text{in }\bar{B}_{r_{\varepsilon,\lambda}}.
\label{eq:mah}
\end{equation}
 Thanks to the  transformation
\[
v(x) = u(\frac{\varepsilon}{\tau}x) + 8\log
\varepsilon-4\log({\tau(1+\varepsilon^2)}/{2}),
\]
Equation \eqref{eq:mah} can be written as
\begin{equation}
\Delta^2 v +  \mathscr{Q}_\lambda (v)
-24e^{v} =0\quad \text{in }\bar{B}_{R_{\varepsilon,\lambda}}.\label{eq:mah1}
\end{equation}

Now, we look for a solution of \eqref{eq:mah1} of the form
$v(x)=u_1(x)+h(x)$; this amounts to solving
\begin{equation}
\mathscr{L}h = \frac{384}{(1 + |x|^{2})^4}(e^{h} - h -
1)- \mathscr{Q}_\lambda(u_1+h)\quad \text{in }
\bar {B}_{R_{\varepsilon,\lambda}}.\label{equ:y}
\end{equation}
We will need the following definition.

\begin{definition} \rm \label{de:6.1}
Given $ \bar r \geqslant 1$, $k \in {\mathbb N}$, $\alpha\in (0,1)$ and
$\delta\in \mathbb{R}$, the weighted space $\mathcal{C}^{k,
\alpha}_{\delta} (B_{\bar r})$ is defined to be the space of
functions $w \in {\mathcal{C}}^{k, \alpha} (B_{\bar r})$ endowed
with the norm
\[
\| w \|_{{\mathcal{C}}^{k, \alpha}_{\delta} (\bar B_{\bar r})}
:= \| w \|_{{\mathcal{C}}^{k, \alpha} (B_1)} + \sup_{1 \leqslant r
\leqslant \bar r}   ( r^{-\delta}  \| w (r  \cdot)
\|_{{\mathcal{C} }^{k, \alpha} ( \bar B_1 - B_{1/2})} ).
\]
\end{definition}

For $\sigma \geqslant 1$, we denote by
 \[
\mathscr{E}_{\sigma} : \mathcal{C}^{0, \alpha}_\delta (\bar B_{\sigma}) \to
\mathcal{C}^{0, \alpha}_\delta (\mathbb{R}^4) ,
\]
the extension operator defined by
\[
\mathscr{E}_{\sigma}  (f) (x) = \chi ( \frac{|x|}{\sigma})
f ( \sigma  \frac{x}{|x|} ),
\]
where $t \mapsto \chi (t)$ is a smooth nonnegative cutoff
function identically equal to $1$ for $t \geqslant 2$ and identically
equal to $0$ for $t \leqslant 1$. It is easy to check that there exists
a constant $c = c (\delta)
>0$, independent of $\sigma \geqslant 1$, such that
\begin{equation}
\| \mathscr{E}_{\sigma} (w ) \|_{\mathcal{C}^{0,
\alpha}_{\delta} (\mathbb{R}^4)} \leqslant
 c  \| w \|_{\mathcal{C}^{0, \alpha}_{\delta} (\bar B_{\sigma})} . \label{eq:6.3}
\end{equation}

We fix $\delta \in (0,1)$,
and denote by $\mathscr{G}_{\delta}$ to be a right inverse of
$\mathscr{L}_{\delta}$ assured by Proposition~\ref{pr:surj-1}.
 Now, we use the result of Proposition~\ref{pr:surj-1} to rephrase the
nonlinear equation  \eqref{equ:y}  as a fixed point problem.
Hence, to obtain a solution of \eqref{equ:y}, it is enough to
find a fixed point $h$ in a small ball of
$\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)$ for the mapping
\begin{equation}
 h \mapsto{\mathscr N}(h):= \mathscr{G}_{\delta}\circ\mathscr{E}_{\delta}
\circ \mathscr{R}(h),\label{eq:T}
\end{equation}
where
\[
\mathscr{R}(h):=\frac{384}{(1 + |x|^{2})^4}(e^{h} - h -1)
- \mathscr{Q}_\lambda(u_1+h).
\]
We have
\begin{align*}
\mathscr{R}(0)
&= - \lambda\big[ (\Delta u)^2+\Delta(|\nabla u|^2)
+2 \nabla u \cdot \nabla( \Delta u)\big]\\
&\quad - 2\lambda^2 \big[ \Delta u|\nabla u|^{2}+ \nabla
u\cdot\nabla (|\nabla u|^2)\big] - \lambda^3|\nabla u|^4.
\end{align*}
Recall that
\begin{equation*}
u_1=
 4  \log(2)- 4  \log(1 + r^2).
\end{equation*}
Then
$$
|\nabla u_1|^2=64\frac{r^2}{(1 + r^2)^2}, \quad
\Delta u_1=-16\frac{2+r^2}{(1 + r^2)^2}, \quad
\Delta(|\nabla u_1|^2)=512\frac{1-2\; r^2}{(1 + r^2)^4}.
$$
Hence,
\begin{gather*}
 (1 + r^2)^{2-\frac{\delta}{2}}|(\Delta u_1)^2+\Delta(|\nabla u_1|^2)
+2 \nabla u_1 \cdot \nabla( \Delta u_1)|
\leqslant  c(1 + r^2)^{-\frac{\delta}{2}},
\\
(1 + r^2)^{2-\frac{\delta}{2}}|\Delta u_1|\nabla u_1|^{2}+ \nabla
u_1\cdot\nabla (|\nabla u_1|^2)| \leqslant c(1 +
r^2)^{-1-\frac{\delta}{2}},
\\
(1 + r^2)^{2-\frac{\delta}{2}}|\nabla u_1|^4 \leqslant c(1 +
r^2)^{-\frac{\delta}{2}}.
\end{gather*}
This implies that given $ \kappa > 0$, there exists $c_{\kappa}>0$
(which can depend only on ${\kappa}$), such that for $\delta\in(0,1)$ and
$|x| = r$, we have
\[
\sup_{r \leqslant R_{\varepsilon,\lambda}}
 (1 + r^2)^{2-\frac{\delta}{2}} |\mathscr{R}(0)|
\leqslant  c_{\kappa}\lambda\,.
\]
So
\begin{equation}
\|{\mathscr N}(0)\| _ {\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}
\leqslant c_{\kappa}r^{2}_{\varepsilon,\lambda}.
\label{eq:T4}
\end{equation}
Using Proposition \ref{pr:surj-1} and \eqref{eq:6.3}, we deduce that
\begin{equation}
\| h\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},
\delta}(\mathbb{R}^4)}\leqslant 2 c_\kappa
r_{\varepsilon,\lambda}^2.
\label{eq:h}
\end{equation}
Now  let $h_1,
h_2\;\text{in}\;B (0, 2 c_\kappa\;
r_{\varepsilon,\lambda}^2)$ of ${\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}$
and for $\delta\in(0,1)$, then
\begin{equation*}
|\mathscr{R}(h_2)-\mathscr{R}(h_1)|\leqslant {|e^{h_2}-e^{h_1}+
h_1-h_2 |}+  {{|\mathscr{Q}_\lambda(u_1+h_2)
- \mathscr{Q}_\lambda(u_1+h_1)|}}.
\end{equation*}
Furthermore,
\begin{align*}
 r^{4-\delta}{|e^{h_2}-e^{h_1}+h_1-h_2 |}
&\leqslant c r^{4-\delta} |h_2-h_1||h_2+h_1|\\
&\leqslant c_\kappa r^{\delta}r^2_{\varepsilon,\lambda}
\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
\begin{align*}
r^{4-\delta}{| (\Delta (u_1+h_1))^2 -(\Delta( u_1+h_2))^2  |}
&= r^{4-\delta}{| (\Delta (h_1-h_2)) (\Delta (2u_1+h_1+h_2)) |}\\
&\leqslant c_\kappa {\big( 1+r^{\delta}r^2_{\varepsilon,\lambda}
\big)}\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
\begin{align*}
r^{4-\delta}{|\Delta|\nabla(u_1+h_2)|^2- \Delta|\nabla (u_1+h_1)|^2 |}
&= r^{4-\delta}{| \Delta (\nabla(h_1-h_2)\cdot\nabla(2u_1+h_1+h_2)) |}\\
&\leqslant c_\kappa {\big( 1+r^{\delta}r^2_{\varepsilon,\lambda}
\big)}\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
\begin{align*}
&r^{4-\delta}\big|\nabla(\Delta(u_1+h_2))\cdot\nabla( u_1+h_2)
 - \nabla(\Delta(u_1+h_1))\cdot\nabla( u_1+h_1) \big|\\
&=r^{4-\delta}\big| \nabla(\Delta(h_1-h_2))\cdot\nabla(2u_1+h_1+h_2)
 + \nabla(h_2-h_1)\cdot\nabla(\Delta(2u_1+h_1+h_2))\big| \\
&\quad\times r^{4-\delta}\big|\nabla(\Delta(u_1+h_2))\cdot\nabla( u_1+h_2)
- \nabla(\Delta(u_1+h_1))\cdot\nabla( u_1+h_1) \big|\\
&\leqslant c_\kappa {\big( 1+r^{\delta}r^2_{\varepsilon,\lambda}\big)}
\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
Since
\begin{align*}
&|\nabla(u_1+h_1)|^2\Delta(u_1 + h_1)
 - |\nabla(u_1+h_2)|^2\Delta(u_1+h_2)\\
& = \Delta(h_1-h_2)[ |
\nabla(u_1+h_1)|^2+|\nabla(u_1+h_2)|^2]\\
&\quad + \Delta(2 u_1+h_1+h_2)[ |
\nabla(u_1+h_1)|^2-|\nabla(u_1+h_2)|^2 ],
\end{align*}
it follows that
\begin{align*}
& r^{4-\delta}\big||\nabla(u_1+h_1)|^2\Delta(u_1+h_1)
 - |\nabla(u_1+h_2)|^2\Delta(u_1+h_2)\big|\\
&\leqslant c_\kappa {\big(1+r^{\delta}r^2_{\varepsilon,\lambda}+r^{2\delta}r_{\varepsilon,\lambda}^4
\big)}\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
 Its easy to see that
\begin{align*}
&\nabla(|\nabla(u_1+h_2)|^2)\nabla(u_1+h_2)
 - \nabla(|\nabla(u_1+h_1)|^2)\nabla(u_1+h_1)\\
& = \nabla(h_2-h_1)\nabla( | \nabla(u_1+h_2)|^2+|\nabla(u_1+h_1)|^2 )\\
&\quad + \nabla(2 u_1+h_1+h_2)\nabla(|\nabla(u_1+h_2)|^2-|\nabla(u_1+h_1)|^2 );
\end{align*}
hence
\begin{align*}
&r^{4-\delta}\big|\nabla(|\nabla(u_1+h_2)|^2)\nabla(u_1+h_2)
 - \nabla(|\nabla(u_1+h_1)|^2)\nabla(u_1+h_1)\Bigr|\\
&\leqslant c_\kappa {\big(1+r^{\delta}r^2_{\varepsilon,\lambda}+r^{2\delta}r_{\varepsilon,\lambda}^4
\big)}\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
Finally, since
\begin{align*}
&|\nabla(u_1+h_2)|^4- |\nabla(u_1+h_1)|^4\\
& = \nabla(h_2-h_1)\nabla(2u_1+h_2+h_1)( |
\nabla(u_1+h_2)|^2+|\nabla(u_1+h_1)|^2 ),
\end{align*}
it follows that
\begin{align*}
& r^{4-\delta}\Bigr||\nabla(u_1+h_2)|^4
- |\nabla(u_1+h_1)|^4\Bigr|\\
&\leqslant c_\kappa {\big(1+r^{\delta}r^2_{\varepsilon,\lambda}
+r^{2\delta}r_{\varepsilon,\lambda}^4+r^{3\delta}r_{\varepsilon,\lambda}^6
\big)}\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
Thanks to condition (A1),
$$
 {\sup_{r \leqslant
R_{\varepsilon,\lambda}}r^{4-\delta}\;|\mathscr{R}(h_2)-\mathscr{R}(h_1)|\leqslant
{c}_{\kappa}r^{2}_{\varepsilon,\lambda}
\|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}}.
$$
Similarly, by Proposition \ref{pr:surj-1} and \eqref{eq:6.3},
we conclude that given $\kappa>0$, then there exist
$\bar{c}_{\kappa} >0$ (independent of $\varepsilon$ and $\lambda$), $\lambda_\kappa $ and
$ \varepsilon_\kappa $ such that
\begin{equation}
\|{\mathscr N}(h_2)-{\mathscr N}(h_1)\|_{\mathcal{C}
 ^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}
\leqslant \bar c_\kappa  r^{2}_{\varepsilon,\lambda}
 \|h_2-h_1\|_{\mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\label{eq:vol}
\end{equation}
Reducing $\lambda_\kappa >0$ and $\varepsilon_{\kappa} > 0$ if necessary, we
can assume that
 $\bar c_\kappa  r^{2}_{\varepsilon,\lambda} \leqslant 1/2$  for all
$\lambda \in (0, \lambda_\kappa )$ and $\varepsilon \in (0, \varepsilon_\kappa )$. Then,
\eqref{eq:vol} and \eqref{eq:h} are enough
to show that
$h  \mapsto {\mathscr N} (h)$ is a contraction from
$\{ h \in \mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4):
 \|h \|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)} \leqslant 2
c_{\kappa}r^{2}_{\varepsilon,\lambda} \}$
into itself and hence has a unique fixed point $h$ in this set. This
fixed point is solution of \eqref{eq:T} in $\bar{B}_{R_{\varepsilon,\lambda}}$.
We summarize
this in the following proposition.

