\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 187, pp. 1--26.\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 - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/187\hfil Singular limit solutions]
{Singular limit solutions for 4-dimensional stationary Kuramoto-Sivashinsky
equations with exponential nonlinearity}

\author[S. Baraket, M. Khtaifi, T. Ouni \hfil EJDE-2015/187\hfilneg]
{Sami Baraket, Moufida Khtaifi, Taieb Ouni}

\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{Moufida Khtaifi \newline
D\'epartement de Math\'ematiques,
Facult\'e des Sciences de Tunis Campus Universitaire,
Universit\'e Tunis Elmanar, 2092 Tunis, Tunisia}
\email{moufida180888@gmail.com}

\address{Taieb Ouni \newline
D\'epartement de Math\'ematiques,
Facult\'e des Sciences de Tunis Campus Universitaire,
Universit\'e Tunis Elmanar, 2092 Tunis, Tunisia}
\email{Taieb.Ouni@fst.rnu.tn}

\thanks{Submitted January 28, 2015. Published July 13, 2015.}
\subjclass[2010]{58J08, 35J40, 35J60, 35J75}
\keywords{Singular limits; Green's function; Kuramoto-Sivashinsky equation; 
\hfil\break\indent domain decomposition method}

\begin{abstract}
 Let $\Omega$ be a bounded domain in $\mathbb{R}^4$ with smooth boundary, and
 let $x_1, x_2, \dots, x_m $ be points in $\Omega$.
 We are concerned with the singular stationary non-homogenous
 Kuramoto-Sivashinsky equation
 $$
 \Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2 = \rho^4f(u),
 $$
 where $f$ is a function that depends only the spatial variable. We
 use a nonlinear domain decomposition method to give sufficient
 conditions for the existence of  a positive weak solution satisfying
 the Dirichlet-like boundary conditions $u =\Delta u =0$, and being
 singular at each $x_i$ as the parameters $\lambda, \gamma$ and
 $\rho$ tend to $0$. An analogous problem in two-dimensions was
 considered in \cite{BIO} under condition (A1) below. However we do 
 not assume this condition.
\end{abstract}

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

\section{Introduction and statement of results}

First, we introduce a model arising in the growth of amorphous
surfaces which is a partial differential equation, called the
non-homogenous Kuramoto-Sivashinsky (KS) equation,
$$
\partial_tu+\Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2=f(u).
$$
on $\mathbb{R}^{d}$ with $d\geq 1$, where $ \lambda$ and $\gamma$ are real
parameters and $f(u)$ is a nonlinear function.
The Kuramoto-Sivashinsky equation was independently created by Kuramoto
and Tsuzuki \cite{Y.T}, and by Sivashinsky \cite{Sivashinsky} in
the study of a reaction-diffusion system and flame front
propagation, respectively. This equation is also found in the study
of $2D$ Kolmogorov fluid flows \cite{Sivashinsky2}. This form of the
Kuramoto-Sivashinsky equation is sometimes called the integrated
version of the Kuramoto-Sivashinsky equations (KSE), which arises in
several models for surface growth. Most mathematical results concern
the case $n\leq 3$ and essentially for $n=1$. Subject to appropriate
initial and boundary conditions has been introduced in \cite{Kuramoto} and some reference therein in studying phase turbulence
and the flame front propagation in combustion theory. This type of
version equation is suggested in \cite{Raible,Raible2} (and
some reference therein) as a phenomenological model for the
growth of an amorphous surface $(Z_{r65}Al_{7,5},Cu_{27,5})$.
Winkler and Stein \cite{Stein} used Rothe's method to verify the
existence of a global weak solution, this result has been recently
extended by Winkler \cite{Winkler} to the two-dimensional case of
(KS) equation, using energy type estimates for $ \int e^u dx$.

The non homogeneous Kuramoto-Sivashinsky equation with exponential
nonlinearity is a generalization of the fourth-order one-dimensional
semilinear parabolic equation arises also in several other models
for surface growth see for example \cite{V. A. Galak} for the
equation
$$
u_t+ u_{xxxx} -\beta [(u_{x})^3]_{x}=e^u
$$
with a parameter $\beta\geq0$, which is a model equation from explosion-convection
theory of which the fourth-order extension of the Frank-Kamenetskii
equation,
$$
u_t+ u_{xxxx} =e^u
$$
(a solid fuel model) is a limiting case.

Recently Chen and McKenna \cite{Y.Chen}  suggested to investigate the  equation
\begin{equation}
u_{xxxx}+cu_{xx} =e^u, \label{eq:refffff}
\end{equation}
where they give some existence and nonexistence results. In a note
on an exponential semilinear equation of the fourth order,
Mugnai \cite{Mugnai} considered the related problem to
\eqref{eq:refffff}. More precisely he considered, without
non linear gradient term, the problem
\begin{equation}
\begin{gathered}
\Delta^2 u +c\Delta u = b( e^u -1) \quad\text{in } \Omega\\
u =\Delta u =0 \quad \text{on } \partial\Omega
\end{gathered} \label{eq:refff}
\end{equation}
where $ \Omega$ is a bounded and smooth domain of $\mathbb{R}^n$, 
$c \in \mathbb{R}$ and
$b \in \mathbb{R}$. The author prove some existence and nonexistence 
results for
\eqref{eq:refff} via variational techniques. Such equations may occur 
while studying  traveling waves in suspension bridges.
 For more general problem see \cite{MRuichang}, for the  Navier
 boundary-value problem
\begin{equation}
\begin{gathered}
\Delta^2 u +c\Delta u =f( x,u) \quad\text{in } \Omega\\
u =\Delta u =0 \quad \text{on } \partial\Omega
\end{gathered} \label{eq:reffff}
\end{equation}
in $\mathbb{R}^n$, $n\geq 4$ and $f$ is non linear growth function. In
conformal dimensional i.e $n=4$ and $f$ has the subcritical
(exponential) growth on $\Omega$, i.e.,
$$
\lim_{t\to+\infty} \frac{|f (x,t)|}{\exp(\alpha t)}= 0
$$
uniformly on $x \in \Omega$  for all $\alpha > 0$ and in some cases
and hypothesis and using Adams inequality, (see \cite{Liu}), for the
fourth-order derivative, namely,
\[
 \sup_{\{u\in H^2(\Omega)\cap H^{1}_{0}(\Omega), \|u\|\leq1\}}
 \int_{\Omega} e^{32\pi^2u^2}dx\leq C|\Omega|,
\]
the authors show that the problem \eqref{eq:reffff} has at least two
nontrivial solutions (for more details see Theorem 1.3 in \cite{Liu})
or infinitely many nontrivial solutions (for more details see
\cite[Theorem 1.4]{Liu}).


A fundamental goal in the study of non-linear initial boundary value problems
involving partial differential equations is to determine
whether solutions to a given equation develop a singularity.
Resolving the issue of blow-up is important, in part because
it can have bearing on the physical relevance and validity of the underlying model.
However, determining the answer to
this question is notoriously difficult for a wide range of equations such
fourth order equation like stationary non homogenous Kuramoto-Sivashinsky
equation with strong nonlinearity like exponential $e^u$.
One route is to try to simplify or modify the boundary conditions in an
attempt to gain evidence for or against the occurrence of blow-up.
A second route is to modify the equations in some way,
and to study the modified equations with the hope of gaining insight
into the blow-up of solutions to the original equations: see problems
\eqref{eq:1.1}-\eqref{eq:BDPO}
 bellow  and the effect of the presence of  the second-order backward
 diffusion term $-\gamma\Delta u $ and
the nonlinear term $- \lambda|\nabla u|^2$ in \eqref{eq:1.1}.
 The occurrence and type of blow-up depends on the parameters $
\lambda,\gamma $ and the domain. Studding this type of equations, we
will answer for different basic questions. We concentrate next on
the analysis of the main questions raised in the study of blow-up
for such equations. This list can be suitably adapted to other
singularity formation problems. We will examine several case studies
related to such approaches where basic list includes the questions
of, where and how. We propose here an expanded list of three items:
(i) Does blow-up occur?
(ii) Where?
(iii) How?
For the first question, the blow-up problem is properly formulated only when a
suitable class of solutions is chosen for all solutions in the given
class or only for some solutions (which should be identified) or
other kinds of generalized solutions can be more natural to a given
problem and which equations and problems do exhibit blow-up. The
second question, is concerned with where finite number of points, or
regional blow-up, are localized: The set of blow-up is defined by
$$
S := \{x\in \Omega:\exists  x_{n}\to x \text{ such that } u_{n}(x_{n})\to +\infty\}.
$$
For the third question, we are concerned just by calculate the rate at which
solution diverges as $x$ approaches to the set $S$ of blow-up point and to
calculate the blow-up profiles as limits of solution at the non-blowing points.
 A major aim of the present work is to provide examples which demonstrate that
one must be extremely cautious in generalizing claims about the blow-up of
problems studied in idealized settings to
claims about the blow-up of the original problem and to the nonlinearity of a
problem which can cause the formation of a singularity,
where no such singularity is present in the unaltered equation.
However, many such studies have tried to search for singularities of
the solutions of the equations in the setting of different types of boundary
conditions like periodic boundary conditions related to the solution of
Kuramoto-Sivashinsky equation. The question of blow up of of solutions of
stationary Kuramoto-Sivashinsky equation is still an open question in dimensions
fourth and in higher cases.

For the stationary Kuramoto-Sivashinsky equation, the reader is referred to
\cite{A} and some refrence therein, where the author give some explicit estimates
for the $L^{\infty}$-norm of the periodic solutions of the time-independent
non homogeneous Kuramoto-Sivashinsky equation
$$
\Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2 =f(u)
$$
in $\mathbb{R}^n$  and its dependence on $f(u)$.
In particular, they give an estimate of the Michelson's upper bound of all
periodic solutions on space $x$ in $\mathbb{R}$ of
the time-independent homogeneous Kuramoto-Sivashinsky equation which is the
case with non linearity exponential i.e: a
solutions of such equation under steady  with $f(u)=e^u$ is invariant under the
group of translations $a \to u( \cdot +a)$.

One of the purposes of this article is to present a rather
efficient method to solve such singularly perturbed problems of the
time-independent Kuramoto-Sivashinsky equation called also
the integrated version of the homogeneous steady state KSE. This
method has already been used successfully in geometric
context (constant mean curvature surfaces, constant scalar curvature
metrics, extremal K\"ahler metrics, manifolds with special holonomy,
\ldots) The techniques developed and used here are inspired by the
work of \cite{{Bdpo}}. Motivated by the above discussion, we felt that,
given the interest in singular
perturbation problems, it was worth illustrating this on the non Homogenous
stationary Kuramoto-Sivashinsky equation:
$\Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2 =\rho^2f(u)$
in $\Omega\subset \mathbb{R}^4$
 under the physical Dirichlet-like boundary conditions
$u =\Delta u =0$ on $\partial\Omega$, given by the following problem.

Let $\Omega \subset {\mathbb{R}}^4$ be a regular bounded open
domain in ${\mathbb{R}}^4$. We are interested in the positive
solution of
\begin{equation}
\begin{gathered}
\Delta^2 u -\gamma\Delta u- \lambda|\nabla u|^2
= \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}
which is singular at each point $x_i$ as the
parameters $\lambda, \gamma$ and $\rho$ tend to $0$.
This  problem in some way a generalization of a fourth order Liouville 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:BDPO}
\end{equation}
in the case $(\gamma,\lambda)=(0,0)$, when the parameters $\rho$
tends to $0$. (See for example \cite{Bdpo}). Also problem
\eqref{eq:1.1} can be considered as a higher order counterpart of
the problem
 \begin{equation}
\begin{gathered}
-\Delta u -\lambda|\nabla u|^2 = \rho^2 e^u \quad\text{in }
\Omega\subset\mathbb{R}^2\\
u =0 \quad\text{on }\partial\Omega
\end{gathered} \label{eq:Tgra}
\end{equation}
when the parameter $\rho$ tends to 0 
($\rho\sim \epsilon$ as $\epsilon$ tends to 0).
This is a particular case of non homogenous
viscous Hamilton-Jacobian equation \cite{Souplet},
\begin{gather*}
\partial_tu-\Delta u -\lambda|\nabla u|^{p} =f(u) \quad\text{in } \Omega\\
u =0 \quad \text{on }\partial\Omega
\end{gather*}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{p}$, $p\geq 1$.

Problem \eqref{eq:Tgra} was studied by Baraket et al. in \cite{BIO}
for the existence of  $v_{\varepsilon,\lambda}$ a sequence of solutions
which converges to some singular function as the parameters
$\varepsilon$ and $\lambda$ tend to $0$, under the assumption
\begin{itemize}
\item[(A1)] If $0<\varepsilon<\lambda$,
then $\lambda^{1+\delta/2}\varepsilon^{-\delta}\to
0$ as $\lambda \to 0$, for any $\delta\in(0,1)$.
\end{itemize}
In particular, if we take $\lambda = \mathcal{O}(\varepsilon^{2/3})$, 
then  condition
(A1) is satisfied. With assumption (A1), problem
\eqref{eq:Tgra}, can be treated as a perturbation of the Liouville equation
 $$
-\Delta u = \rho^2 e^u\quad \text{in }\Omega\subset\mathbb{R}^2.
$$
This last equation was studied by Baraket and frank in \cite{Bar-Pac} as
$\rho$ tends to $0$.
 As observed by Ren and Wei \cite{Ren3},
 problem \eqref{eq:Tgra}, can be reduced to
a problem without gradient term. Indeed, if $u$ is a solution of
\eqref{eq:Tgra}, then the function
\[
w = (\lambda \rho^2 e^u)^{\lambda},
\]
satisfies
\begin{equation}
\begin{gathered}
-\Delta w = w^{\frac{\lambda+1}{\lambda}} \quad\text{in } \Omega \\
w  = (\lambda \rho^2)^{\lambda}\quad \text{on } \partial\Omega,
\end{gathered} \label{eq:eq}
\end{equation}
since the exponent $p = (\lambda+1)/\lambda$
tends to infinity as $\lambda$ tends to $0$, see also \cite{Esp-Mus-Pis}.

Note that Ghergu and Radulescu \cite{Gher} studied a more general problem
on a domain $\Sigma \subset \mathbb{R}^n$, $n \geq 2$:
\begin{equation}
\begin{gathered}
-\Delta u - \lambda |\nabla u|^a = g(u) + \mu f(x,u)\quad \text{in } \Sigma\\
u  = 0 \quad \text{on } \partial\Sigma,
\end{gathered}\label{eq:radu}
\end{equation}
with $0 < a \leq 2$, $\lambda, \mu > 0$ and some assumptions on $f$ and $g$.
Problems of the type  \eqref{eq:radu} arise in the study of non-Newtonian fluids,
boundary layer phenomena for viscous fluids, chemical heterogeneous catalysts,
as well as in the theory of heat conduction in electrically
conducting materials. See also \cite{Gherr}. It includes also some simple
prototype models from boundary-layer theory of viscous fluids \cite{Won}.

The question we would like to study is concerned with the existence
of other branches of solutions of \eqref{eq:1.1} as $\rho, \lambda$ and
$\gamma$ tend to $0$. To describe our result, let us denote by  $G(x, \cdot)$
the solution of
\begin{equation}
\begin{gathered}
\Delta^2 G(x, \cdot )  =  64  \pi^2  \delta_{x} \quad \text{in } \Omega \\
G(x, \cdot ) =  \Delta  G(x, \cdot)  =  0 \quad \text{on } \partial\Omega.
\end{gathered} \label{eq:1.2}
\end{equation}
It is easy to check that the function
\begin{equation}
R(x, y) := G(x, y) + 8  \log |x-y| \label{eq:1.3}
\end{equation}
is a smooth function.

We define
\begin{equation}
W (x^{1},\ldots ,x^m)  : = \sum_{j=1}^{m} R(x^{j},x^{j})
+ \sum_{j \neq \ell} G(x^{j},x^{\ell}). \label{eq:1.4}
\end{equation}

In dimension 4, Wei \cite{wei}, studied the behavior of
solutions to the nonlinear eigenvalue problem for the
biharmonic operator $\Delta^2$ in $\mathbb{R}^4$,
\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}
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{thm1.0}
Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^4$ and $f$ a smooth
nonnegative increasing function such that
\begin{equation}
e^{-u}f(u)\quad\text{and}\quad
e^{-u}\int_{0}^u f(s) ds\;\text{tend to $1$,  as $u\to+\infty$}.\label{cdwei}
\end{equation}
For $u_\lambda$ solution of \eqref{eq:weichap3}, denote by
${\Sigma_{\lambda} = \lambda
\int_{\Omega}f(u_\lambda) dx}$. Then many cases occur:
\begin{itemize}
\item[(i)] $\Sigma_{\lambda} \to 0$ therefore,
$\|u_\lambda\|_{L^{\infty(\Omega)}} \to 0$ as $\lambda  \to 0$.

\item[(ii)] $\Sigma_{\lambda} \to +\infty$ then
$u_\lambda \to +\infty$ as $\lambda \to 0$.

\item[(iii)] $\Sigma_{\lambda} \to 64 \pi^2 m$, 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 $W$.
\end{itemize}
\end{theorem}

Our main result reads as follows.

\begin{theorem} \label{th:1.1}
Let $\alpha\in (0,1)$ and $\Omega$  be an open smooth bounded domain of
$\mathbb{R}^4$. Assume that $(x^1,\ldots ,x^m)\in \Omega^{m} $ is a
\emph{nondegenerate} critical point of $W$, then there exist
$\rho_{0} >0$, $\lambda_{0} > 0$, $\gamma_{0} > 0$ and
$\big\{ u_{\rho, \lambda, \gamma}\big\}$ with
$0<\rho <\rho_{0}$, $ 0<\lambda<\lambda_{0}$, $0<\gamma <\gamma_{0}$,  a
one parameter family of solutions of \eqref{eq:1.1}, such that
\[
\lim_{\rho \to 0,\, \lambda \to 0, \, \gamma \to 0} u_{\rho,\lambda,\gamma}
= \sum_{j=1}^m G(x_j , \cdot)
\]
in ${\mathcal{C}}_{\rm loc}^{4,\alpha}(\Omega - \{x^1 , \ldots ,x^m \} )$.
\end{theorem}

Our result reduces the study of nontrivial branches of solutions of
\eqref{eq:1.1} to the search for critical points of the function $W$
defined in \eqref{eq:1.4}. Observe that the assumption on the
nondegeneracy of the critical point is a rather mild assumption
since it is certainly fulfilled for generic choice of the open
domain $\Omega$.

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 on
$4$-dimensional Riemannian manifolds \cite{Cha-1}, \cite{Chang-Yang}
\[
Q_g = \frac{1}{12}  \big( -  \Delta_g S_g + S^2_g - 3  
|\operatorname{Ric}_g|^2\big)
\]
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
\begin{equation}
P_{g} : =  \Delta_g^2 +  \delta  \Big( \frac{2}{3}  S_g   I - 2
  \operatorname{Ric}_g \Big)  d \label{eq:1.6}
\end{equation}
is the Panietz operator, which is an elliptic $4$-th order partial
differential operator \cite{Chang-Yang} and which transforms
according to
\begin{equation}
e^{4 w}  P_{e^{2w} g}  =  P_g, \label{eq:1.7}
\end{equation}
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
\[
P_{g_{eucl}} = \Delta^2
\]
in which case \eqref{eq:1.5} reduces to
\[
\Delta^2  w =  \tilde{Q}  e^{4  w}
\]
the solutions of which give rise to conformal metric $g_w = e^{2  
w}   g_{eucl}$ 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.

We briefly describe the plan of the paper~: In Section 2 we discuss
rotationally symmetric solutions of \eqref{eq:1.1}.
In Section 3 we study the linearized operator about the radially 
symmetric solution defined in the previous section.
In Section 4, we recall some Known results about the
analysis of the bi-Laplace operator in weighted spaces.
Both section strongly use the b-operator which has been developed by Melrose
\cite{Mel} in the context of weighted Sobolev spaces and by Mazzeo
\cite{Maz} in the context of weighted H\"older spaces (see also
\cite{Pac-Riv}).


A first nonlinear problem is studied in Section 5  where the
existence of an infinite dimensional family of solutions of
\eqref{eq:1.1} which are defined on a large ball and which are close
to the rotationally symmetric solution is proven. In Section 6, we
prove the existence of an infinite dimensional family of solutions
of \eqref{eq:1.1} which are defined on $\Omega$ with small ball
removed. Finally, in Section 7, we show how elements of these
infinite dimensional families can be connected to produce solutions
of \eqref{eq:1.1} described in Theorem~\ref{th:1.1}. In Section 7,
we patch these pieces, in the two last sections, together via a
nonlinear version of the Cauchy data matching. Throughout the paper,
the symbol $c_\kappa>0$ (which can depend only on $\kappa$) denotes
always a positive constant independent of $\varepsilon, \lambda$ and $\gamma$
which might change from one line to another.


\section{Rotationally symmetric solutions}

We first describe the rotationally symmetric approximate solutions of
\begin{equation}
\Delta^2 u -\gamma\Delta u- \lambda |\nabla u|^2 =\rho^4e^u\label{eq:111}
\end{equation}
in $\mathbb{R}^4$ which will play a central role in our analysis.
For this raison given
$\varepsilon>0$, we define
\[
u_{\varepsilon} (x) : =  4  \log(1 + \varepsilon^2) - 4
\log(\varepsilon^2 + |x|^2).
\]
which is clearly a solution of
\begin{equation}
\Delta^2u - \rho^4e^u = 0 ,\label{eq:2.1}
\end{equation}
when
\begin{equation}
\rho^4 = \frac{{384}
{\varepsilon^4}}{(1+\varepsilon^2)^4}. \label{eq:2.2}
\end{equation}

Let us notice that \eqref{eq:2.1} is invariant under some
dilation in the following sense: If $u$ is a solution of
\eqref{eq:2.1} and $\tau > 0$, then $u(\tau  \cdot ) + 4  \log  \tau$ is also a
solution of \eqref{eq:2.1}. With this observation in mind, we
define, for all $\tau > 0$
\begin{equation}
u_{\varepsilon,\tau} (x) : = 4 \log  (1 + \varepsilon^2)+ 4
\log  \tau - 4 \log  (\varepsilon^2 + \tau^2 |x|^2). \label{eq:2.3}
\end{equation}

\section{A linear fourth-order elliptic operator on $\mathbb{R}^4$}

We define the linear fourth-order elliptic operator
\begin{equation}
\mathbb{L} : = \Delta^2 - \frac{384}{(1 + |x|^2)^4} \label{3.1}
\end{equation}
which corresponds to the linearization of
\eqref{eq:2.1} about the solution $u_1 (= u_{\varepsilon =1})$ which has been
defined in the previous section.

We are interested in the classification of bounded solutions of
$\mathbb{L}  w =0$ in $\mathbb{R}^4$. 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:2.1} is invariant under the group of
dilations $ \tau \to u(\tau  \cdot ) + 4 \log
\tau $. We also define, for $i=1, \ldots, 4$
\[
\phi_i(x) : = -{\partial_{x_i}} u_{1}(x) =\frac{8  
x_i}{1+|x|^2},
\]
which are also solutions of $\mathbb{L}    \phi_j =0$ since these
solutions correspond to the invariance of the equation under the
group of translations $a \to u( \cdot +a)$.

The following result classifies all bounded solutions of $\mathbb{L}  w =0$
which are defined in $\mathbb{R}^4$.

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

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

\begin{definition} \label{def3.1} \rm
Given $k \in {\mathbb N}$, $\alpha \in (0,1)$ and
$\mu \in \mathbb{R}$, we introduce the H\"older weighted spaces
$\mathbb{C}^{k,\alpha}_\mu(\mathbb{R}^4)$ as the space of functions $w \in
\mathbb{C}^{k,\alpha}_{\rm loc}(\mathbb{R}^4)$ for which the
norm
\[
\| w \|_{\mathbb{C}^{k,\alpha}_\mu(\mathbb{R}^4)} : = \|
w\|_{\mathcal{C}^{k, \alpha} (\bar B_1)} +
{\sup_{r\geq 1}}  \Big( (1+r^2)^{-\mu/2} \|w
(r \cdot) \|_{\mathbb{C}^{k,\alpha}_\mu(\bar B_1-B_{1/2})} \Big),
\]
is finite. \label{de:3.1}
\end{definition}

We also define
\[
\mathcal{C}^{k, \alpha}_{{\rm rad},\mu}(\mathbb{R}^4)=\{f\in
\mathcal{C}^{k, \alpha}_{\mu}(\mathbb{R}^4);\; f(x)=f(|x|), \forall x\in
\mathbb{R}^4\}.
\]
As a consequence of the result in  Lemma~\ref{le:3.1}, we have:

\begin{proposition}[\cite{Bdpo}]
(i) Assume that $\mu > 1$ and $\mu \not\in \mathbb{N}$, then
\begin{align*}
L_\mu :  \mathbb{C}^{4,\alpha}_\mu(\mathbb{R}^4) & \to
 \mathbb{C}^{0,\alpha}_{\mu-4}(\mathbb{R}^4)\\
 w & \mapsto  \mathbb{L}  w
\end{align*}
is surjective.