\begin{proposition} \label{prop2}
 Given $\delta \in (0,1)$ and $\kappa>0$, then there exist
$\varepsilon_\kappa > 0,\;\lambda_\kappa > 0$ and $\bar{c}_{\kappa}>0$ (depending on $\kappa$)
such that for all $\lambda \in (0 , \lambda_\kappa)$, and for $\varepsilon \in (0,  \varepsilon_\kappa)$,
there exists a unique solution
$h \in \mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)$
solution of \eqref{eq:T}  such that
\[
v(x)=u_1(x) +h(x)
\]
solves \eqref{eq:mah1} in $\bar{B}_{R_{\varepsilon,\lambda}}$. In addition
$$
\|h \|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)} \leqslant 2
c_{\kappa}r^{2}_{\varepsilon,\lambda}.
$$
\end{proposition}

\subsection{Analysis of the Bi-Laplace operator in weighted spaces}

 In this section, we prove a surjectivity result of the bi-laplace operator 
in some weighted spaces  and recall some estimations concerning the bi-harmonic
extensions.
  First, given $x^1, \ldots, x^m \in \Omega$ we
define
\[
\bar \Omega^* : = \bar \Omega - \{x^1, \ldots x^m\},
\]
and  we choose $r_0>0$ so that the balls $B_{r_0}(x^i)$ of center $x^i$ and radius
$r_0$ are mutually disjoint and included in $\Omega$. For 
$k \in \mathbb{N}$, $\alpha \in (0,1)$ and $\nu\in \mathbb{R}$, we
introduce the H\"older weighted space
${\mathcal{C}}^{k,\alpha}_{\nu}(\bar \Omega^*)$ as the space of
functions $w \in {\mathcal{C}}^{k,\alpha}_{\rm loc}(\bar \Omega^*)$
 endowed with the norm
\[
\| w \|_{{\mathcal{C}}^{ k, \alpha}_{\nu}(\bar \Omega^*)}
 : = \| w \|_{{\mathcal{C}}^{ k,\alpha}(\bar \Omega -
\cup_{j=1}^{m} B_{r_0/2}(x^j))}+ \sum_{j=1}^m \sup_{0 < r \leqslant
r_0/2} r^{-\nu}  \| w(x^j + r  \cdot ) \|_{{\mathcal{C}}^{
k,\alpha} (B_2-B_1)}.
\]
When $k \geqslant 2$, we let $[ \mathcal{C}^{k,\alpha}_\nu (\bar
\Omega^*) ]_0$ be the subspace of functions 
$w \in  \mathcal{C}^{k,\alpha}_\nu (\bar \Omega^*)$ satisfying
 $ w = \Delta w = 0$.

In this article, we need the following mapping properties
of $\Delta^2$.

\begin{proposition}[\cite{Bdop}] \label{pr:surj-2}
Assume that $\nu < 0$ and $\nu \not\in \mathbb{Z}$, then $$
\Delta^{2} : [\mathcal{C}^{4,\alpha}_\nu (\bar \Omega^*)]_0
\to \mathcal{C}^{0,\alpha}_{\nu-4} (\bar \Omega^*)$$
is surjective. 
\end{proposition}

\begin{remark}[\cite{Bdop}] \label{re:1} \rm % \label{remd} 
It is interesting to observe that, when $\nu <0$, $\nu \notin
{\mathbb Z}$, the right inverse even though it is not unique can be
chosen to depend smoothly on the points $x^1, \ldots, x^m$, at least
locally. Once a right inverse is fixed for some choice of the points
$x^1, \ldots, x^m$, a right inverse which depends smoothly on the
points $\tilde x^1, \ldots , \tilde x^m$ close to $x^1, \ldots, x^m$
can be obtained using a simple perturbation argument. 
\end{remark}

\begin{proof}[Proof of Proposition \ref{pr:surj-2}]
Given $(\tilde x^i)$ close enough to $(x^i)$, we  define a family of diffeomorphism 
$D : \Omega \to \Omega$ depending smoothly on $(\tilde x^i)$ by
\[
D (x) = x + \sum_{j=1}^m \chi_{r_0} (x-x^j)  (x^j- \tilde x^j),
\]
where $\chi_{r_0}$ is a cut-off function identically equal to $1$ in
$B_{r_0/2}$ and identically equal to $0$ outside $B_{r_0}$. 
Hence $D (\tilde x^j) = x^j$ for each $j$. Then the equation 
$\Delta^2 \tilde w  = \tilde f$ where  
$\tilde f \in \mathcal{C}^{0,\alpha}_{\nu-4} 
(\bar \Omega - \{\tilde x^i, 1 \leqslant i\leqslant m\})$ can be solved by
 considering $\tilde w  = w \circ D$ where $w$
is a solution of the problem
\begin{equation}
\Delta^2  w  + \big[ \Delta^2 (w \circ D)  - \Delta^2 w \circ D
\big] \circ D^{-1} = \tilde f \circ D^{-1} \label{eq:D^-1}
\end{equation}
and this time  $\tilde f \circ D^{-1} \in \mathcal{C}^{0,
\alpha}_\nu (\bar \Omega - \{x^1, \ldots, x^m\})$. It should be
clear that
\[
\left\| \big[ \Delta^2 (w \circ D)  - \Delta^2 w \circ D \big] \circ
D^{-1} \right\|_{\mathcal{C}^{0, \alpha}_{\nu -4} (\bar \Omega^*)}
\leqslant  C\| w\|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*)}
\sup_{j=1, \ldots, m} |\tilde x^j- x^j|.
\]
Since we have a fixed right inverse for 
$\Delta^2 :\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*) \to \mathcal{C}^{0,
\alpha}_{\nu -4} (\bar \Omega^*)$, a perturbation argument shows
that \eqref{eq:D^-1} is solvable provided the $\tilde x^j$ are
close enough to the $x^j$. This provides a right inverse which
depends smoothly on the choice of the points $\tilde x^i$.
\end{proof}

\subsection{Bi-harmonic extensions}
Now, we  give some estimates.
More precisely, given $\varphi \in \mathcal{C}^{4,\alpha}(S^3)$
and  $\psi \in \mathcal{C}^{2,\alpha}(S^3)$, we define 
$H^i (=H^i_{\varphi, \psi})$ to be the solution of
\begin{gather*}
\Delta^{2}  H^i= 0 \quad \text{in } B_1\\
H^i  = \varphi  \quad \text{on } \partial B_1\\
\Delta  H^i  = \psi \quad \text{on } \partial B_1,
\end{gather*}
where, as already mentioned, $B_1$ denotes the unit ball in
$\mathbb{R}^4$. Given $k\in {\mathbb N}$, $\alpha \in (0,1)$ and
$\nu \in \mathbb{R}$, we introduce the H\"older weighted spaces
$\mathcal{C}^{k,\alpha}_\nu(\bar B_1^*)$ as the space of function
in $\mathcal{C}^{k,\alpha}_{\rm loc}(\bar B_1^*)$ for which the
following norm
\[
\|u \|_{\mathcal{C}^{k,\alpha}_\nu(\bar B_1^*)} =
 {\sup_{r\leqslant 1/2}} r^{-\nu}  \| u(r  \cdot)
\|_{\mathcal{C}^{k,\alpha}(\bar B_2-B_1)}
\]
is finite. Here $\bar B_1^* = \bar B_1 - \{0\}$, therefore, this norm
corresponds to the norm already defined in the previous section
when $\Omega = B_1$, $m=1$ and $x^1=0$.  We denote by 
$e_1, \ldots, e_4$ the coordinate functions on $S^3$.

\begin{lemma}[\cite{Bdop}] \label{le:HI}
Assume that
\begin{equation}
\int_{S^{3}} (8  \varphi  - \psi ) \,d\sigma=0 \quad
\text{and} \quad \int_{S^{3}} (12  \varphi - \psi )
 e_\ell \,d\sigma=0 ,\label{eq:ortho-1}
\end{equation}
for $\ell = 1, \ldots , 4$. Then there exists $c> 0$ such that
\[
\| H^i_{\varphi, \psi} \|_{{\mathcal{C}}_{2}^{ 4,\alpha}( \bar
B_1^*)}\leqslant c  ( \| {\varphi} \|_{{\mathcal{C}}^{ 4,\alpha}(
S^3)} + \| {\psi} \|_{{\mathcal{C}}^{ 2,\alpha}(S^3)} ).
\]
\end{lemma}

Given $\varphi \in \mathcal{C}^{4,\alpha}(S^3)$ and 
$\psi \in \mathcal{C}^{2,\alpha}(S^3)$ we define (when it exists!)
$H^e(=H^e_{\varphi, \psi})$ to be a solution of
\begin{gather*}
\Delta^{2}  H^e = 0 \quad \text{in }  \mathbb{R}^4 - B_1\\
H^e  =  \varphi \quad  \text{on } \partial B_1\\
\Delta H^e = \psi \quad \text{on } \partial B_1.
\end{gather*}
which decays at infinity. Given $k\in {\mathbb N}$, 
$\alpha \in (0,1)$ and $\mu \in \mathbb{R}$, we introduce the H\"older
weighted spaces $\mathcal{C}^{k,\alpha}_\mu (\mathbb{R}^4 -B_1)$
as the space of function  
$w \in \mathcal{C}^{k,\alpha}_{\rm loc}(\mathbb{R}^4 -B_1)$ for which the 
norm
\[
\| w \|_{\mathcal{C}^{k,\alpha}_\mu(\mathbb{R}^4 -B_1)} =
 {\sup_{r\geq 1}}  r^{-\mu}  \|w(r
\cdot)\|_{\mathcal{C}^{k,\alpha}_\mu(\bar B_2-B_1)},
\]
is finite.

\begin{lemma}[\cite{Bdop}] \label{le:HE}
Assume that
\begin{equation}
\int_{S^{3}} \psi  \,d\sigma = 0. \label{eq:ortho-2}
\end{equation}
Then there exists $c > 0$ such that \[ \| H^e_{\varphi, \psi}
\|_{{\mathcal{C}}_{-1}^{ 4,\alpha} (\mathbb{R}^4 -B_1)}\leqslant c
(\|{\varphi}\|_{{\mathcal{C}}^{ 4,\alpha}( S^3)}+
\|{\psi}\|_{{\mathcal{C}}^{ 2,\alpha}(  S^3)} ).
\]
\end{lemma}

Observe that, under the hypothesis of the Lemma, there is
uniqueness of the bi-harmonic extension of the boundary data which
decays at infinity.

If $E\subset L^2(S^3)$ is a space of functions defined on $S^3$,
we define the space $E^\perp$ to be the subspace of functions
which are $L^2$-orthogonal to the functions $1, e_1, \ldots, e_4$.

\begin{lemma}[\cite{Bdop}] \label{lem4}
The mapping
\begin{align*}
\mathcal{P}:\; \mathcal{C}^{4,\alpha}(S^3)^\perp \times
\mathcal{C}^{2,\alpha}(S^3)^\perp  & \to 
\mathcal{C}^{3,\alpha}(S^3)^\perp \times \mathcal{C}^{1,\alpha}(S^3)^\perp  \\
 (\varphi ,\psi) & \mapsto  (\partial_{r}(H^i_{{{\varphi}}, { {\psi}}}-
 H^e_{{ \varphi}, {\psi}}),\partial_{r}(\Delta H^i_{{{\varphi}}, {
{\psi}}}-\Delta H^e_{{ \varphi}, { \psi}}))
\end{align*}
is an isomorphism. 
\end{lemma}

\section{Nonlinear interior problem}

We are interested in studying equations of type
\begin{equation}
\Delta^2  w + \mathscr{Q}_{\lambda }( w) - 24e^w = 0 \label{eq:03001}
\end{equation}
in $\bar{B}_{R_{\varepsilon,\lambda}}$.
Given $\varphi \in \mathcal{C}^{4,\alpha}(S^3)$ and  
$\psi \in \mathcal{C}^{2,\alpha}(S^3)$.
Let $\kappa >0$ (whose value will be fixed later on),
we further assume that the functions $\varphi$, $\psi$ satisfy
\begin{equation}
\|\varphi\|_{\mathcal{C}^{4, \alpha}}\leqslant \kappa
r_{\varepsilon,\lambda}^2 \quad \text{and} \quad \|\psi\|_{\mathcal{C}^{2,
\alpha}}\leqslant \kappa  r_{\varepsilon,\lambda}^2.
 \label{eq:cdfipsii}
\end{equation}
Define 
\[
{\bf v} : = u_1+ H^i(\varphi,\psi, \cdot/R_{\varepsilon,\lambda})+h,
\]
then we look for a solution of \eqref{eq:03001} of the form
$w={\bf v}+v$ and using the fact that $H^{i}$ is biharmonic, this
amounts to solving
\begin{equation}
\begin{aligned} 
\mathscr{L} v &= {\frac{384}{(1 + |x|^{2})^4}}\;e^{h}( e^{H^i(\varphi,\psi,
\cdot/R_{\varepsilon,\lambda})+ v} -
v - 1)+ {\frac{384}{(1 + |x|^{2})^4}}\;( e^{h} - 1)v\\
&\quad + \mathscr{Q}_{\lambda}(u_1 + h)
 - \mathscr{Q}_{\lambda}\Bigr(u_1+H^i(\varphi,\psi,\cdot/R_{\varepsilon,\lambda})+h+v\Bigr).
\end{aligned} \label{eq:zi}
\end{equation}
We fix $\mu \in (1,2)$
and denote by $\mathscr{G}_{\mu}$ the right inverse of
$\mathscr{L}_{\mu}$ provided by Proposition~\ref{pr:surj-1}. To
obtain a solution of \eqref{eq:zi} it is sufficient to find $v \in
\mathcal{C}^{4, \alpha}_{\mu} (\mathbb{R}^4)$ solution of
\begin{equation}
v = {\mathscr N}(v):=\mathscr{G}_{\mu}\circ{\mathcal
E}_{\mu}\circ{\mathscr S}(v),
\label{eq:302}
\end{equation}
where
\begin{equation}
\begin{aligned} 
{\mathscr S}(v)&:= {\frac{384}{(1 + |x|^{2})^4}}\;e^{h}
 ( e^{H^i(\varphi,\psi,\cdot/R_{\varepsilon,\lambda})+ v} -  v - 1)
 + {\frac{384}{(1 + |x|^{2})^4}}\;( e^{h} -1)v\\
&\quad +\mathscr{Q}_{\lambda}(u_1+h)
 - \mathscr{Q}_{\lambda}\Bigr(u_1+H^i(\varphi,\psi,\cdot/R_{\varepsilon,\lambda})+h+v\Bigr).
\end{aligned}\label{eq:zi1}
\end{equation}
We denote by ${\mathscr N}(=  {\mathscr N}_{\varepsilon,
\lambda,\varphi,\psi})$ the nonlinear operator appearing on the right-hand 
side of\eqref{eq:302}; then we have the following result.