(ii) Assume that $\delta > 0$ and $\delta \not\in \mathbb{N}$ then
\begin{align*}
L_\delta:  \mathcal{C}^{4,\alpha}_{{\rm rad},\delta}(\mathbb{R}^4) & \to
 \mathcal{C}^{0,\alpha}_{{\rm rad},\delta-4}(\mathbb{R}^4)\\
 w & \mapsto  \mathbb{L}  w
\end{align*}
is surjective.\label{pr:surj-1'}
\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 space $\mathbb{C}^{k,\alpha}_\mu(\bar B_1^*)$
 as the space of functions in $\mathbb{C}^{k, \alpha}_{\rm loc}(\bar B_1^*)$
for which the  norm
\[
\|u \|_{\mathbb{C}^{k, \alpha}_\mu(\bar B_1^*)} =
{\sup_{r\leq 1/2}}   \left( r^{-\mu} \| u(r  
\cdot) \|_{\mathbb{C}^{k, \alpha}(\bar B_2-B_1)} \right)
\]
is finite.
\end{definition}

Then we define the subspace of radial functions in
$\mathcal{C}^{k, \alpha}_{{\rm rad},\delta}(\bar B_1^*)$ by
\[
\mathcal{C}^{k, \alpha}_{{\rm rad},\delta}(\bar B_1^*)
=\{f\in \mathcal{C}^{k, \alpha}_{\delta}(\mathbb{R}^4);  f(x)=f(|x|),
 \forall x\in \bar B_1^*\}.
\]
For  $\varepsilon , \tau, \lambda > 0 $, we define
\[
R_{\varepsilon,\lambda, \gamma}: = \tau r_{\varepsilon,\lambda, 
\gamma}/\varepsilon
\]
where
\begin{equation}
r_{\varepsilon,\lambda,\gamma}:=\max(\sqrt{\varepsilon},
\sqrt{\lambda},\sqrt{\gamma})
%\label{eq:small bool}.
\end{equation}
 We would like to find a solution $u$ of
\begin{equation}
\Delta^2 u -\gamma\Delta u-\lambda|\nabla u|^2
-\rho^4e^u =0\label{eq:mah}
\end{equation}
in $\bar{B}_{r_{\varepsilon,\lambda, \gamma}}$.
Recall that in the polar coordinates if we assume that $\varphi$ is a
radially symmetric function, we get the usual formulas
$|\nabla\varphi|=(\nabla\varphi, \nabla\varphi)^{1/2}$
where  $(\cdot,\cdot)$ is the usual Euclidian dot product in $\mathbb{R}^n$. Then
\begin{gather*}
|\nabla\varphi|^2=(\frac{\partial \varphi}{\partial r})^2, \\
\Delta\varphi=\frac{\partial^2 \varphi}{\partial r^2}
 +\frac{n-1}{r}\frac{\partial \varphi}{\partial r}, \\
\Delta^2\varphi=\frac{\partial^4 \varphi}{\partial r^4}
+\frac{2(n-1)}{r}\frac{\partial^3 \varphi}{\partial r^3}
+\frac{(n-1)(n-3)}{r^2}\frac{\partial^2 \varphi}{\partial r^2}
-\frac{(n-1)(n-3)}{r^3}\frac{\partial \varphi}{\partial r}.
\end{gather*}
Using the transformation
\[
v(x) = u\big(\frac{\varepsilon}{\tau}x\big)
 + 8\log \varepsilon-4\log\left({\tau(1+\varepsilon^2)}/{2}\right),
\]
 then  \eqref{eq:mah} is equivalent to
\begin{equation}
\Delta^2 v -\left(\frac{\varepsilon}{\tau}\right)^2\left(\gamma\Delta v 
+\lambda|\nabla v|^2\right) -24e^{v} =0\label{eq:mah1}
\end{equation}
in $\bar{B}_{R_{\varepsilon,\lambda, \gamma}}$. 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}
\mathbb{L}h = \frac{384}{(1 +
| x|^2)^4}(e^{h} - h - 1)+\left(\frac{\varepsilon}{\tau}\right)^2
\left(\gamma\Delta (u_{1}+h)+\lambda|\nabla(u_{1}+h)|^2\right)\label{equ:y}
\end{equation}
in $\bar{B}_{R_{\varepsilon,\lambda, \gamma}}$.

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

For $\sigma \geq 1$, we denote
$\mathcal{E}_{\sigma} :
\mathcal{C}^{0, \alpha}_\mu (\bar B_{\sigma}) \to
\mathcal{C}^{0, \alpha}_\mu (\mathbb{R}^4)$
the extension operator defined by
\begin{equation}
\mathcal{E}_{\sigma}  (f) (x) = \begin{cases}
f (x) &\text{for } |x|\le \sigma\\
\chi \left( \frac{|x|}{\sigma}
\right)  f \left( \sigma  \frac{x}{|x|} \right)
 &\text{for } |x|\geq \sigma,
\end{cases} \label{eq:9.31}
\end{equation}
 where $t \mapsto \chi (t)$ is a smooth nonnegative cutoff
function identically equal to $1$ for $t \leq 1$ and identically
equal to $0$ for $t \geq 2$. It is easy to check that there exists a
constant $c = c (\mu)>0$, independent of $\sigma \geq 1$, such that
\begin{equation}
\| \mathcal{E}_{\sigma} (w ) \|_{\mathcal{C}^{0, \alpha}_{\mu}
(\mathbb{R}^4)} \leq  c  \| w \|_{\mathcal{C}^{0,
\alpha}_{\mu} (\bar B_{\sigma})}. \label{eq:6..3}
\end{equation}
We fix $\delta \in (0,1)$
and denote by $\mathcal{G}_{\delta}$ to be a right inverse of
$\mathbb{L}_{\delta}$ provided by Proposition~\ref{pr:surj-1'}. To
find a solution of \eqref{equ:y} it is enough to find a fixed point
$h$, in a small ball of $\mathcal{C}^{4,
\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)$, solution of
\begin{equation}
 h =\aleph(h)\label{eq:T}
\end{equation}
where
\begin{gather*}
 \aleph(h) :=\mathcal{G}_{\delta}\circ\mathcal{E}_{\delta} 
\circ \mathfrak{R}(h), \\
\mathfrak{R}(h)=\frac{384}{(1 +
| x|^2)^4}(e^{h} - h - 1)+\left(\frac{\varepsilon}{\tau}\right)^2
\Big(\gamma\Delta(u_{1}+h)
+\lambda|\nabla(u_{1}+h)|^2\Big).
\end{gather*}
We have
$$
{|\mathfrak{R}(0)| =\left(\frac{\varepsilon}{\tau}\right)^2
\Big(\gamma\Delta u_{1}+
\lambda|\nabla u_{1}|^2}\Big).
$$
For $|x|=r$, we have
\[
 {\sup_{r \leq R_{\varepsilon,\lambda, \gamma}}r^{4-\delta}\;|\mathfrak{R}(0)|}
 \leq \left(\frac{\varepsilon}{\tau}\right)^2\sup_{r \leq
R_{\varepsilon,\lambda, \gamma}}r^{4-\delta}\Big(\gamma\Delta u_{1} +
\lambda|\nabla u_{1}|^2\Big).
\]
Using
\begin{align*}
 \gamma\Delta u_{1}+\lambda|\nabla u_{1}|^2
&=-16\gamma\frac{2+r^2}{(1+r^2)^2}+64\lambda\frac{r^2}{(1+r^2)^2}\\
&=\frac{-32\gamma}{(1+r^2)^2}+16\left(4\lambda-\gamma\right)
\frac{r^2}{(1+r^2)^2}\,,
\end{align*}
this implies that for each $\kappa>0$,
there exist $c_\kappa>0$ (which can depend only on $\kappa$), such that for
$\delta\in(0,1)$, we have
\begin{align*}
{\sup_{r \leq
R_{\varepsilon,\lambda, \gamma}}r^{4-\delta}\;|\mathfrak{R}(0)|}
&\leq c_\kappa\gamma\varepsilon^2 + c_{\kappa}(4\lambda+\gamma)
\varepsilon^2 R^{2-\delta}_{\varepsilon,\lambda, \gamma}\\
&\leq c_\kappa\gamma\varepsilon^2 + c_\kappa(4\lambda+\gamma)
 \varepsilon^{\delta} r^{2-\delta}_{\varepsilon,\lambda, \gamma}\\
&\leq c_{\kappa}\varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}.
\end{align*}
Then there exist $\bar{c}_\kappa>0$ (which can depend only on $\kappa$),
such that
\[
\| \aleph(0) \|_{C^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}
\le  \bar{c}_\kappa \varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}
 \]
Using Proposition \ref{pr:surj-1'} and
\eqref{eq:6..3}, we conclude that
\begin{equation}
\| h \|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}
\leq 2\bar{c}_{\kappa}\varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma} .
\label{eq:T4.}
\end{equation}
Now, let $h_1, h_2$ be in $B(0,2c_{\kappa}\varepsilon^{\delta}r^2_{\varepsilon,
\lambda, \gamma})$ of $\mathcal C_{{\rm rad},\delta}^{4, \alpha}(\mathbb{R}^4)$.
Then for each $\kappa>0$, there exist $c_\kappa>0$ (which can depend
only on $\kappa$), such that for $\delta\in(0,1)$, we have
\begin{align*}
&{\sup_{r \leq R_{\varepsilon,\lambda, \gamma}}r^{4-\delta}}\;|\mathfrak{R}(h_2)
-\mathfrak{R}(h_1)|\\
&\leq c_{\kappa}{\sup_{r \leq R_{\varepsilon,\lambda, \gamma}}
r^{4-\delta}{(1 + | x|^2)^{-4}}\left|e^{h_2}-e^{h_1}+ h_1-h_2
\right|}\\
&\quad +c_{\kappa}\lambda\varepsilon^2{\sup_{r \leq R_{\varepsilon,\lambda, 
\gamma}}
r^{4-\delta}{(|\nabla(u_{1}+h_2)|^2-|\nabla(u_{1}+h_1)|^2)}}\\
&\quad +c_{\kappa}\gamma\varepsilon^2{\sup_{r \leq R_{\varepsilon,\lambda, 
\gamma}}
r^{4-\delta}{\Big|\Delta(u_{1}+h_2)-\Delta(u_{1}+h_1)\Big|}}\\
&\leq c_{\kappa}{\sup_{r \leq R_{\varepsilon,\lambda, \gamma}}}
 r^{-4-\delta}|h_2-h_1||h_2+h_1|\\
&\quad + c_{\kappa}\lambda\varepsilon^2{\sup_{r \leq R_{\varepsilon,\lambda, 
\gamma}}
}r^{4-\delta}{\Big(|\nabla ( h_2-h_1)| \left(|\nabla (
h_2+h_1)|+2|\nabla u_1|\right)}\Big)\\
&\quad +c_{\kappa}\gamma \varepsilon^2{\sup_{r \leq R_{\varepsilon,\lambda, 
\gamma}}   r^{4-\delta}{\Big|\Delta(h_2-h_1)\Big|}} \\
 &\leq c_{\kappa}\sum_{i=1}^2\|h_i\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}
(\mathbb{R}^4)}
\|h_2-h_1\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}
+c_{\kappa}\gamma \varepsilon^2 R^4_{\varepsilon,\lambda,
\gamma}\|h_2-h_1\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}
\\
&\quad +c_\kappa\lambda \varepsilon^2 R^4_{\varepsilon,\lambda, \gamma}
 \Big(R^{\delta}_{\varepsilon,\lambda, \gamma}\sum_{i=1}^2
\|h_i\|_{\mathcal{C}^{4,
\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}+1\Big)\|h_2-h_1\|_{\mathcal{C}^{4,
\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
Provided $h_{i} \in \mathcal{C}^{4, \alpha}_{{\rm rad}, \delta} (\mathbb{R}^4)$
satisfies $\| h_{i} \|_{\mathcal{C}^{4, \alpha}_{{\rm rad},
\delta} (\mathbb{R}^4)} \leq 2   c_\kappa  
\varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}$, then the last estimate, is
given by
\begin{align*}
& \sup_{r \leq R_{\varepsilon,\lambda,
\gamma}}r^{4-\delta} |\mathfrak{R}(h_2)-\mathfrak{R}(h_1)|\\
&\leq c_{\kappa}\varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}
\|h_2-h_1\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}+
c_{\kappa}r^2_{\varepsilon,\lambda, \gamma} \|h_2-h_1\|_{\mathcal{C}^{4,
\alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}\\
&\quad +c_{\kappa}r^2_{\varepsilon,\lambda, \gamma}
\|h_2-h_1\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\end{align*}
Similarly, using Proposition \ref{pr:surj-1'} and
\eqref{eq:6.3}, we conclude that for each $\kappa>0$, there exist
$\varepsilon_\kappa, \lambda_\kappa$, $\gamma_\kappa$  and $\bar{c}_\kappa>0$
(only depend on $\kappa$) such that
\begin{equation}
\|\aleph(h_2)-\aleph(h_1)\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},
\delta}(\mathbb{R}^4)}
\leq \bar{c}_{\kappa}r^2_{\varepsilon,\lambda,
\gamma}\|h_2-h_1\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)}.
\label{eq:vol}
\end{equation}
Reducing $\varepsilon_\kappa, \lambda_\kappa$ and $\gamma_\kappa$ if necessary, 
we can assume that
\[
\bar c_\kappa  r^2_{\varepsilon,\lambda, \gamma}\leq \frac{1}{2}
\]
for all $\varepsilon \in (0, \varepsilon_\kappa )$, 
$\lambda \in (0, \lambda_\kappa )$
and $\gamma \in (0, \gamma_\kappa )$.
Then,
\eqref{eq:T4.} and \eqref{eq:vol} are enough to show that
$h  \mapsto \aleph (h)$
is a contraction from the ball
\[
\{ h \in \mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4): \|h
\|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}({\mathbb{R}}^4)} \leq
2c_{\kappa}\varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma} \}
\]
into itself and hence has a unique fixed point $h$ in this set. This
fixed point is a solution of \eqref{eq:T} in $\bar{B}_{R_{\varepsilon,\lambda, 
\gamma}}$.
 We summarize this in the following proposition.