\begin{lemma} \label{le:6.1}
For $\mu \in (1,2)$ and $\kappa > 0$, then there exist 
$\lambda_\kappa > 0$, $\varepsilon_\kappa >0$, $c_\kappa >0$ and $ \bar c_\kappa >0$ 
(depending on $\kappa$) such that for all 
$\lambda \in (0 , \lambda_\kappa)$ and $\varepsilon \in (0, \varepsilon_\kappa)$,
\begin{equation}
\| {\mathscr N}( 0) \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}
\leqslant c_\kappa r_{\varepsilon,\lambda}^2. \label{eq:6.6}
\end{equation}
Moreover,
\begin{equation}
\|  {\mathscr N}( v_2) - {\mathscr
N}( v_1) \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)}\leqslant \bar c_\kappa
r_{\varepsilon,\lambda}^2  \| v_2 - v_1 \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)}, \label{eq:6.7}
\end{equation}
provided that $ v_1, v_2 \in \mathcal{C}^{4, \alpha}_\mu
(\mathbb{R}^4)$,  satisfy
\[
\|  v_i \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} \leqslant
2  c_\kappa  r_{\varepsilon,\lambda}^2.
\]
\end{lemma}

\begin{proof}
The proof of the first estimate follows from the
asymptotic behavior of $H^i$ together with the assumption on the norm
of boundary data $\varphi$ and $\psi$ given by $\eqref{eq:cdfipsii}$. Indeed, let
$c_\kappa$ be a constant
depending only on $\kappa$ (provided $\varepsilon$ and $\lambda$ are chosen
small enough) it follows from the estimate of $H^i$, given by Lemma
\ref{le:HI}, that
\[
\| H^i (\varphi, \psi  ,  \cdot / R_{\varepsilon,\lambda}) \|_{\mathcal{C}^{4,
\alpha} _{2} ( \bar B_{R_{\varepsilon,\lambda}})} 
\leqslant  c  R_{\varepsilon,\lambda}^{-2}  (\|
\varphi \|_{\mathcal{C}^{4, \alpha}(S^3)} 
+ \| \psi \|_{\mathcal{C}^{2, \alpha}(S^3)}) 
\leqslant c_\kappa  \varepsilon^2.
\]
Since for each $x \in \bar B_{R_{\varepsilon,\lambda}}$, we have
\[
|h(x)|
\leqslant c_{\kappa} r_{\varepsilon,\lambda}^{2+\delta}\varepsilon^{-\delta}
\leqslant \begin{cases}
\varepsilon^{1-\delta/2} &\text{for } \varepsilon \geqslant \lambda \\
\lambda^{1+\delta/2}\varepsilon^{-\delta} &\text{for }\lambda > \varepsilon.
\end{cases}
\]
Then, using condition (A1) , we prove that $| h(x)| \to 0$ as $\varepsilon$ and
 $\lambda$ tend to $0$.
Given $\kappa > 0$, there exist $c_\kappa > 0$ such that
\[
\| (1+ |\cdot|^2)^{-2} e^h \big( e^{H^i(\varphi, \psi ,
\cdot / R_{\varepsilon,\lambda})} -1\big) \|_{\mathcal{C}^{0, \alpha}_{\mu-2}
(\bar B_{R_{\varepsilon, \lambda}})} \leqslant c_\kappa  \varepsilon^{2}.
\]
On the other hand, using  condition (A1), we obtain
\[
\sup_{r\leqslant R_{\varepsilon,\lambda}}(1 + r^{2})^{2-\frac{\mu}{2}}|
\mathscr{Q}_{\lambda}(u_1 + h)
- \mathscr{Q}_{\lambda}\big(u_1+H^i(\varphi,\psi,\cdot/R_{\varepsilon,\lambda})+h\big)|
\leqslant c_\kappa r^2_{\varepsilon,\lambda}.
\]
By Proposition~\ref{pr:surj-1} and
\eqref{eq:6.3},  for $\mu \in ( 1, \;2)$, we obtain
\begin{equation*}
\| {\mathscr N}( 0) \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}
\leqslant c_\kappa r_{\varepsilon,\lambda}^2.
\end{equation*}
To derive the second estimate, let 
$v_i\in \mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)$ satisfy 
$\| v_i \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} 
\leqslant 2  c_\kappa  r_{\varepsilon,\lambda}^2$, 
$i = 1, 2$, $ \mu \in ( 1, 2)$ and condition (A1). 
Hence there exist $c_\kappa > 0$ such that
\begin{gather*}
\begin{aligned}
&\| (1+ |\cdot|^2)^{-4}
 e^{H^i({\varphi, \psi}, \cdot / R_{\varepsilon,\lambda}) }  ( e^{v_2} -
e^{v_1} - (v_2 - v_1) ) \|_{\mathcal{C}^{0,
\alpha}_{\mu-4} (\bar B_{R_{\varepsilon,\lambda}})} \\
&\leqslant c_\kappa   \varepsilon^2
\| v_2 - v_1 \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}\,,
\end{aligned} \\
\|  ( e^{h}  -1 )  (v_2  - v_1 )\|_{\mathcal{C}^{0, \alpha}_{\mu-4} 
(\bar B_{R_{\varepsilon,\lambda}})} 
\leqslant c_\kappa   r_{\varepsilon,\lambda}^2 \| v_2 - v_1 \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)},
\\
\begin{aligned}
&\big\|\mathscr{Q}_{\lambda}\big(u_1+ H^i(\varphi,\psi,\cdot/R_{\varepsilon,\lambda})
+h+v_2\big)\\
&- \mathscr{Q}_{\lambda} \big(u_1+H^i(\varphi,\psi,\cdot/R_{\varepsilon,\lambda}) 
+ h + v_1\big)\big\|_{\mathcal{C}^{0, \alpha}_{\mu-4} (\bar B_{R_{\varepsilon,\lambda}})}\\
&\leqslant c_\kappa   r_{\varepsilon,\lambda}^2
 \big\| v_2 - v_1 \big\|_{\mathcal{C}^{4,\alpha}_\mu (\mathbb{R}^4)}.
\end{aligned}
\end{gather*}
So
$$
 {\sup_{r\leqslant R_{\varepsilon,\lambda}}(1 + r^{2})^{2-\frac{\mu}{2}}
 |{\mathscr S}(v_2)-{\mathscr S}(v_1)|
\leqslant {c}_{\kappa}r^{2}_{\varepsilon,\lambda}\|v_2-v_1\|
 _{\mathcal{C}^{4,\alpha}_{{\rm rad},\mu}(\mathbb{R}^4)}}.
$$

Similarly, using Proposition~\ref{pr:surj-1} and \eqref{eq:6.3}, we 
conclude that there exist $\bar c_\kappa > 0$ such that
$$
\|  {\mathscr N}( v_2) - {\mathcal
N}( v_1) \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)}\leqslant \bar c_\kappa
r_{\varepsilon,\lambda}^2  \| v_2 - v_1 \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)}.
$$
\end{proof}

Reducing $\lambda_\kappa >0$ and $\varepsilon_\kappa >0$ if necessary, we can assume that
\begin{equation}
\bar c_\kappa  r_{\varepsilon,\lambda}^2 \leqslant \frac{1}{2},\label{eq:6.9}
\end{equation} 
for all $\lambda \in  (0, \lambda_\kappa )$ and $\varepsilon \in (0, \varepsilon_\kappa )$. 
Then, \eqref{eq:6.6} and \eqref{eq:6.7} in Lemma~\ref{le:6.1} are sufficient
to show that
$v  \mapsto { \mathscr N} ( v)$ is a contraction from
\[
\big\{ v \in \mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)\; :\;
 \|v \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} \leqslant
2  c_\kappa  \varepsilon^{2} \big\}
\]
into itself and hence has a unique fixed point $ v = v (\varepsilon, \tau,
\varphi, \psi  ;  \cdot )$ in this set. This fixed point is
 a solution of \eqref{eq:302} in $\mathbb{R}^4$. We summarize
this in the following proposition.

\begin{proposition} \label{propint}
 For $\mu \in (1,2)$ and $\kappa >0$ there exist $\varepsilon_\kappa >0$, 
$\lambda_\kappa >0$ and $c_\kappa>0$ (depending on $\kappa$) such that 
for all $\varepsilon \in (0, \varepsilon_\kappa)$, $\lambda \in (0, \lambda_\kappa)$ 
satisfying {\rm (A1)}, for all $\tau$ in some fixed compact subset of 
$[\tau_-, \tau^+] \subset (0, \infty)$ and for a given $\varphi$ and 
$\psi$ satisfying \eqref{eq:ortho-1}-\eqref{eq:cdfipsii}, then there exists
 a unique $v (:= \bar{v}_{\varepsilon, \tau, \varphi, \psi})$ solution of \eqref{eq:302} such
that
\[ 
w : = u_1+ H^i(\varphi, \psi,\cdot/R_{\varepsilon,\lambda})+h+\bar{v}_{\varepsilon, \tau,
\varphi, \psi}
\]
solve \eqref{eq:03001} in $\bar{B}_{R_{\varepsilon,\lambda}}$. In addition
\[
\| v \|_{\mathcal{C}^{4,\alpha}_{\mu}(\mathbb{R}^4)} \leqslant 2
c_\kappa r^{2}_{\varepsilon,\lambda}.
\]
\end{proposition}


\section{Nonlinear exterior problem}

Denote $G_{\tilde x} = G(x, \tilde x)$ where $G$ is the Green
function given by \eqref{Green} and $H (x, \tilde x)$ its regular
part. Clearly $x \mapsto H(x, \tilde x)$ is a smooth function.

Let ${\tilde{\mathbf{x}}} = (\tilde x^j)\in \Omega^m$ close to 
$\mathbf{x} = (x^j)$, $\tilde{\boldsymbol{\eta}}
= (\tilde \eta^j)\in \mathbb{R}^m$ close
to $0$. Let $\tilde{\boldsymbol{\varphi}} = (\tilde \varphi^j) \in
(\mathcal{C}^{4, \alpha}(S^3))^m$ and $\tilde{\boldsymbol{\psi}}
= (\tilde \psi^j) \in (\mathcal{C}^{2, \alpha} (S^3))^m$
satisfy \eqref{eq:ortho-2}. We define
\begin{equation*} \tilde{\mathbf{u}} = \tilde u_{\varepsilon, \tilde{\boldsymbol{\eta}},
{\tilde{\mathbf{x}}}, \tilde{\boldsymbol{\varphi}}, \tilde{\boldsymbol{\psi}}} :=
\sum_{j=1}^m (1 + \tilde \eta^j ) G_{{\tilde x}^j} + \sum_{j=1}^m
\chi_{r_0} (x - {\tilde x}^j) H^{e}_{\tilde \varphi^j, \tilde
\psi^j}(\frac{x - {\tilde x}^j}{r_{\varepsilon,\lambda}}),
\end{equation*}
 where $\chi_{r_0}$ is a cut-off function identically equal to $1$ in
$B_{r_0/2}$ and identically equal to $0$ outside $B_{r_0}$. We would
like to solve the equation
\begin{equation}
\Delta^2 u + \mathscr{Q}_\lambda (u)
-\rho^4e^{u} =0,\quad \text{in } \Omega-\cup_{1\leqslant j\leqslant m}
 B_{r_{\varepsilon,\lambda}(\tilde x^j)},
\label{eq: ext}
\end{equation}
with $u = \tilde{\mathbf{u}} + \tilde v$ is a perturbation of 
$\tilde{\mathbf{u}}$. This amounts to solve
$$
\Delta^2 \tilde v
=\rho^4 e^{\bf\tilde{u}}\;e^{ \tilde v} 
- \mathscr{Q}_\lambda ( {\bf \tilde{u}} +  \tilde v)- \Delta^2 {\bf\tilde{u}} =
{\tilde{\mathscr S}}( \tilde v).
$$
Denote $ {\Omega_{R, \tilde x} = \Omega
- \cup_{1\leqslant j\leqslant m} B_R(\tilde x^j)}$ for any
$R > 0$. We denote by 
$\tilde{{\mathscr\xi}}_R : \mathcal{C}^{0, \alpha}_\nu
(\bar \Omega_{R, \tilde x}) \to \mathcal{C}^{0,
\alpha}_\nu (\bar \Omega^*)$ the extension operator defined by
\begin{gather*}
\tilde {\mathscr \xi}_R (f) \equiv  f \quad \text{in  } \Omega_{R, \tilde x},\\
\tilde {\mathscr \xi}_R (f)(x_i+x) =   \frac{2 |x|- R}{R}  f
( x_i + \frac{Rx }{|x|} ) \quad \text{in  } {B_{R}(\tilde x^j)
\backslash B_{R/2}(\tilde x^j)}, \quad \forall 1\leqslant j\leqslant m,\\
\tilde {\mathscr \xi}_R (f) \equiv  0 \quad \text{in  }  {\cup_j B_{R/2}(\tilde x^j)}.
\end{gather*}
It easy to check that there exist a constant $c = c(\nu)> 0$, 
only depending on $\nu$ such that
\begin{equation}
\| \tilde {\mathscr {\xi}_R } (w) \|_{\mathcal{C}^{0, \alpha}_\nu (\bar \Omega^*) }
\leqslant c \|w\|_{\mathcal{C}^{0, \alpha}_\nu (\bar{\Omega}_{R, \tilde x} )} .
\label{eq:est xi}
\end{equation}


We fix $ \nu \in (-1,0)$, and denote by $\tilde{\mathscr{G}}_\nu$
the right inverse provided by Proposition~\ref{pr:surj-2}. Clearly,
it is enough to find $\tilde v \in \mathcal{C}^{4, \alpha}_\nu
(\Omega^*)$ solution of
\begin{equation}
\tilde{v} = \tilde{\mathscr{G}}_\nu \circ
\tilde{\xi}_{r_{\varepsilon,\lambda}} \circ \tilde{\mathscr S}
 (\tilde v).\label{eq:invvext}
\end{equation}
We denote by $\tilde {\mathscr N}(\tilde v)$
$( = \tilde {\mathscr{N}}_{\varepsilon, \boldsymbol{\eta},
 \tilde{\mathbf{x}}, \tilde{\boldsymbol{\varphi}},
\tilde{\boldsymbol{\psi}} }(\tilde v))
= \tilde{\mathscr{G}}_\nu \circ \tilde{\xi}_{r_{\varepsilon,\lambda}} 
\circ \tilde{\mathscr{S}}(\tilde v)$, 
the nonlinear operator on the right-hand side.
 Even though this is not notified in the notation,
 $\tilde{\mathscr{G}}_\nu : \mathcal{C}^{0, \alpha}_{\nu-4} (\bar \Omega^*)
\to \mathcal{C}^{4, \alpha}_{\nu} (\bar \Omega^*)$ is
the right inverse defined in Remark \ref{re:1} with 
$\bar \Omega^* = \bar \Omega - \{\tilde x^1, \ldots, \tilde x^m\}$.

Given $\kappa >0$ (whose value will be fixed later on), we further
assume that, the functions $\tilde \varphi^j$ and $\tilde \psi^j$
satisfy
\begin{equation}
\| \tilde{\varphi}^j \|_{\mathcal{C}^{4, \alpha}}\leqslant \kappa
r^2_{\varepsilon, \lambda} \quad \text{and} \quad \|\tilde {\psi}^j \|_{\mathcal{C}^{2, \alpha}}\leqslant \kappa r^2_{\varepsilon,\lambda}, \quad \forall j =
1,\ldots, m. \label{eq:cdfipsie}
\end{equation}
Moreover, we assume that the parameters $\tilde \eta^j$ and the
points $\tilde x^j$ are chosen to verify
\begin{equation}
|\tilde \eta^j|\leqslant \kappa r^2_{\varepsilon,\lambda} \quad
\text{and} \quad r_{\varepsilon, \lambda}|\tilde x^j -x^j |
\leqslant \kappa r^2_{\varepsilon,\lambda}.
\label{eq:cdetaxj}
\end{equation}
Then the following result holds.

\begin{lemma}\label{lemext}
Given $ \nu \in (-1,0)$ and $\kappa>0$, there exist $\varepsilon_\kappa >0$ and 
$c_\kappa >0$ (depending on $\kappa$) such that for all $\varepsilon \in (0,\varepsilon_\kappa)$ 
and under the assumptions \eqref{eq:cdfipsie} and \eqref{eq:cdetaxj}, we have
\begin{gather*}
\| \tilde {\mathscr N} (0) \|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*
) }\leqslant c_\kappa r_{\varepsilon, \lambda}^{2}, 
\\
\| \tilde {\mathscr N} (\tilde v_2) - \tilde {\mathscr N} ( \tilde v_1)
\|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^* ) }\leqslant \bar{c}_{\kappa}
r_{\varepsilon, \lambda}^{2} \| \tilde v_2 - \tilde v_1 \|_{\mathcal{C}^{4,
\alpha}_\nu (\bar \Omega^*)},
\end{gather*}
provided that ${\tilde v}_1, {\tilde v}_2 
\in \mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*)$ and 
$\|{\tilde v}_i \|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*)}
\leqslant 2c_\kappa r_{\varepsilon, \lambda}^{2}$.
\end{lemma}

\begin{proof} 
The proof of the first estimate follows from the asymptotic behavior of 
$H_e$ together with the assumption on the norm of boundary data 
${\tilde{\varphi}_j}$ and ${\tilde{\psi}_j}$ given by \eqref{eq:cdfipsie}. 
Indeed, let $c_{\kappa}$ be a constant depending only on $\kappa$ 
(provided $\varepsilon$ and $\lambda$ are chosen small enough), 
it follows from the estimate of $H_e$, given by Lemma \ref{le:HE}, that
\begin{equation}
\big|H^e_{\tilde{\varphi}_j,\tilde{\psi}_j}
\big(\frac{x-\tilde{x}^j}{r_{\varepsilon,\lambda}}\big)\big|
\leqslant c_\kappa\;r^3_{\varepsilon,\lambda}\;r^{-1}.
\label{eq:He ext}
\end{equation}
Recall that $\tilde {\mathscr N}  (\tilde v) 
= \tilde{\mathscr{G}}_\nu \circ {\tilde{\xi}}_{r_\varepsilon} 
\circ \tilde{\mathscr S}(\tilde v)$, we
will estimate $\tilde {\mathscr N}(0)$ in different subregions of
$\bar\Omega^*$.

 $\bullet$ In $B_{r_0}(\tilde{x}^j)$ for $1\leqslant j \leqslant m$,
we have  $\chi_{r_0}(x-\tilde{x}^j)=1$ and $\Delta^2{\bf\tilde{u}}=0$,
 so that
\begin{align*}
  |\tilde{{\mathscr S}}(0)| 
&\leqslant c \varepsilon^4 {\prod_{j=1}^{m}\big[e^{(1 + \tilde \eta^j )
  G_{{\tilde x}^j}(x) +
H^{e}_{\tilde \varphi^j, \tilde \psi^j}((x - {\tilde x}^j)/
r_\varepsilon )}} - \mathscr{Q}_\lambda ( \bf\tilde{u} )\big]\\
&\leqslant  c\varepsilon^4\prod_{j=1}^m|x-\tilde{x}^j|^{-8(1+\tilde{\eta}^j)} 
 +|\mathscr{Q}_\lambda ( \bf\tilde{u} )|.
\end{align*}
So, by an easy computation, for $ \nu \in (-1,0)$ and $\tilde{\eta}^j$ 
small enough, we obtain
\begin{equation*}
 {\|\tilde{{\mathscr S}}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}
\big(\cup_{j=1}^{m}B(\tilde{x}^j,r_0)\big)}}
\leqslant \sup_{r_{\varepsilon,\lambda} \leqslant r\leqslant
r_0/2}r^{4-\nu}|\tilde{\mathscr{S}}(0)|
\leqslant c_{\kappa}\big(\varepsilon^4 r_{\varepsilon,\lambda}^{-4} +  \lambda\big).
\end{equation*}

$\bullet$ In $  {\Omega- B_{r_0}(\tilde x^j)}$, we have
$\chi_{r_0}(x-\tilde{x}^j)=0$ and $\Delta^2{\bf \tilde{u}} =0$, then
$$|\tilde{{\mathscr S}}(0)| 
\leqslant c \big(\varepsilon^4 \prod_{j=1}^m e^{(1 + \tilde \eta^j)G_{{\tilde
  x}^j}}+|\mathscr{Q}_\lambda ( \bf\tilde{u} )|\big).
$$
Thus
$$
\|\tilde{{\mathscr S}}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}(\Omega_{r_0,
\tilde x})} 
\leqslant c_\kappa \sup_{ r\geqslant r_0}r^{4-\nu}  |\tilde{{\mathscr S}}(0)|
 \leqslant c_\kappa\big(\varepsilon^4  +  \lambda\big).
$$