\begin{proposition} \label{prop3.2}
For each $\kappa>0$, there exist $\varepsilon_\kappa >0$, $\lambda_\kappa >0$,
$\gamma_\kappa >0$ and $c_\kappa>0$ (which can depend only on
$\kappa$) such that for all  for all $\varepsilon \in (0, \varepsilon_\kappa )$,
$\lambda \in (0, \lambda_\kappa )$ and $\gamma \in (0, \gamma_\kappa
)$ and for $\delta\in (0,1)$, there exists a unique solution $h \in
\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)$ of \eqref{eq:T} such
that
\[v(x)= u_{1}(x) +   h(x)
\]
 solves \eqref{eq:mah1} in $\bar{B}_{R_{\varepsilon,\lambda, \gamma}}$.
In addition
\[
\| h \|_{\mathcal{C}^{4, \alpha}_{{\rm rad},\delta}(\mathbb{R}^4)} \leq
2c_{\kappa}\varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}.
\]
\end{proposition}

\section{Known results \cite{Bdpo}}
\subsection{Analysis of the bi-Laplace operator in weighted spaces}
\quad\\
Given $x^1, \ldots, x^m \in \Omega$ we define $X: = (x^1, \ldots,
x^m)$ and
\[
\bar \Omega^*   (X): = \bar \Omega - \{x^1, \ldots , x^m\},
\]
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 all $r \in (0, r_0)$ we define
\[
\bar \Omega_r  (X) : = \bar \Omega - \cup_{j=1}^{m} B_{r}(x^j)
\]
With these notation, we have the following definition.

\begin{definition}  \label{de:4.1} \rm
Given $k \in \mathbb{R}$, $\alpha \in (0,1)$ and
$\nu\in \mathbb{R}$, we introduce the H\"older weighted space
${\mathcal{C}}^{k,\alpha}_{\nu}(\bar \Omega^* (X))$ as the space
of functions $w \in {\mathcal{C}}^{k,\alpha}_{\rm loc}(\bar \Omega^* (X))$
which is endowed with the norm
\[
\| w \|_{{\mathcal{C}}^{ k, \alpha}_{\nu}(\bar \Omega^*  (X))}
 : = \| w \|_{{\mathcal{C}}^{ k,\alpha}( \bar \Omega_{r_0/2}  (X))}
+ \sum_{j=1}^m  \sup_{r  \in (0, r_0/2)}   \left( r^{-\nu}  \| w(x^j
 + r  \cdot )
\|_{{\mathcal{C}}^{ k,\alpha} (\bar B_2-B_1)} \right),
\]
is finite.
\end{definition}

When $k \geq 2$, we denote by $[ \mathbb{C}^{k,\alpha}_\nu (\bar
\Omega^*  (X)) ]_0$ be the subspace of functions
$w \in \mathbb{C}^{k,\alpha}_\nu (\bar \Omega^*  (X))$ satisfying
$ w = \Delta w = 0$.

\begin{proposition}[\cite{Bdpo}]
Assume that $\nu < 0$ and $\nu \not\in \mathbb{Z}$, then
\begin{align*}
 {\mathcal L}_\nu  : [ \mathbb{C}^{4,\alpha}_\nu (\bar \Omega^*  (X))]_0
&\to   \mathbb{C}^{0,\alpha}_{\nu-4} (\bar \Omega^*  (X))\\
 w & \mapsto  \Delta^2  w
\end{align*}
is surjective. \label{pr:4.1}
\end{proposition}


\subsection{Bi-harmonic extensions}

Given $\varphi \in \mathbb{C}^{4,\alpha}(S^3)$ and
$\psi \in \mathbb{C}^{2,\alpha}(S^3)$ we define
$H^i (= H^i (\varphi, \psi ; \cdot) )$ as the solution of
\begin{equation}
\begin{gathered}
\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{gathered} \label{eq:5.1}
\end{equation}
where, as already mentioned, $B_1$ denotes the unit ball in
$\mathbb{R}^4$.

We set $B_1^* = B_1 - \{0\}$. As in the previous section, we
have a definition.

\begin{definition} \label{def4.2} \rm
Given $k\in {\mathbb N}$, $\alpha \in (0,1)$ and $\mu \in \mathbb{R}$,
 we introduce the H\"older weighted spaces $\mathbb{C}^{k,\alpha}_\mu(\bar B_1^*)$
as the space of function in $\mathbb{C}^{k,\alpha}_{\rm loc}(\bar B_1^*)$
for which the  norm
\[
\|u \|_{\mathbb{C}^{k,\alpha}_\mu(\bar B_1^*)} =
{\sup_{r\leq 1/2}}   \left( r^{-\mu} \| u(r  
\cdot) \|_{\mathbb{C}^{k,\alpha}(\bar B_2-B_1)} \right),
\]
is finite. 
\end{definition}

This corresponds to the space and norm already defined in the previous section
 when $\Omega = B_1$, $m=1$ and $x^1=0$.


Let $e_1, \ldots, e_4$ be the coordinate functions on $S^3$.

\begin{lemma}\cite{Bdpo} \label{le:5.1}
Assume that
\begin{equation}
\int_{S^3} (8  \varphi  - \psi )    dv_{S^3} =0 \quad \text{and} \quad
\int_{S^3} (12  \varphi - \psi ) e_\ell   dv_{S^3}=0 \label{eq:5.2}
\end{equation}
for $\ell = 1, \ldots , 4$. Then there exists $c> 0$ such that
\[
\| H^i(\varphi, \psi  ; \cdot ) \|_{{\mathcal{C}}_{2}^{ 4,\alpha}(
\bar B_{1}^*)}\leq c  ( \| {\varphi} \|_{{\mathcal{C}}^{
4,\alpha}( S^3)} + \| {\psi} \|_{{\mathcal{C}}^{ 2,\alpha}(S^3)} ).
\]
\end{lemma}

Given $\varphi\in \mathcal{C}^{4,\alpha}(S^3)$ and 
$\psi\in \mathcal{C}^{2,\alpha}(S^3)$,
we define (when it exsits)
$H^e$ ($=H^e(\varphi,\psi;\cdot)$) to be the 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.



\begin{definition} \label{def4.3} \rm
Given $k\in {\mathbb N}$, $\alpha \in (0,1)$ and $\nu \in \mathbb{R}$,
we define the space $\mathbb{C}^{k,\alpha}_\nu (\mathbb{R}^4 -B_1)$
as the space of functions
$w \in \mathbb{C}^{k,\alpha}_{\rm loc}(\mathbb{R}^4 -B_1)$ for which the norm
\[
\| w \|_{\mathbb{C}^{k,\alpha}_\nu(\mathbb{R}^4 -B_1)} =
{\sup_{r\geq 1}}  \left(  r^{-\nu} \|w(r  
\cdot)\|_{\mathbb{C}^{k,\alpha}_\nu(\bar B_2-B_1)} \right),
\]
is finite. \label{de:5.2}
\end{definition}

\begin{lemma}[\cite{Bdpo}] \label{le:5.2}
Assume that
\begin{equation}
\int_{S^3} \psi     dv_{S^3} = 0.\label{eq:5.6}
\end{equation}
Then there exists $c > 0$ such that
\[
\| H^e (\varphi, \psi ;
\cdot) \|_{{\mathcal{C}}_{-1}^{ 4,\alpha} (\mathbb{R}^4 -B_1)}\leq
c  (\|{\varphi}\|_{{\mathcal{C}}^{ 4,\alpha}( S^3)}+
\|{\psi}\|_{{\mathcal{C}}^{ 2,\alpha}(  S^3)} ).
\]
\end{lemma}

\begin{lemma}[\cite{Bdpo}]
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 - \partial_r H^e  ,
\partial_{r}  \Delta H^i  - \partial_r  \Delta H^e )
\end{align*}
where $H^i = H^i ( \varphi , \psi  ; \cdot )$ and
 $H^e = H^e( \varphi , \psi  ; \cdot )$, is an isomorphism. \label{le:5.3}
\end{lemma}


\section{First nonlinear Dirichlet problem}

Recall  for  $\varepsilon , \tau, \lambda, \gamma > 0 $, we define
$R_{\varepsilon,\lambda, \gamma}: = \tau r_{\varepsilon,\lambda, \gamma}/\varepsilon$,
where
\begin{equation}
r_{\varepsilon,\lambda, \gamma}:=\max(\sqrt{\varepsilon},\sqrt{\lambda},\sqrt{\gamma})
\label{eq:small bool}.
\end{equation}
Given $\varphi \in \mathcal{C}^{4, \alpha}(S^3)$ and
$\psi \in \mathcal{C}^{2, \alpha}(S^3)$ satisfying \eqref{eq:5.2}, we define
\[
\mathbf{u}: = u_1+h + H^i ({ \varphi}, { \psi}  ;  ( \cdot /
R_{\varepsilon,\lambda, \gamma})).
\]
We would like to find a solution $u$ of
\begin{equation}
\Delta^2  u -\gamma\big(\frac{\varepsilon}{\tau}\big)^2\Delta u
-\lambda\big(\frac{\varepsilon}{\tau}\big)^2|\nabla u|^2- 24  e^u = 0  \label{eq:6.1}
\end{equation}
which is defined in $B_{R_{\varepsilon,\lambda, \gamma}}$ and which is a perturbation
of $\mathbf{u}$. Writing $u = \mathbf{u} + v$, this amounts to solve the equation
\begin{equation}
\begin{aligned}
\mathbb{L}  v
&= \frac{384}{(1 + r^2)^4}e^{h}( e^{H^{i} ({ \varphi}, {
\psi}  ;  (\cdot / R_{\varepsilon,\lambda, \gamma} )) + v}
- 1 - v)+\frac{384}{(1 + r^2)^4}( e^{h} - 1)v \\
&\quad + \gamma\big(\frac{\varepsilon}{\tau}\big)^2\Delta\Big( u_1+h
 + H^i ({ \varphi}, { \psi}  ;  ( \cdot /
R_{\varepsilon,\lambda, \gamma})) + v\Big)-\gamma\big(\frac{\varepsilon}{\tau}\big)^2\Delta( u_{1}
 + h) \\
&\quad + \lambda\big(\frac{\varepsilon}{\tau}\big)^2\Big|\nabla\Big( u_1+h
 + H^i ({ \varphi}, { \psi}   ;  ( \cdot /
R_{\varepsilon,\lambda, \gamma})) + v\Big)\Big|^2
 -\lambda\big(\frac{\varepsilon}{\tau}\big)^2|\nabla(u_{1} + h)|^2,
\end{aligned} \label{eq:66.2}
\end{equation}
since $H^i$ is bi-harmonic.
In the following, we will denote by $\mathcal{K}(v)$ the right hand side of
\eqref{eq:66.2}.