$\bullet$ In $B_{r_0}(\tilde{x}^j)-B_{r_0/2}(\tilde{x}^j)$, using 
 estimate \eqref{eq:He ext},  we have
\begin{align*}
|\tilde{{\mathscr S}}(0)|
&\leqslant c_\kappa
\varepsilon^4\prod_{j=1}^m|x-\tilde{x}^j|^{-8-\tilde{\eta}^j}
 + |  \mathscr{Q}_\lambda ( \bf\tilde{u} )| \\
&\quad + c\varepsilon^4 \sum_{j=1}^{m}
 |[\Delta^2,\chi_{r_0}](x-\tilde{x}^j)||H^{\rm ext}_{\tilde{\varphi}_j,
\tilde{\psi}_j}  ((x-\tilde{x}^j)/r_{\varepsilon,\lambda})|.
 \end{align*}
Here
$$
[\Delta^2,\chi_{r_0}]w=2\Delta
\chi_{r_0}\Delta w+w\Delta^2 \chi_{r_0}+4\nabla
\chi_{r_0}\cdot\nabla(\Delta w)+4\nabla w\cdot\nabla(\Delta
\chi_{r_0})+4\nabla^2\chi_{r_0}\cdot\nabla^2 w.
$$
 So,
$$
\|\tilde{\mathscr S}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}(B(\tilde{x}^j,r_0)
-B(\tilde{x}^j,r_0/2))}
\leqslant c_\kappa \sup_{r_0/2\leqslant r\leqslant
r_0}r^{4-\nu}|\tilde{\mathscr S}(0)| 
\leqslant c_\kappa \big( r^{2}_{\varepsilon, \lambda} + \lambda \big).
$$
 Finally, using Proposition \ref{pr:surj-2} with \eqref{eq:est xi}, we conclude that
\begin{equation}
\|\tilde{\mathscr N}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}
(\bar\Omega^*)}\leqslant c_\kappa
r^{2}_{\varepsilon, \lambda}.
\label{eq:next0}
\end{equation}

For the proof of the second estimate, let $\tilde{v}_1$ and 
$\tilde{v}_2\ \in C^{4, \alpha}_\nu(\bar\Omega^*)$ satisfying 
$\|\tilde {v}_i\|_{\mathcal{C}^{4,\alpha}_{\nu}} 
\leqslant c_\kappa r^2_{\varepsilon, \lambda}$, so
\begin{equation*}
|\big(\tilde{\mathscr S}(\tilde v_2) - \tilde{\mathscr S}(\tilde v_1)\big)|
\leqslant c_{\kappa}|\rho^4e^{{\bf\tilde{u}}}(e^{\tilde v_2}-
e^{\tilde v_1})- \big(\mathscr{Q}_\lambda ( {\bf\tilde{u}} + \tilde{v}_2 ) 
- \mathscr{Q}_\lambda ( {\bf\tilde{u}} + \tilde{v}_1 ) \big)|.
\end{equation*}
Then, for $\tilde \eta^j$ small enough and using estimate \eqref{eq:est xi}, 
there exist $\bar c_\kappa > 0$ (depending on $\kappa$) such that
\begin{equation}
\|\tilde {\mathscr  N} (\tilde v_2)-\tilde {\mathscr N} (\tilde
v_1)\|_{\mathcal{C}^{4,\alpha}_{\nu}(\bar\Omega^*)}
\leqslant \bar c_\kappa r^2_{\varepsilon, \lambda}\|\tilde v_2-\tilde
v_1\|_{\mathcal{C}^{4,\alpha}_{\nu} (\bar\Omega^*)}.
\label{eq:next}
\end{equation}
Then we get the second estimate.
\end{proof}


Reducing $\lambda_\kappa >0$ and $\varepsilon_\kappa >0$ if necessary, we can assume that
\begin{equation}
\bar c_\kappa  r_{\varepsilon,\lambda}^2  \leqslant \frac{1}{2}\,,\label{eq:6.9b}
\end{equation} 
for all $\lambda \in  (0, \lambda_\kappa )$ and $\varepsilon \in (0, \varepsilon_\kappa )$. Then,
\eqref{eq:next} and \eqref{eq:next0} are sufficient
to show that
${\tilde v} \mapsto {\tilde { \mathscr N}} ({\tilde v})$
is a contraction from
\[
\Big\{{\tilde v} \in \mathcal{C}^{4, \alpha}_\nu (\mathbb{R}^4)  \; :
\; \|{\tilde v} \|_{\mathcal{C}^{4, \alpha}_\nu (\mathbb{R}^4)} \leqslant
2  c_\kappa  r^2_{\varepsilon, \lambda} \Big\}
\]
into itself and hence has a unique fixed point 
$ {\tilde v} = {\tilde v} (\varepsilon, \tau, \varphi, \psi  ;  \cdot )$ in this set. 
This fixed point is  a solution of \eqref{eq:invvext} in $\mathbb{R}^4$.
We summarize this in the following proposition.

\begin{proposition}\label{CD2}
Given $ \nu \in (-1,0)$ and $\kappa >0$, there exist 
$\varepsilon_\kappa >0$, $\lambda_\kappa >0$ and
$c_\kappa>0$ (depending on $\kappa$) such that for all
$\varepsilon \in (0, \varepsilon_\kappa)$ and $\lambda \in (0, \lambda_\kappa)$, for all set of
parameters $\tilde \eta^j$ and points $\tilde x^j$ satisfying
\eqref{eq:cdetaxj}, all functions $\tilde \varphi^j$, 
$\tilde \psi^j$ satisfying \eqref{eq:ortho-2} and \eqref{eq:cdfipsie}, there
exists a unique $\tilde v$ 
$(= \tilde v_{\varepsilon, \boldsymbol{\eta} , {\tilde{\mathbf{x}}},
 \tilde{\boldsymbol{\varphi}} , \tilde{\boldsymbol{\psi}}})$ solution of
\eqref{eq:invvext}, such that
 \[ 
\tilde u_{\varepsilon, \boldsymbol{\eta} , {\tilde{\mathbf{x}}}, 
\tilde{\boldsymbol{\varphi}} , \tilde{\boldsymbol{\psi}}} 
:= \sum_{j=1}^m (1 + \tilde \eta^j )  G_{{\tilde
x}^j} + \sum_{j=1}^m \chi_{r_0} (\cdot - {\tilde x}^j)
H^{e}_{\tilde \varphi^j, \tilde \psi^j}(\frac{x - {\tilde
x}^j}{r_\varepsilon})
 +  \tilde v_{\varepsilon, \tilde \eta , \tilde x,
\tilde \varphi, \tilde \psi}
\]
solves \eqref{eq: ext} in $\bar\Omega^*$. In addition
$$
\| \tilde v \|_{\mathcal{C}^{4,\alpha}_{\nu}(\bar \Omega^*)} \leqslant 2c_\kappa 
r_{\varepsilon, \lambda}^{2}.
$$
\end{proposition}

As in the previous section, observe that the function $ \tilde v_{\varepsilon, \tilde
\eta, \tilde x, \tilde \varphi, \tilde \psi}$ being obtained as a
fixed point for contraction mapping, it depends smoothly on the
parameters $\tilde \eta^j $, the points $\tilde x^j$ and the
boundary data $\tilde \varphi^j$ and $\tilde \psi^j$, for $j=1,
\ldots, m$. Moreover, as in the previous section, the mapping
$$
(\boldsymbol{\eta}, {\tilde{\mathbf{x}}}, 
\tilde{\boldsymbol{\varphi}}, \tilde{\boldsymbol{\psi}}) \mapsto 
\tilde v_{\varepsilon, \boldsymbol{\eta}, {\tilde{\mathbf{x}}}, 
\tilde{\boldsymbol{\varphi}}, \tilde{\boldsymbol{\psi}}} 
\circ D^{-1} |_{\Omega_{r_{\varepsilon,\lambda}, \tilde x}} 
\in \mathcal{C}^{4, \alpha} (\Omega_{r_{\varepsilon,\lambda}, \tilde x})
$$ 
is compact (here $D$ is the diffeomorphism defined in \S 2.2). Again
this follows from the fact that the equation we solve is semilinear
and in \eqref{eq:invvext} the right hand side belongs to  
$\mathcal{C}^{8, \alpha}(\bar \Omega^* )$.

\section{Nonlinear Cauchy-data matching}

We will gather the results of the previous sections, keeping the
notations, applying the result of \S~2, \S~3, as well as the
results of \S~4.
Assume that ${\tilde{\mathbf{x}}} = (\tilde x^i) \in \Omega^m$ are given
close enough to $\mathbf{x} = (x^i)$ such that it satisfies
\eqref{eq:cdetaxj}, assume also 
$\boldsymbol{\tau} = (\tau^i) \in [\tau^-,\tau^+ ]^m \subset (0, \infty)^m$ 
(the values of $\tau^-$ and
$\tau^+$ will be fixed shortly). First, we consider some set of
boundary data ${\bf \varphi} = (\varphi^i) \in (\mathcal{C}^{4,\alpha}(S^3))^m$ 
and ${\bf \psi} = (\psi^i) \in (\mathcal{C}^{2,\alpha}(S^3))^m$ satisfying
\eqref{eq:ortho-1} and \eqref{eq:cdfipsii}. According to 
Proposition~\ref{propint}, and provided $\varepsilon \in (0, \varepsilon_\kappa)$, we
can find a solution of
\[
\Delta^{2}u  + \mathscr{Q}_\lambda (u) - \rho^4e^{u} = 0 \quad
\text{in } B_{r_{\varepsilon,\lambda}}(\tilde x^j) \; \forall 1\leqslant
j\leqslant m.
\]
These solutions can be decomposed (in each $B_{r_{\varepsilon,\lambda}}(\tilde
x^j)$) as
\begin{align*}
u_{int,j}(x) 
&= u_{\varepsilon,\tau^j}(x-{\tilde x}^j) +
h\Big(\frac{R_{\varepsilon,\lambda}^j (x - {\tilde x}^j)}{r_{\varepsilon,\lambda}}\Big) 
+ H^{i}_{ \varphi^j, \psi^j}\Big(\frac{x-{\tilde x}^j}{r_{\varepsilon,\lambda}} \Big)\\
&\quad + v_{\varepsilon, \tau^j, \varphi^j, \psi^j}\Big(\frac{R_{\varepsilon,\lambda}^j (x - {\tilde
x}^j)}{r_{\varepsilon,\lambda}}\Big)
\end{align*}
 where $R_{\varepsilon,\lambda}^j = \tau^j r_{\varepsilon, \lambda} / \varepsilon$ and the function 
$v^j = v_{\varepsilon, \tau^j, \varphi^j, \psi^j} $ satisfies
\begin{equation}
\| v^j \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} \leqslant
2\; c_\kappa r_{\varepsilon, \lambda}^{2}. \label{Est1}
\end{equation}

Similarly, given some boundary data $\tilde{\boldsymbol{\varphi}} : =
(\tilde \varphi^i) \in (\mathcal{C}^{4,\alpha}(S^3))^m$ and
$\tilde{\boldsymbol{\psi}} = (\tilde \psi^i) \in (\mathcal{C}^{2,\alpha}(S^3))^m$ 
satisfying \eqref{eq:ortho-2} and
\eqref{eq:cdfipsie}, some parameters $\tilde{\boldsymbol{\eta}} : = (\tilde
\eta^i) \in \mathbb{R}^m $ satisfying \eqref{eq:cdetaxj}, we can
use  Proposition \ref{CD2} to find a solution $u_{ext}$
 (provided $\varepsilon \in (0, \varepsilon_\kappa)$) of
\[
\Delta^{2}u  + \mathscr{Q}_\lambda (u) - \rho^4e^{u} = 0, \quad
\text{in  } B_{r_{\varepsilon,\lambda}}(\tilde x^j), \; \forall 1\leqslant
j\leqslant m.
\]
Here the solution can be decomposed as  
\[ 
u_{ext}(x) = \sum_{j=1}^m (1+
\tilde \eta^j)  G_{\tilde{x}^j}(x) + \sum_{j=1}^m \chi_{r_0} (x -
{\tilde x}^j)H^{e}_{\tilde{{\varphi}}^j, {
\tilde{\psi}}^j}\Big(\frac{x-{\tilde x}^j}{r_{\varepsilon,\lambda}} \Big) 
+ {\tilde v}_{\varepsilon, \tilde{\boldsymbol{\eta}}, {\tilde{\mathbf{x}}},  
 \tilde{\boldsymbol{\varphi}}, \tilde{\boldsymbol{\psi}}}(x),
\]
where the function $\tilde v^j : ={\tilde v}_{\varepsilon, \tilde{\boldsymbol{\eta}},
{\tilde{\mathbf{x}}},  \tilde{\boldsymbol{\varphi}}, \tilde{\boldsymbol{\psi}}} \in
\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*)$ satisfies
\begin{equation}
\| \tilde v^j \|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*)}
\leqslant c_\kappa r^2_{\varepsilon, \lambda}. \label{Est2}
\end{equation}
It remains to determine the parameters and the functions is such a
way that the function which is equal to $u_{int, j}$ in $B_{r_{\varepsilon,\lambda}}
(\tilde x^j)$ and which is equal to $u_{ext}$ in $\Omega_{r_{\varepsilon,\lambda},
\tilde x}$ will become a smooth function. This amounts to find the
boundary data and the parameters so that, for each $j=1, \ldots, m$
\begin{equation}
u_{int,j} = u_{ext}, \quad 
\partial_r u_{int, j} = \partial_r u_{ext}, \quad 
\Delta u_{int,j} = \Delta u_{ext},\quad
\partial_r \Delta u_{int, j} = \partial_r \Delta u_{ext}
\label{eq:mcd}
\end{equation}
on $\partial B_{r_{\varepsilon,\lambda}} (\tilde x^j)$. Assuming we have already
\eqref{eq:mcd} (for all $\varepsilon$ small enough), the function 
$u_\varepsilon \in \mathcal{C}^{4, \alpha}$ obtained by patching together the
functions $u_{int, j}$ and the function $u_{ext}$, is a solution of
our equation. Then the elliptic regularity theory implies that this
solution is in fact smooth. This will complete the proof of our
result. Because when as $\varepsilon$ tends to $0$, the sequence of solutions
constructed will satisfy the required properties, namely, away from
the points $x^j$ the sequence $u_\varepsilon$ converges to $\sum_j G_{x^j}$.
Before we proceed, the following remarks are important. It will be
convenient to observe that the functions $u_{\varepsilon, \tau^j}$ can be
expanded as
\begin{equation}
u_{\varepsilon, \tau^j} (x) = - 8 \log |x| - 4 \log \tau^j + \mathcal{O}(r_{\varepsilon,\lambda}^2) \label{exp1} \end{equation}
 near $\partial B_{r_{\varepsilon,\lambda}}$.
Moreover, the function
\[
\sum_{1 \leqslant j \leqslant m} (1+ \tilde \eta^j)
G_{\tilde{x}^j}(x)
\]
which appears in the expression of $u_{ext}$ can be expanded as
 \begin{equation} \sum_{\ell=1}^m (1+ \tilde \eta^\ell)
 G_{\tilde{x}^\ell}(\tilde x^j+x ) = - 8(1+ \tilde \eta^j)\log
|x| + E_j (\tilde x^j, {\tilde{\mathbf{x}}}) + \nabla_x E_j (\tilde x^j,
{\tilde{\mathbf{x}}}) \cdot x + \mathcal{O} (r_{\varepsilon,\lambda}^2) \label{exp2}
\end{equation} near $\partial B_{r_{\varepsilon,\lambda}}$, where we define
\[
E_j(x, {\tilde{\mathbf{x}}}) := H(x, \tilde x^j )+ \sum_{\ell \neq j} G
(x, \tilde x^\ell).
\]

Next, in \eqref{eq:mcd}, all functions are defined on 
$\partial B_{r_{\varepsilon,\lambda}} (\tilde x^j)$, nevertheless, 
it will be convenient to solve, instead
of \eqref{eq:mcd} the following set of equations
\begin{equation}
\begin{gathered}
(u_{int,j} -u_{ext} )(\tilde x^j + r_{\varepsilon,\lambda} y)   = 0,  \quad
\partial_r (u_{int,j} -u_{ext}) (\tilde x^j + r_{\varepsilon,\lambda} y)  = 0,
\\
\Delta (u_{int,j} -u_{ext}) (\tilde x^j + r_{\varepsilon,\lambda} y)  = 0 ,  \quad
\partial_r \Delta (u_{int,j} -u_{ext} ) (\tilde x^j + r_{\varepsilon,\lambda} y)  =
0,
\end{gathered} \label{eq:mcdm}
\end{equation}
on $S^3$.

Also we decompose 
\[
\varphi^j = \varphi^j_0 + \varphi^j_1 + \varphi^{j}_{\perp}, \quad
\psi^j = 8 \varphi^j_0 + 12\varphi^j_1 + \psi^{j}_{\perp}, \quad
\tilde \varphi^j = \tilde \varphi^j_0 + \tilde \varphi^j_1 + \tilde
\varphi^{j}_{\perp} \quad
\tilde \psi^j = \tilde \psi^j_1 + \tilde \psi^{j}_{\perp}
\]
where $\varphi^j_0, \tilde \varphi_0 \in \mathbb{E}_0 = \mathbb{R}$, 
$\varphi^j_1, \tilde \varphi_1^j, \tilde \psi^j_1 \in 
\mathbb{E}_1 = \operatorname{span} \{e_1, \ldots, e_4\}$ and 
$\varphi^{j}_{\perp} , \psi^{j}_{\perp}, \tilde
\varphi^{j}_{\perp}, \tilde \psi^{j}_{\perp} \in L^2 (S^3)^\perp$,
the subspace of functions which are orthogonal to $\mathbb{E}_0$ and $\mathbb{E}_1$.

Projecting the set of equations \eqref{eq:mcdm} over $\mathbb{E}_0$ will
yield the system
\begin{equation} 
\begin{gathered} 
- 4 \log \tau^j - 8 \log r_{\varepsilon,\lambda} + \varphi_0^j +
8  (1+ \tilde \eta^j) \log r_{\varepsilon,\lambda} -\tilde
\varphi^j_0 - E_j (\tilde x^j, {\tilde{\mathbf{x}}}) + \mathcal{O}
(r_{\varepsilon,\lambda}^2) = 0
\\
- 8 + 2\varphi_0^j  +  8  (1+ \tilde \eta^j)  + 2 \tilde
\varphi^j_0  + \mathcal{O} (r_{\varepsilon,\lambda}^2)  =0
\\
- 16  + 8 \varphi_0^j + 16  (1+ \tilde \eta^j) + \mathcal{O}(r_{\varepsilon,\lambda}^2)  =0
\\
32 - 32(1+ \tilde \eta^j) + \mathcal{O} (r_{\varepsilon,\lambda}^2)  =0.
\end{gathered} \label{eq:mode-0}
\end{equation}
For the rest of this article, the terms $\mathcal{O} (r_{\varepsilon,\lambda}^2)$ depend
nonlinearly on the variables $\tau^\ell, \tilde x^\ell,
\varphi^\ell, \psi^\ell$, $\tilde\varphi^\ell, \tilde\psi^\ell$,
but it is bounded (in the appropriate norm) by a constant
(independent of $\varepsilon$ and $\kappa$) time $r_{\varepsilon,\lambda}^2$. 
Let us comment briefly on how these equations are obtained. These equations
simply come from \eqref{eq:mcdm} when expansions \eqref{exp1} and
\eqref{exp2} are used, together with the expression of $H^i$ and
$H^{e}$ given in Lemma~\ref{le:HI} and Lemma~\ref{le:HE}, and
also the estimates \eqref{Est1} and \eqref{Est2}.