\begin{definition} \label{de:6.2} \rm
Given $ \bar r \geq 1$, $k \in {\mathbb N}$, $\alpha \in (0,1)$ and
$\mu \in \mathbb{R}$, the weighted space $\mathcal{C}^{k,
\alpha}_{\mu} (B_{\bar r})$ is defined to be the space of functions
$w \in {\mathcal{C}}^{k, \alpha} (B_{\bar r})$ endowed with the
norm
\[
\| w \|_{{\mathcal{C}}^{k, \alpha}_{\mu} (\bar B_{\bar r})}  : = \|
w \|_{{\mathcal{C}}^{k, \alpha} (B_{1})} + \sup_{1 \leq r \leq \bar
r}   \left( r^{-\mu}  \| w (r  \cdot) \|_{{\mathcal{C} }^{k,
\alpha} ( \bar B_{1} - B_{1/2})} \right).
\]
\end{definition}

For $\sigma \geq 1$, we denote by $\mathcal{E}_{\sigma} :
\mathcal{C}^{0, \alpha}_\mu (\bar B_{\sigma}) \to
\mathcal{C}^{0, \alpha}_\mu (\mathbb{R}^4)$ the extension
operator defined by
\[
\mathcal{E}_{\sigma}  (f) (x) = \chi \Big( \frac{|x|}{\sigma}
\Big)  f \Big( \sigma  \frac{x}{|x|} \Big),
\]
where $t \mapsto \chi (t)$ is a smooth nonnegative cutoff
function identically equal to $1$ for $t \geq 2$ and identically
equal to $0$ for $t \leq 1$. It is easy to check that there exists a
constant $c = c (\mu)>0$, independent of $\sigma \geq 1$, such that
\begin{equation}
\| \mathcal{E}_{\sigma} (w ) \|_{\mathcal{C}^{0, \alpha}_{\mu}
(\mathbb{R}^4)} \leq  c  \| w \|_{\mathcal{C}^{0,
\alpha}_{\mu} (\bar B_{\sigma})} . \label{eq:6.3}
\end{equation}
We fix $\mu \in (1,2)$
and denote by $\mathcal{G}_\mu$ a right inverse provided by
Proposition~\ref{pr:surj-1'}. To find {\underline a}  solution of
\eqref{eq:66.2}, it is enough to find $v \in \mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)$ solution of
\begin{equation}
v = N( \varepsilon, \lambda,\gamma, \tau, \varphi, \psi ; v) \label{eq:6.4}
\end{equation}
where we have defined
\[
 N (\varepsilon, \lambda,\gamma, \tau, \varphi, \psi  ;  v)
: =\mathcal{G}_\mu \circ \mathcal{E}_{R_{\varepsilon,\lambda, \gamma}}
 \left(\mathcal{K}(v) \right)
\]
Given $\kappa >1$ (whose value will be fixed later on), we now
further assume that the functions $\varphi \in \mathcal{C}^{4,
\alpha} (S^3)$, $\psi \in \mathcal{C}^{2, \alpha} (S^3)$ and the
constant $\tau >0$ satisfy
\begin{equation}
\begin{gathered}
\frac{1}{\log 1/r^2_{\varepsilon,\lambda, \gamma}}|\log (\tau /
\tau_*)|\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} ,  \quad
\|\varphi \|_{\mathbb{C}^{4, \alpha} (S^3)}\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} , \\
 \| \psi \|_{\mathbb{C}^{2, \alpha} (S^3)}\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} ,
\end{gathered}\label{eq:6.5}
\end{equation}
where $\tau_* >0$ is fixed later.

\begin{lemma} \label{le:6.1}
For each $\kappa > 0$, $\mu\in(1,2)$ and $\delta\in(0,1)$, there exist
$\varepsilon_\kappa >0$, $\lambda_{\kappa}>0$, $\gamma_{\kappa} >0$,
$c_\kappa >0$ and $\bar c_\kappa >0$ such that, for all
$\varepsilon \in (0, \varepsilon_\kappa )$, $\lambda \in (0, \lambda_\kappa )$ and
$\gamma \in (0, \gamma_\kappa )$,
\begin{equation}
\| N (\varepsilon, \lambda,\gamma, \tau, \varphi, \psi  ;  0) \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)}\leq c_\kappa  \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}}.
\label{eq:6.6}
\end{equation}
Moreover,
\begin{equation}
\| N (\varepsilon, \lambda,\gamma, \tau, \varphi, \psi  ;  v_2)
 - N (\varepsilon, \lambda,\gamma, \tau, \varphi,
\psi ; v_1) \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}
\leq \bar c_\kappa  {r^2_{\varepsilon,\lambda, \gamma}}  \| v_2 - v_1
\|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} \label{eq:6.7}
\end{equation}
provided $\tilde v= v_1, v_2 \in \mathcal{C}^{4, \alpha}_\mu
(\mathbb{R}^4)$, $\varphi \in \mathcal{C}^{4, \alpha} (S^3)$,
$\psi \in \mathcal{C}^{2, \alpha} (S^3)$ satisfy
\begin{gather*}
\| \tilde v \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} \leq 2
 c_\kappa  \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}} , \quad
 \|\varphi \|_{\mathbb{C}^{4, \alpha} (S^3)}\leq \kappa  r^2_{\varepsilon,\lambda, \gamma},\\
 \| \psi \|_{\mathbb{C}^{2, \alpha} (S^3)} \leq \kappa  r^2_{\varepsilon,\lambda, \gamma} ,
\quad |\log (\tau / \tau_*)|\leq \kappa  r^2_{\varepsilon,\lambda, \gamma}
\log 1/r^2_{\varepsilon,\lambda, \gamma}.
\end{gather*}
\end{lemma}

\begin{proof}  The  estimates follow from  Lemma \ref{le:5.1}
together with the assumption on the norms of
$\varphi$ and $\psi$. Let $c^{(i)}_\kappa$ denote constants which
only depend on $\kappa$ (provided $\varepsilon$, $\lambda$ and $\gamma$
are chosen small enough).

It follows from Lemma \ref{le:5.1} and the estimates given by \eqref{eq:6.5} that
\begin{equation}
\begin{aligned}
\| H^i (\varphi, \psi ;  \cdot / R_{\varepsilon,\lambda, \gamma}) \|_{\mathcal{C}^{4,
\alpha} _{2} ( \bar B_{R_{\varepsilon,\lambda, \gamma}})} \\
&\leq  c_\kappa R_{\varepsilon,\lambda, \gamma}^{-2}  (\| \varphi
\|_{\mathcal{C}^{4, \alpha}(S^3)} + \| \psi \|_{\mathcal{C}^{2,
\alpha}(S^3)}) \\
&\leq c_\kappa  \varepsilon^2
\end{aligned} \label{hii}
\end{equation}
Therefore, using the fact that for each $x \in \bar{B}_{R_{\varepsilon,\lambda, \gamma}}$, 
we have
$|h(x)|\leq c_{\kappa}\; r_{\varepsilon,\lambda,\gamma}^{2+\delta}$, which tends to $0$ as
$\varepsilon,\lambda$ and $\gamma$ tend to $0$, we obtain
\[
\| (1+ |\cdot|^2)^{-4} e^{h}  \big( e^{H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma})} -1 \big) \|_{\mathcal{C}^{0, \alpha}_{\mu-4}
(\bar B_{R_{\varepsilon,\lambda, \gamma}})}
\leq c_\kappa   \varepsilon^2,
\]
for $\mu\in(1,2)$, we have
\begin{align*}
&\big\|   \varepsilon^2 \big[|\nabla\left(u_1+h+ H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma}) \right)|^2-|\nabla( u_{1} + h)|^2\big] 
 \big\|_{\mathcal{C}^{0, \alpha}_{\mu-4}
(\bar B_{R_{\varepsilon,\lambda, \gamma}})}\\ 
&\leq \big\|   \varepsilon^2 |\nabla\left( H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma}) \right)|
 \big[2|\nabla(u_1+h)| \\
&\quad + |\nabla\left(H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma}) \right)|\big]  
 \big\|_{\mathcal{C}^{0, \alpha}_{\mu-4}
(\bar B_{R_{\varepsilon,\lambda, \gamma}})}
\\ 
&\leq  c_\kappa \varepsilon^{\mu}r^{4-\mu}_{\varepsilon,\lambda, \gamma}\Big[1+ r^{\delta}_{\varepsilon,\lambda, \gamma}\varepsilon^{-\delta}
 \| h \|_{\mathcal{C}^{4, \alpha}_{{\rm rad}, \delta} 
 (\mathbb{R}^4)}+ r^2_{\varepsilon,\lambda, \gamma}\varepsilon^{-2}
 \| H^i (\varphi, \psi ;  \cdot / R_{\varepsilon,\lambda, \gamma}) \|_{\mathcal{C}^{4,
\alpha} _{2} ( \bar B_{R_{\varepsilon,\lambda, \gamma}})}\Big]
\end{align*}

Provided $h\in \mathcal{C}^{4, \alpha}_{{\rm rad}, \delta} (\mathbb{R}^4)$
satisfy  $\| h \|_{\mathcal{C}^{4, \alpha}_{{\rm rad}, \delta} 
(\mathbb{R}^4)}\leq 2  c_\kappa  \varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}$, 
and from the asymptotic behavior of $H^i$ given by the estimate \eqref{hii}
and $\mu\in(1,2)$, we deduce that
\[
\left\|   \varepsilon^2 \Big[|\nabla\left(u_1+h+ H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma}) \right)|^2-|\nabla( u_{1} + h)|^2\Big]  
\right\|_{\mathcal{C}^{0, \alpha}_{\mu-4}
(\bar B_{R_{\varepsilon,\lambda, \gamma}})}
\leq c_\kappa     \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}}
\]
and
\[
\left\|  \varepsilon^2 \Delta\left( H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma})  \right) \right\|_{\mathcal{C}^{0, \alpha}_{\mu-4}
(\bar B_{R_{\varepsilon,\lambda, \gamma}})} \leq c_\kappa  
 \varepsilon^{\mu}{r^{4-\mu}_{\varepsilon,\lambda, \gamma}}\leq c_\kappa 
 \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}}.
\]
Using Proposition~\ref{pr:surj-1'} and \eqref{eq:6.3}, we conclude that
\[
\| N (\varepsilon, \lambda,\gamma, \tau, \varphi, \psi ;  0) \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)}\leq c_\kappa    \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}}.
\]
To derive the second estimate, we use the fact that for each 
$x \in \bar{B}_{R_{\varepsilon,\lambda, \gamma}}$, we have 
$|h(x)|\leq c_{\kappa} r_{\varepsilon,\lambda, \gamma}^{2+\delta}$,
 which tends to $0$ as $\varepsilon, \lambda$ and $\gamma$ tend to $0$.  
For each $\kappa>0$, there exists $c_\kappa>0$ such that
\begin{align*} 
&\big\| (1+ |\cdot|^2)^{-4}
 e^{H^i({\varphi, \psi}; \cdot / R_{\varepsilon,\lambda, \gamma}) +h}
 \left( e^{v_2} - e^{v_1} - v_2 + v_1 \right) \big\|_{\mathcal{C}^{0,
\alpha}_{\mu-4} (\bar B_{R_{\varepsilon,\lambda, \gamma}})} \\
&\leq c_\kappa  \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}} \| v_2 
- v_1 \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}
\end{align*}
and
\begin{align*}
&\big\| (1+ |\cdot|^2)^{-4}  e^{h}\left( e^{H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma}) }  -1 \right)  \left(v_2  - v_1 \right)
\big\|_{\mathcal{C}^{0, \alpha}_{\mu-4} (\bar B_{R_{\varepsilon,\lambda, \gamma}})} \\
&\leq c_\kappa    \varepsilon^2\| v_2 - v_1 \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)} ,
\end{align*}
and for $\mu\in(1,2)$, we obtain
\[
\| \varepsilon^2 \Delta(v_2-v_1)\|_{\mathcal{C}^{0, \alpha}_{\mu-4}
(\bar B_{R_{\varepsilon,\lambda, \gamma}})} 
\leq c_\kappa  r^2_{\varepsilon,\lambda, \gamma}\| v_2 - v_1 \|_{\mathcal{C}^{4,
\alpha}_\mu (\mathbb{R}^4)}.
\]
Provided $h\in \mathcal{C}^{4, \alpha}_{{\rm rad}, \delta} (\mathbb{R}^4)$
satisfy  $\| h \|_{\mathcal{C}^{4, \alpha}_{{\rm rad}, \delta} (\mathbb{R}^4)}
\leq 2  c_\kappa  \varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}$, we deduce that
\begin{align*}
\left\| (1+ |\cdot|^2)^{-4}  ( e^{h} - 1)  \left(v_2  - v_1
\right) \right\|_{\mathcal{C}^{0, \alpha}_{\mu-4} (\bar
B_{R_{\varepsilon,\lambda, \gamma}})} 
&\leq c_\kappa   \| h \|_{\mathcal{C}^{4, \alpha}_\delta (\mathbb{R}^4)}
 \| v_2 - v_1 \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}\\
&\leq c_\kappa \varepsilon^{\delta}r^2_{\varepsilon,\lambda, \gamma}
 \| v_2 - v_1 \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} ,
 \end{align*}
and
\begin{align*}
&\Big\|  | \varepsilon^2 \Big(\Big|\nabla(u_1+h+ H^i(\varphi, \psi ;
\cdot / R_{\varepsilon,\lambda, \gamma})+v_2)\Big|^2\\
&-\Big|\nabla(u_1+h+ H^i(\varphi, \psi ; \cdot / R_{\varepsilon,\lambda,
\gamma})+v_1)\Big|^2\Big) \Big\|_{\mathcal{C}^{0,
\alpha}_{\mu-4} (\bar B_{R_{\varepsilon,\lambda, \gamma}})} 
\\
&\leq \|\varepsilon^2 {|\nabla ( v_2-v_1)| \Big(|\nabla (
v_2+v_1)|+2|\nabla (u_1+h+  H^i(\varphi, \psi ; \cdot /
R_{\varepsilon,\lambda, \gamma}))}|\Big) \|_{\mathcal{C}^{0,
\alpha}_{\mu-4} (\bar B_{R_{\varepsilon,\lambda, \gamma}})}  
\\
&\leq c_\kappa \varepsilon^2R^2_{\varepsilon,\lambda, \gamma} \Big(R^{\mu}_{\varepsilon,\lambda, \gamma}
\sum_{i=1}^2\| v_i \|_{\mathcal{C}^{4, \alpha}_\mu 
 (\mathbb{R}^4)}+1\\
&\quad +R^2_{\varepsilon,\lambda, \gamma} \| H^i (\varphi, \psi ;
 \cdot / R_{\varepsilon,\lambda, \gamma}) \|_{\mathcal{C}^{4, \alpha} _{2} 
 (\bar B_{R_{\varepsilon,\lambda, \gamma}})}\Big)\| v_2 - v_1 
 \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}\\
&\leq c_\kappa   r^2_{\varepsilon,\lambda, \gamma} 
 \Big(r^{\mu}_{\varepsilon,\lambda, \gamma}\varepsilon^{-\mu}
\sum_{i=1}^2\| v_i \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}
 +1\\
&\quad +r^2_{\varepsilon,\lambda, \gamma}\varepsilon^{-2}
\| H^i (\varphi, \psi ;  \cdot / R_{\varepsilon,\lambda, \gamma}) \|_{\mathcal{C}^{4,
\alpha} _{2} ( \bar B_{R_{\varepsilon,\lambda, \gamma}})}\Big)\| v_2 - v_1
\|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}.
\end{align*}
Provided $v_1, v_2 \in \mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)$
satisfy  $\| v_i \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)}
\leq 2  c_\kappa  \varepsilon^{\mu}r^2_{\varepsilon,\lambda, \gamma}$ and from
the asymptotic behavior of $H^i$ given by the estimate
\eqref{hii}, we derive the desired estimate,
using Proposition~\ref{pr:surj-1'} and \eqref{eq:6.3}.
\end{proof}

Reducing $\varepsilon_\kappa, \lambda_\kappa$ and $\gamma_\kappa$ if necessary, 
we can assume that
\begin{equation}
\bar c_\kappa   {r^2_{\varepsilon,\lambda, \gamma}} \leq \frac{1}{2}. \label{eq:6.9}
\end{equation}  
Then there exist $\varepsilon_\kappa >0, \lambda_{\kappa}>0$, $\gamma_{\kappa} >0$, 
$c_\kappa >0$ and $\bar c_\kappa >0$ such that, for all $\varepsilon \in (0, \varepsilon_\kappa )$, 
$\lambda \in (0, \lambda_\kappa )$
and $\gamma \in (0, \gamma_\kappa )$. Then,
\eqref{eq:6.6} and \eqref{eq:6.7} in Lemma~\ref{le:6.1} are sufficient 
to show that
\[
v  \mapsto N (\varepsilon, \lambda,\gamma, \tau, \varphi, \psi ; v)
\]
is a contraction from
$\{ v \in \mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4) :
\|v \|_{\mathcal{C}^{4, \alpha}_\mu (\mathbb{R}^4)} \leq 2
 c_\kappa   \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}} \}$
into itself and hence has a unique fixed point 
$ v (\varepsilon, \lambda, \gamma, \tau, \varphi, \psi ; \cdot )$ in this set.
 This fixed point is a solution of \eqref{eq:6.4} in 
$B_{R_{\varepsilon,\lambda, \gamma}}$.
We summarize this as follows.

\begin{proposition} \label{pr:6.1}
For each $\kappa >1$, there exist $\varepsilon_\kappa >0$, $\lambda_\kappa >0$, 
$\gamma_\kappa >0$ and $c_\kappa >0$
(only depending on $\kappa$) such that given 
$\varphi \in \mathcal{C}^{4, \alpha} (S^3)$,
$\psi \in \mathcal{C}^{2, \alpha} (S^3)$ satisfying \eqref{eq:5.2} and
 $\tau >0$ satisfying
\[
|\log (\tau / \tau_*)|\leq \kappa   r^2_{\varepsilon,\lambda, \gamma}
\log 1/{ r^2_{\varepsilon,\lambda, \gamma}}, \quad
 \|\varphi \|_{\mathbb{C}^{4,\alpha} (S^3)}\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} ,\quad
\| \psi \|_{\mathbb{C}^{2, \alpha} (S^3)}\leq \kappa r^2_{\varepsilon,\lambda, \gamma} , 
\]
the function
\[
u (\varepsilon, \lambda,\gamma, \tau, \varphi, \psi ; \cdot ) : = u_1 +h+ H^i ({
\varphi}, { \psi} ;  \cdot / R_{\varepsilon,\lambda, \gamma}) + v(\varepsilon, \lambda,\gamma, \tau,
\varphi, \psi ; \cdot ),
\]
solves \eqref{eq:6.1} in $B_{R_{\varepsilon,\lambda, \gamma}}$. In addition
\begin{equation}
\| v (\varepsilon, \lambda,\gamma, \tau, \varphi, \psi ; \cdot ) 
\|_{\mathbb{C}^{4,\alpha}_{\mu}(\mathbb{R}^4)} \leq 2  c_\kappa 
 \varepsilon^{\mu}{r^2_{\varepsilon,\lambda, \gamma}}.
\label{eq:6.11} 
\end{equation} 
\end{proposition}

Observe that the function $ v (\varepsilon, \lambda,\gamma,\tau, \varphi, \psi ; \cdot)$ 
being obtained as a fixed point for contraction mapping, it
depends continuously on the parameter $\tau$.

\section{Second nonlinear Dirichlet problem}

For  $(\varepsilon,\lambda,\gamma) \in (0, r_0^2)^3$, we recall that
\[
r_{\varepsilon,\lambda,\gamma}:=\max(\sqrt{\varepsilon},\sqrt{\lambda},\sqrt{\gamma}) .
\]
Recall that $G ( x , \cdot)$ denotes the unique solution of
\[
\Delta^2  G ( x , \cdot)  = 64   \pi^2   \delta_{x}
\]
in $\Omega$, with $G ( x, \cdot) = \Delta  G (x, \cdot) =0$ on
$\partial \Omega$. In addition, the following decomposition holds
\[
G (x, y) = - 8   \log |x- y| + R (x, y)
\]
where $ y \mapsto R(x, y)$ is a smooth function.

We recall in this section a result which concerns the properties of the
Green’s function in the following lemma.

\begin{lemma} \label{lem6.1}
There exists $C > 0$ such that for all $x, y \in \Omega, x \neq y$, we have 
$$
|\nabla^{i}G (x, y)|\leq  C|x-y|^{-i}, i\geq 1.
$$
\end{lemma}

The  estimate in the above lemma is originally due to Krasovski\v{i} 
\cite{Krasovskii}.

Given $x^1, \ldots, x^m \in \Omega$. We  need the
following data:
\begin{itemize}

\item[(i)] Points $Y := (y^1, \ldots, y^m )\in \Omega^m$ close enough
to $X : = (x^1, \ldots, x^m)$.

\item[(ii)] Parameters $ \tilde{\eta} : = (\tilde{\eta}^1, \ldots, 
\tilde{\eta}^m )\in \mathbb{R}^m$ close to $0$.

\item[(iii)] Boundary data $\Phi : = (\varphi^1,
\ldots, \varphi^m ) \in (\mathcal{C}^{4, \alpha} (S^3))^m$ and
$\Psi  : = (\psi^1, \ldots, \psi^m ) \in (\mathcal{C}^{2, \alpha}
(S^3))^m$ each of which satisfies \eqref{eq:5.6}.
\end{itemize}
With all these data, we define
\begin{equation} 
\tilde {\mathbf{u} } : =
\sum_{j=1}^m (1 + \tilde{\eta}^j )  G ( y^j , \cdot ) +
\sum_{j=1}^m \chi_{r_0} (\cdot - y^j)  H^{e} ( \varphi^j, \psi^j
  ;  (\cdot - y^j) / r_{\varepsilon,\lambda, \gamma} ) \label{eq:7.1}
\end{equation}
where  $\chi_{r_0}$ is a cutoff function identically equal to $1$ in
$B_{r_0/2}$ and identically equal to $0$ outside $B_{r_0}$.

We define $\rho >0$ by
\[
\rho^4 = \frac{384   \varepsilon^4}{(1+\varepsilon^2)^4} .
\]
We would like to find a solution of the equation
\begin{equation}
\Delta^2  u -\gamma\Delta u -\lambda|\nabla u|^2 - \rho^4  e^u = 0 ,\label{eq:7.2}
\end{equation}
which is defined in $\bar \Omega_{r_{\varepsilon,\lambda, \gamma}} (Y)$ and which is a
perturbation of $\tilde{\mathbf{u}}$. Writing $u = \tilde{\mathbf{u}} +
\tilde v$, this amounts to solve
\begin{equation}
\Delta^2  \tilde v = \rho^4  e^{\tilde{\mathbf{u}}  + \tilde{v}}
- \Delta^2 \tilde{\mathbf{u}} + \gamma\Delta (\tilde{\mathbf{u}} 
+ \tilde{v}) + \lambda|\nabla (\tilde{\mathbf{u}}+ \tilde{v})|^2 . \label{eq:7.3}
\end{equation}
We need to define an auxiliary weighed space.

\begin{definition} \label{de:7.1} \rm
Given $\bar r \in (0, r_0/2)$, $k \in \mathbb{R}$, 
$\alpha \in (0,1)$ and $\nu\in \mathbb{R}$, we define the H\"older weighted
space ${\mathcal{C}}^{k,\alpha}_{\nu}(\bar \Omega_{\bar r}  (X))$
as the space of functions $w \in {\mathcal{C}}^{k,\alpha}(\bar
\Omega_{\bar r}  (X))$ which is endowed with the norm
\[
\| w \|_{{\mathcal{C}}^{ k, \alpha}_{\nu}(\bar \Omega_{\bar r}  (X))} 
: = \| w \|_{{\mathcal{C}}^{ k,\alpha}( \bar \Omega_{r_0/2}
  (X) )}+ \sum_{j=1}^m \sup_{r \in [\bar r, r_0/2)}  \left(
r^{-\nu} \| w(x^j + r  \cdot ) \|_{{\mathcal{C}}^{ k,\alpha}
(\bar B_2-B_1)} \right).
\]
\end{definition}