Observe that the projection of the term $\nabla_x E_j (\tilde x^j,
{\tilde{\mathbf{x}}}) \cdot y$ arising in \eqref{exp2}, as well as the
projection of its partial derivative with respect to $r$, over the
set of constant function is equal to $0$, while its Laplacian
vanishes identically. The system \eqref{eq:mode-0} can be readily
simplified to
\begin{gather*}
\frac{1}{\log r_{\varepsilon, \lambda}} 
[4 \log \tau^j + E_j (\tilde x^j, {\tilde{\mathbf{x}}}) ]
 = \mathcal{O}(r_{\varepsilon,\lambda}^2), \quad 
\tilde \eta^j = \mathcal{O}(r_{\varepsilon,\lambda}^2), \\
\varphi^j_0 = \mathcal{O}(r_{\varepsilon,\lambda}^2),\quad
\tilde \varphi_0^j = \mathcal{O}(r_{\varepsilon,\lambda}^2).
\end{gather*}
We are now in a position to define $\tau^-$ and
$\tau^+$ since, according to the above, as $\varepsilon$ tends to $0$ we
expect that $\tilde x^j$ will converge to $x^j$ and that $\tau^j$
will converge to $\tau^j_*$ satisfying
\[
4 \log \tau^j_* = - E_j(x^j, \mathbf{x})
\]
and hence it is enough to choose $\tau^-$ and $\tau^+$ in such a way
that
\[
4 \log(\tau^-)  <  -\sup_{j} E_j(x^j, \mathbf{x}) \leqslant  - \inf_j
E_j(x^j, \mathbf{x})  < 4 \log(\tau^+).
\]
We now consider the $L^2$-projection of \eqref{eq:mcdm} over $\mathbb{E}_1$.
Given a smooth function $f$ defined in $\Omega$, we identify its
gradient $\nabla f = (\partial_{x_1} f, \ldots, \partial_{x_4} f)$ with the
element of $\mathbb{E}_1$
\[
\bar \nabla f = \sum_{i=1}^4 \partial_{x_i} f  e_i.
\]
With these notation in mind, we obtain the system of equations
\begin{equation}
\begin{gathered} 
\varphi^j_1 - \tilde \varphi^j_1 - \bar \nabla E_j
(\tilde x^j, {\tilde{\mathbf{x}}}) + \mathcal{O}(r_{\varepsilon,\lambda}^2)  =0 \\
3\varphi^j_1 + 3\tilde \varphi^j_1 + \frac{1}{2}\tilde \psi^j_1 -
\bar \nabla E_j (\tilde x^j, {\tilde{\mathbf{x}}}) 
 + \mathcal{O}(r_{\varepsilon,\lambda}^2)  =0 \\
12 \varphi^j_1 - \tilde \varphi^j_1 + \mathcal{O}(r_{\varepsilon,\lambda}^2)  =0 \\
12 \varphi^j_1 + 3\tilde \varphi^j_1 + \mathcal{O}(r_{\varepsilon,\lambda}^2)  =0.
\end{gathered} \label{eq:mode-1}
\end{equation}
Again, let us comment briefly on how these equations are obtained.
This time, the only important observation is that the term
 $\nabla_x E_j (\tilde x^j, {\tilde{\mathbf{x}}}) \cdot y$ 
projects identically over
$\mathbb{E}_1$ as well as its derivative with respect to $r$.

The system \eqref{eq:mode-1} simplifies to
$$
\varphi^j_1 
= \mathcal{O}(r_{\varepsilon,\lambda}^2), \quad 
\psi^j_1 = \mathcal{O}(r_{\varepsilon,\lambda}^2), \quad 
\tilde \psi^j_1  = \mathcal{O}(r_{\varepsilon,\lambda}^2), \quad
 \bar \nabla E_j (\tilde x^j, {\tilde{\mathbf{x}}}) =
\mathcal{O}(r_{\varepsilon,\lambda}^2).
$$
Finally, we consider the $L^2$-projection onto $L^2(S^3)^\perp$.
This yields the system
\begin{equation}
 \begin{gathered}
\varphi^{j}_{\perp} -\tilde \varphi^{j}_{\perp} 
 + \mathcal{O}(r_{\varepsilon,\lambda}^2)  =0 
\\
\partial_r \big(H^i_{\varphi^{j}_{\perp} , \psi^{j}_{\perp}} -
H^e_{\tilde \varphi^{j}_{\perp}, \tilde \psi^{j}_{\perp}}\big) +
\mathcal{O}(r_{\varepsilon,\lambda}^2) = 0 
\\
\psi^{j}_{\perp} - \tilde \psi^{j}_{\perp} + \mathcal{O}(r_{\varepsilon,\lambda}^2)  =0 
\\
\partial_r \Delta \big(H^i_{\varphi^{j}_{\perp}, \psi^{j}_{\perp}}
- H^e_{\tilde \varphi^{j}_{\perp} ,\tilde \psi^{j}_{\perp}}\big) +
\mathcal{O}(r_{\varepsilon,\lambda}^2)  =0.
\end{gathered}
\label{eq:mode-j}
\end{equation}
Thanks  the Lemma \ref{lem4}, this last system can be
re-written as
\[
\varphi^{j}_{\perp} = \mathcal{O}(r_{\varepsilon,\lambda}^2), \quad
\psi^{j}_{\perp} = \mathcal{O}(r_{\varepsilon,\lambda}^2).
\]
If we define the parameters $\mathbf{t} = (t^j) \in \mathbb{R}^m$ by
\[
t^j = \frac{1}{\log r_{\varepsilon, \lambda}} \big[ 4 \log \tau^j + E_j (\tilde x^j,
{\tilde x}) \big], \quad \forall 1\leqslant j\leqslant m.
\]
Then the system we have to solve reads
\begin{equation}
\Big(\mathbf{t}, \tilde{\boldsymbol{\eta}}, \boldsymbol{\varphi}_0  ,{\bf  \tilde
\varphi_0}, \boldsymbol{\varphi}_1 , {\bf \tilde \varphi_1}, {\bf \tilde
\psi_1}, \bar \nabla E ({\tilde{\mathbf{x}}}), {\bf  \varphi_\perp} , {\bf
\tilde \varphi_\perp}, {\bf \psi_\perp}, {\tilde \psi_\perp}\Big) =
\mathcal{O}(r_{\varepsilon,\lambda}^2), 
\label{theequation}
\end{equation}
where as usual, the terms $\mathcal{O}(r_{\varepsilon,\lambda}^2)$ depend
nonlinearly on all the variables on the left side, but is bounded
(in the appropriate norm) by a constant (independent of $\varepsilon$ and
$\kappa$) time $r_{\varepsilon,\lambda}^2$, provided $\varepsilon \in (0, \varepsilon_\kappa)$.

We claim, provided that the degree of the mapping
\begin{equation}
\bar \nabla E : {\tilde{\mathbf{x}}} \mapsto( \bar \nabla E_1 (\tilde
x^1; {\tilde{\mathbf{x}}}), \ldots, \bar \nabla  E_m (\tilde x^m; {\tilde{\mathbf{x}}})), \label{eq:map}
\end{equation}
from a neighborhood of $\mathbf{x} \in \Omega^m$ to a neighborhood of
$0$ in $\mathbb{E}_1^m$ is equal to $1$, this nonlinear system can be solve
using Schauder's fixed point theorem in the ball of radius $\kappa
r_{\varepsilon, \lambda}^2$ in the product space where the entries live, namely
\begin{gather*}
\mathbf{t}, \boldsymbol{\eta} \in  \mathbb{R}^m ; \quad 
r_{\varepsilon, \lambda} ({\tilde{\mathbf{x}}} - \mathbf{x}) \in (\mathbb{R}^4)^m ; \quad 
\boldsymbol{\varphi}_0 , \tilde{\boldsymbol{\varphi}_0} \in \mathbb{R}^m
\\
 \boldsymbol{\varphi}_1 ,  \tilde{\boldsymbol{\varphi}_1} ,
\tilde{\boldsymbol{\psi}_1} \in \mathbb{E}_1^m ;\quad 
\boldsymbol{\varphi}_\perp, \tilde{\boldsymbol{\varphi}_\perp},
\boldsymbol{\psi}_\perp, \tilde{\boldsymbol{\varphi}_\perp}
\in (\mathcal{C}^{2, \alpha}(S^3)^\perp )^m. 
\end{gather*} 
Indeed, the nonlinear mapping which appears on the right hand side of
\eqref{theequation} is continuous, compact. In addition, this
nonlinear mapping sends the ball of radius $\kappa  r_{\varepsilon,\lambda}^2$ (for
the natural product norm) into itself, provided $\kappa$ is fixed
large enough.

To obtain the precise statement of our Theorem, we simply
observe that
\[
2\nabla_{x} E_j (\tilde x^j, {\tilde{\mathbf{x}}}) 
 = \nabla_{\tilde x^j} E({\tilde{\mathbf{x}}}).
\]
where $E$ is the functional defined by \eqref{eq:fr4}, then a
sufficient condition for the mapping \eqref{eq:map} to have degree
$1$ is just that the point $\mathbf{x} = (x^1, \ldots, x^m)$ is a
nondegenerate critical point of the functional $E$. This completes
the proof of our Theorem. 

\subsection*{Acknowledgments}
This projet was founded by the national plan for science, technology and 
innovation (MAARIFAH), King Abdulaziz city for science and Technology,
kingdom of Saudi Arabia,  Award number12-MAT2880-02.


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\end{document}