For all $\sigma \in (0, r_0/2)$ and all $Y \in \Omega^m$ such that
$\| X-Y\|\leq r_0/2$, we denote by
\[
\tilde{\mathcal E}_{\sigma,Y} : \mathcal{C}^{0, \alpha}_\nu (\bar
\Omega_\sigma
  (Y)) \to \mathcal{C}^{0, \alpha}_\nu (\bar \Omega^*  (Y)),
\]
the extension operator defined by $\tilde{\mathcal{E}}_{\sigma,Y}
(f) = f$ in $\bar \Omega_\sigma  (Y)$
\[
\tilde{\mathcal{E}}_{\sigma,Y} (f)  (y^i+x) 
= \tilde{\chi} \left( \frac{|x|}{\sigma} \right)  f \left(y^i 
+ \sigma  \frac{x}{|x|} \right)
\]
for each $j=1, \ldots, m$ and $\tilde{\mathcal{E}}_{\sigma,Y} (f) =
0$ in each $B_{\sigma/2} (y^j)$, where $t \mapsto \tilde \chi(t)$ 
is a cutoff function identically equal to $1$ for $t \geq 1$
and identically equal to $0$ for $t \leq 1/2$. It is easy to check
that there exists a constant $c =  c(\nu) >0$ only depending on
$\nu$ such that
\begin{equation}
\| \tilde{\mathcal{E}}_{\sigma,Y}  (w) \|_{\mathcal{C}^{0,
\alpha}_{\nu} (\bar \Omega^*  (X))} 
\leq c \| w \|_{\mathcal{C}^{0, \alpha}_{\nu} (\bar \Omega_{\sigma}  (X))}. 
\label{le:7.2}
\end{equation}
We fix
$\nu \in (-1,0)$,
and denote by $\tilde{\mathcal{G}}_{\nu, Y}$ the right inverse provided by
Proposition \ref{pr:4.1}. Clearly, it is enough to find $\tilde v
\in \mathcal{C}^{4, \alpha}_\nu (\Omega^*  (Y))$ solution of
\begin{equation}
\tilde v =  \tilde N (\varepsilon, \lambda, \gamma, \tilde{\eta} , Y, \Phi, \Psi ; \tilde v )
\label{eq:7.30}
\end{equation}
 where we have defined
\begin{align*}
\tilde{N}(\tilde{v})
&:=\tilde N (\varepsilon, \lambda, \gamma, \tilde{\eta} , Y, \Phi, \Psi ; \tilde v )\\
&:= \tilde{\mathcal{G}} \circ {\tilde{\mathcal{E}}}_{r_{\varepsilon,\lambda, \gamma} , Y}  
\left( \rho^4  e^{\tilde{\mathbf{u}}  + \tilde{v}}
- \Delta^2 \tilde{\mathbf{u}} + \gamma\Delta (\tilde{\mathbf{u}} + \tilde{v}) 
+ \lambda|\nabla (\tilde{\mathbf{u}}+ \tilde{v})|^2 \right)\\
&:=\tilde{\mathcal{G}}_{\nu, Y} \circ {\tilde{\mathcal{E}}}_{r_{\varepsilon,\lambda, \gamma} ,
 Y}  
\big(\tilde{S}(v)\big)
\end{align*}

Given $\kappa >0$ (whose value will be fixed later on), we further
assume that $\Phi$ and $\Psi$ satisfy
\begin{equation}
\| \Phi \|_{(\mathbb{C}^{4, \alpha} (S^3))^m}\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} ,
 \quad\text{and} \quad 
\| \Psi \|_{(\mathbb{C}^{2, \alpha} (S^3))^m }\leq \kappa  r^2_{\varepsilon,\lambda, \gamma}. 
\label{eq:7.4}
\end{equation}
Moreover, we assume that the parameters $\tilde{\eta}$ and the points $Y$
are chosen to satisfy
\begin{equation}
| \tilde{\eta} |\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} , \quad \text{and} \quad
\|Y-X \| \leq \kappa  r_{\varepsilon,\lambda, \gamma} . \label{eq:7.5}
\end{equation}
Then, the following result holds.

\begin{lemma} \label{le:7.1}
For each $\kappa >1$, there exist $\varepsilon_\kappa >0$, $\lambda_\kappa >0$, 
$\gamma_\kappa >0$, $c_\kappa>0$ and $\bar c_\kappa >0$ such that, for all 
$\varepsilon \in (0, \varepsilon_\kappa), \lambda \in (0,\lambda_\kappa)$ and
 $\gamma \in (0,\gamma_\kappa)$, we have
\begin{equation}
\| \tilde N (\varepsilon, \lambda, \gamma, \tilde{\eta} , Y, \Phi, \Psi ; 0 ) 
\|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*  (Y))}
\leq c_\kappa  r_{\varepsilon,\lambda, \gamma}^2 . \label{eq:7.6}
\end{equation}
Moreover,
\begin{equation}
\begin{aligned}
&\| \tilde N ( \varepsilon, \lambda, \gamma,\tilde{\eta} , Y, \Phi, \Psi ; \tilde v_2 ) -
\tilde N (\varepsilon, \lambda, \gamma,\tilde{\eta} , Y, \Phi, \Psi ; \tilde v_1 )
\|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*  (Y))}\\
&\leq \bar c_\kappa  r_{\varepsilon,\lambda, \gamma}^2 \| \tilde v_2 - \tilde v_1
\|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*  (Y))}
\end{aligned} \label{eq:7.7}
\end{equation}
provided $\tilde v= v_1, v_2 \in \mathcal{C}^{4, \alpha}_\nu
(\bar \Omega^*  (Y))$, 
$\tilde \Phi = \Phi_1, \Phi_2 \in (\mathcal{C}^{4, \alpha} (S^3))^m$, 
$\tilde \Psi =\Psi_1, \Psi_2 \in (\mathcal{C}^{2, \alpha} (S^3))^m$ satisfy
\begin{gather*}
\| \tilde v \|_{\mathcal{C}^{4, \alpha}_\nu ({\bar \Omega^*}  (Y))} 
 \leq 2  c_\kappa  r_{\varepsilon,\lambda, \gamma}^2 , \quad
\|\tilde \Phi \|_{(\mathbb{C}^{4, \alpha} (S^3))^m}
 \leq \kappa  r^2_{\varepsilon,\lambda, \gamma} , \\
\| \tilde \Psi \|_{(\mathbb{C}^{2, \alpha} (S^3))^m}\leq \kappa
 r^2_{\varepsilon,\lambda, \gamma} , \quad
|\tilde{\eta} | \leq \kappa  r^2_{\varepsilon,\lambda, \gamma} , \quad
\|Y -X\| \leq \kappa  r_{\varepsilon,\lambda, \gamma} .
 \end{gather*}
\end{lemma}


\begin{proof} The  first estimate follows from the
asymptotic behavior of $H^e$ together with the assumption on the norm 
of boundary data $\tilde{\varphi}^{i}$
given by \eqref{eq:7.4}. Indeed, let $c_\kappa$ be a constant
depending only on $\kappa$ (provided $\varepsilon$, $\lambda$  and $\gamma$ are chosen
small enough) it follows from the estimate of $H^e$, given by lemma
\ref{le:5.2}, that
\begin{equation}
|H^e_{\tilde \varphi^j, \tilde \psi^j}((x-y^{j})/r_{\varepsilon,\lambda,\gamma})|
\leq c_\kappa r_{\varepsilon,\lambda, \gamma}^3r^{-1}.\label{eq:he}
\end{equation}
Recall that $\tilde N (\tilde v) = \tilde{\mathcal{G}}_\nu 
\circ {\tilde{\xi}}_{r_{\varepsilon,\lambda, \gamma}} \circ \tilde{S}(\tilde v)$, we
will estimate $\tilde N(0)$ in different subregions of
$\bar\Omega^*$.

 $\bullet$ In $B_{r_0/2}({y}^{j})$ for $1\leq j \leq m$,
we have
 $\chi_{r_0}(x-{y}^{j})=1$ and $\Delta^2{\bf\tilde{u}}=0$ so that
\begin{align*}
  |\tilde{S}(0)| 
&\leq  c_\kappa \varepsilon^4 {\prod_{j=1}^{m}\Big[ e^{(1 + \tilde \eta^j )  G_{{y}^j}(x) +
H^{e}_{\tilde \varphi^j, \tilde \psi^j}((x - {y}^j)/
r_{\varepsilon,\lambda, \gamma} )}}+\gamma|\Delta\tilde{\mathbf{u}}|+\lambda|\nabla
\tilde{\mathbf{u}}|^2\\ 
&\leq c_\kappa\varepsilon^4\prod_{j=1}^m|x-{y}^{j}|^{-8(1+\tilde{\eta}^j)}
+ \gamma|\Delta (\tilde{\mathbf{u}})| + \lambda|\nabla (\tilde{\mathbf{u}})|^2
\\ 
&\leq c_\kappa\varepsilon^4|x-{y}^{j}|^{-8(1+\tilde{\eta}^j)}\prod_{\ell=1,\ell\neq j}^m
 |x-{y}^{\ell}|^{-8(1+\tilde{\eta}^j)}
\\ 
&\quad + c_\kappa\gamma(1+\tilde{\eta}^j)\sum_{j=1}^{m}\Delta R(x,{y}^{j})
+ c_\kappa\gamma(1+\tilde{\eta}^j)|x-{y}^{j}|^{-2}\sum_{\ell=1,\ell \neq j}^{m}
|x-{y}^{\ell}|^{-2}
\\ 
&\quad + c_\kappa\gamma r^3_{\varepsilon,\lambda, \gamma}|x-{y}^{j}|^{-3}
 \sum_{\ell=1,\ell \neq j}^{m}|x-{y}^{\ell}|^{-3}
\\ 
&\quad + c_\kappa\lambda(1+\tilde{\eta}^j)\sum_{j=1}^{m}|\nabla R(x,{y}^{j})|^2
+ c_\kappa\lambda(1+\tilde{\eta}^j)|x-{y}^{j}|^{-2}
 \sum_{\ell=1,\ell \neq j}^{m}|x-{y}^{\ell}|^{-2}
\\ 
&\quad + c_\kappa\lambda r^3_{\varepsilon,\lambda, \gamma}|x-{y}^{j}|^{-4}
 \sum_{\ell=1,\ell \neq j}^{m}|x-{y}^{\ell}|^{-4}.
\end{align*}
Hence, for $\nu\in(-1,0)$ and $\tilde{\eta}^j$ small enough, we get
\begin{align*}
\|\tilde{N}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}(\bigcup_{j=1}^{m}B({y}^{j},r_0/2))}
&\leq  \sup_{r_{\varepsilon,\lambda, \gamma} \leq r\leq
r_0/2}r^{4-\nu}|\tilde{N}(0)|\\
&\leq  c_\kappa\varepsilon^4 r_{\varepsilon,\lambda, \gamma}^{-4}+2c_\kappa\gamma 
 +c_\kappa\gamma r^3_{\varepsilon,\lambda, \gamma} 
+c_\kappa\lambda+c_\kappa\gamma r^3_{\varepsilon,\lambda, \gamma}.
\end{align*}

 $\bullet$ In $\Omega_{r_0, \tilde x}$ (recall that
$\Omega_{r_0, \tilde x} = \Omega\setminus\cup_j B_{r_0}(\tilde
x^j)$), we have $\chi_{r_0}(x-{y}^{j})=0$ and
$\Delta^2\tilde{\mathbf{u}}=0$, then
\begin{align*}
|\tilde{S}(0)| 
&\leq  c_\kappa\varepsilon^4 |x-{y}^{j}|^{-8(1+\tilde{\eta}^j)}
 \prod_{\ell=1,\ell\neq j}^m|x-{y}^{\ell}|^{-8(1+\tilde{\eta}^j)}
\\ 
&\quad + c_\kappa\gamma(1+\tilde{\eta}^j)\sum_{j=1}^{m}\Delta R(x,{y}^{j})
+ c_\kappa\gamma(1+\tilde{\eta}^j)|x-{y}^{j}|^{-2}
 \sum_{\ell=1,\ell \neq j}^{m}|x-{y}^{\ell}|^{-2}
\\ 
&\quad + c_\kappa\lambda(1+\tilde{\eta}^j)\sum_{j=1}^{m}|\nabla R(x,{y}^{j})|^2
+ c_\kappa\lambda(1+\tilde{\eta}^j)|x-{y}^{j}|^{-2}
 \sum_{\ell=1,\ell \neq j}^{m}|x-{y}^{\ell}|^{-2} .
\end{align*}
Thus
$$
\|\tilde{N}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}(\Omega_{r_0,
\tilde x})} \leq c_\kappa \sup_{ r\geq r_0}r^{4-\nu}|\tilde{S}(0)|\leq c_\kappa
\varepsilon^4+c_\kappa\gamma+c_\kappa\lambda.
$$

$\bullet$ In $B_{r_0}({y}^{j})- B_{r_0/2}({y}^{j})$, for 
$j= 1, \ldots, m$, we have
\begin{align*}
|\tilde{S}(0)| 
&\leq c_\kappa\varepsilon^4\Big|\prod_{j=1}^m \;e^{(1 +
 \tilde \eta^j )G_{{y}^j } }e^{\chi_{r_0}(x-{y}^{j})H^{e}_{\tilde
 \varphi^j, \tilde \psi^j}((x - {y}^j)/ r_{\varepsilon,\lambda, \gamma} )} 
 +\Delta^2\tilde{\mathbf{u}}+\Delta\tilde{\mathbf{u}}
 +|\nabla \tilde{\mathbf{u}}|^2\Big|\\
&\leq c_\kappa \varepsilon^4|x-{y}^{j}|^{-8(1+\tilde{\eta}^j)}
 \prod_{\ell=1,\ell\neq 1}^m|x-{y}^{\ell}|^{-8(1+\tilde{\eta}^\ell)}\\ 
&\quad + c_\kappa\varepsilon^4 \sum_{j=1}^{m}
 |[\Delta^2,\chi_{r_0}(x-{y}^{j})]| |H^{\rm ext}_{\tilde{\varphi}_j,\tilde{\psi}_j}
 ((x-{y}^{j})/r_{\varepsilon,\lambda, \gamma})|\\
&\quad + c_\kappa\gamma \sum_{j=1}^{m}
 |[\Delta,\chi_{r_0}(x-{y}^{j})]| |H^{\rm ext}_{\tilde{\varphi}_j,\tilde{\psi}_j}
 ((x-{y}^{j})/r_{\varepsilon,\lambda, \gamma})|\\
&\quad  + c_\kappa\gamma(1+\tilde{\eta}^j)\sum_{j=1}^{m}|\Delta R(x,{y}^{j})|\\
&\quad + c_\kappa \lambda\sum_{j=1}^{m}
 \Big|[\nabla,\chi_{r_0}(x-{y}^{j})]\Big|^2
 |H^{\rm ext}_{\tilde{\varphi}_j,\tilde{\psi}_j}
 ((x-{y}^{j})/r_{\varepsilon,\lambda, \gamma})| \\
&\quad + c_\kappa\lambda(1+\tilde{\eta}^j)\sum_{j=1}^{m}|\nabla R(x,{y}^{j})|^2
\\ 
&\quad+  c_\kappa\gamma(1+\tilde{\eta}^j)|x-{y}^{\ell}|^{-2}
 \sum_{\ell=1,\ell \neq j}^{m}|x-{y}^{j}|^{-2} \\
&\quad + c_\kappa\lambda(1+\tilde{\eta}^j)|x-{y}^{\ell}|^{-2}
 \sum_{\ell=1,\ell \neq j}^{m}|x-{y}^{j}|^{-2}.
\end{align*}
Here
\begin{align*}
&[\nabla,\chi_{r_0}]w\\
&=\nabla \chi_{r_0}\cdot w+ \chi_{r_0}\cdot\nabla w
[\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.
\end{align*}
So,
$$
\|\tilde{N}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}(B({y}^{j},r_0)-B({y}^{j},r_0/2))}
\leq c_\kappa \sup_{r_0/2\leq r\leq
r_0}r^{4-\nu}|\tilde{N}(0)| \leq c_\kappa r^2_{\varepsilon,\lambda, \gamma}.
$$
Finally
$$
\|\tilde{N}(0)\|_{\mathcal{C}^{4,\alpha}_{\nu}
(\Omega-\bigcup_{j=1}^{m}B({y}^{j},r_{\varepsilon,\lambda, \gamma}))}
\leq  c_\kappa (\varepsilon^4+r_{\varepsilon,\lambda,\gamma}^3+\gamma+\lambda).
$$
To derive the second estimate, we use the fact that for
$\tilde{v}_1$ and 
$\tilde{v}_2 \in\tilde{B}_{ r^2_{\varepsilon,\lambda,\gamma}}$ of 
$C^{4, \alpha}_\nu(\bar\Omega^*)$, we obtain
\[
 \|\tilde N (\tilde v_1)-\tilde N (\tilde
 v_2)\|_{\mathcal{C}^{4,\alpha}_{\nu}(\Omega_{r_{\varepsilon,\lambda, \gamma}, {y}^{j}})}
 \leq \left\|\tilde{\mathcal{G}}_{\nu, Y} \circ
{\tilde{\xi}}_{r_{\varepsilon,\lambda, \gamma}} \Big(\tilde{S}(\tilde v_1) 
 - \tilde{S}(\tilde v_2)\Big)\right\|_{\mathcal{C}^{4,\alpha}_{\nu}
 (\Omega_{r_{\varepsilon,\lambda, \gamma}, {y}^{j}})}.
\]
Using\eqref{le:7.2} and Proposition~\ref{pr:4.1}, we conclude that 
\[
\|\tilde N (\tilde v_1)-\tilde N (\tilde
v_2)\|_{\mathcal{C}^{4,\alpha}_{\nu}(\Omega_{r_{\varepsilon,\lambda, \gamma},
{y}^{j}})}\leq c_\kappa r_{\varepsilon,\lambda, \gamma}^2 \|\tilde v_1-\tilde
v_2\|_{\mathcal{C}^{4,\alpha}_{\nu} (\bar\Omega^*)}.
\]
\end{proof}

Reducing $\varepsilon_\kappa, \lambda_\kappa$ and $\gamma_\kappa$ if necessary,
 we can assume that 
\[ 
\bar c_\kappa  r_{\varepsilon,\lambda, \gamma}^2  \leq \frac{1}{2} 
\]
for all $\varepsilon \in (0, \varepsilon_\kappa )$, $\lambda \in (0, \lambda_\kappa )$ and 
$\gamma \in (0, \gamma_\kappa )$. 
Then, \eqref{eq:7.6} and \eqref{eq:7.7} are sufficient to
show that 
$\tilde v  \mapsto \tilde N (\varepsilon, \lambda, \gamma, \tilde{\eta} , Y,
\Phi, \Psi    \tilde v)$
is a contraction from
\[
\{ \tilde{v} \in \mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*  (Y)) 
: \|\tilde{v} \|_{\mathcal{C}^{4, \alpha}_\nu (\bar \Omega^*  (Y))} \leq 2 
 c_\kappa  r_{\varepsilon,\lambda, \gamma}^2 \}
\]
into itself and hence has a unique fixed point 
$\tilde v (\varepsilon,\lambda, \gamma, \tilde{\eta} , Y, \Phi, \Psi ; \cdot )$
in this set. This fixed point is  a solution of
\eqref{eq:7.3}.

We summarize these resutls as follows.

\begin{proposition} \label{pr:7.1}
For each $\kappa > 0$, there exists $\varepsilon_\kappa >0$, $\lambda_\kappa >0$, 
$\gamma_\kappa >0$, and $c_\kappa >0$
(only depending on $\kappa$) such that for all $\varepsilon \in (0, \varepsilon_\kappa)$, 
$\lambda \in (0, \lambda_\kappa)$, $\gamma \in (0, \gamma_\kappa)$ 
 and for all set of parameters $\tilde{\eta}$, points $Y$
satisfying
\[
| \tilde{\eta}|\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} , \quad \text{and} \quad
\|Y-X \| \leq \kappa  r_{\varepsilon,\lambda, \gamma}
\]
and boundary functions $\Phi$ and $\Psi$ satisfying \eqref{eq:5.6}
and
\[
\| \Phi \|_{(\mathbb{C}^{4, \alpha} (S^3))^m}\leq \kappa  r^2_{\varepsilon,\lambda, \gamma} ,
 \quad
\| \Psi \|_{(\mathbb{C}^{2, \alpha} (S^3))^m }\leq \kappa  r^2_{\varepsilon,\lambda, \gamma}.
\]
The function
\begin{align*}
\tilde u(\varepsilon,\lambda, \gamma, \tilde{\eta} , Y, \Phi, \Psi ; \cdot )  
& : =  \sum_{j=1}^m (1 + \tilde{\eta}^j )  G_{y^j} +
\sum_{j=1}^m \chi_{r_0} (\cdot - y^j)  H^{e} (\varphi^j, \psi^j ;  (\cdot - y^j)/
r_{\varepsilon,\lambda, \gamma} ) \\
 &\quad  + \tilde v(\varepsilon, \lambda, \gamma, \tilde{\eta} , Y, \Phi, \Psi ;\cdot ),
\end{align*}
is a solution to \eqref{eq:7.2} in $\bar \Omega_{r_{\varepsilon,\lambda, \gamma}}  (Y)$. 
In addition
\begin{equation}
\| \tilde v (\varepsilon, \lambda, \gamma, \tilde{\eta} , Y, \Phi, \Psi ; 
\cdot ) \|_{\mathbb{C}^{4,\alpha}_{\nu}(\bar \Omega^*)} \leq 2 
 c_\kappa  r^2_{ \varepsilon, \lambda, \gamma }.
\label{eq:7.10} 
\end{equation}
\end{proposition}

Observe that the function $ \tilde v({\varepsilon,\lambda,
\gamma,\tilde{\eta} , Y , \Phi, \Psi})$ being obtained as a fixed
point for contraction mapping, it depends continuously on the
parameters $\tilde{\eta} $ and the points $Y$.

\section{Nonlinear Cauchy-data matching}

Keeping the notations of the previous sections, we gather the
results of the Proposition~\ref{pr:6.1} and
Proposition~\ref{pr:7.1}. From now let $\kappa >1$ is fixed large
enough (we will shortly see how) and assume that 
$\varepsilon \in (0, \varepsilon_\kappa)$, $\lambda \in (0, \lambda_\kappa)$ and 
$\gamma \in (0, \gamma_\kappa)$.



Assume that $X =(x^1, \ldots, x^m) \in \Omega^m$  is a nondegenerate
critical point of the function $W$ defined in the introduction. For
all $j=1, \ldots, m$, we define $\tau^j_*>0$ by
\begin{equation}
- 4  \log \tau^j_* =  R (x^j, x^j) + \sum_{\ell \neq j}
G(x^\ell, x^j). \label{eq:8.0}
\end{equation}

We assume that we are given:
\begin{itemize}

\item[(i)] points $Y : = (y^1, \ldots, y^m) \in \Omega^m$ close to $X : = (x^1,
\ldots, x^m)$ satisfying \eqref{eq:7.5}.

\item[(ii)] parameters $\tilde{\eta} : = (\tilde{\eta}^1, \ldots, \tilde{\eta}^m) \in \mathbb{R}^m
$ satisfying \eqref{eq:7.5}.

\item[(iii)] parameters $T : = (\tau^1, \ldots, \tau^m) \in (0,
\infty)^m$ satisfying \eqref{eq:6.5} (where, for each $j=1, \ldots,
m$, $\tau_*$ is replaced by $\tau_*^j$).
\end{itemize}
We set
\[
R_{\varepsilon,\lambda, \gamma}^j: = \tau^j/ r_{\varepsilon,\lambda, \gamma}
\]
First, we consider the  boundary data
\[
\Phi : = (\varphi^1, \ldots, \varphi^m) \in (\mathbb{C}^{4,\alpha}(S^3))^m 
\quad \text{and} \quad \Psi : = (\psi^1,
\ldots, \psi^m) \in (\mathbb{C}^{2,\alpha}(S^3))^m
\]
satisfying \eqref{eq:5.2} and \eqref{eq:6.5}.

Thanks to the result in Proposition~\ref{pr:6.1}, we can find 
$u_{\rm int}$ a solution of
\[
\Delta^2  u  -\lambda\Delta u-\lambda|\nabla u|^2- \rho^4  e^u = 0
\]
in each $B_{r_{\varepsilon,\lambda, \gamma}} (y^j)$, which can be decomposed as
\begin{align*}
&u_{\rm int} (\varepsilon, \lambda, \gamma, T, Y, \Phi, \Psi ; x) \\
& : =  u_{\varepsilon,\tau^j}(x- y^j) +h(R_{\varepsilon,\lambda, \gamma}^j(x-y^{j})
/r_{\varepsilon,\lambda, \gamma})+ H^{i} ({ \varphi^j}, { \psi^j}   ;
 (x-y^j)/r_{\varepsilon,\lambda, \gamma} ) \\
&\quad +  v (\varepsilon, \lambda, \gamma, \tau^j, \varphi^j, \psi^j ; 
R_{\varepsilon,\lambda, \gamma}^j (x - y^j)/ r_{\varepsilon,\lambda, \gamma} )
\end{align*}
in $B_{r_{\varepsilon,\lambda, \gamma}} (y^j)$.
Similarly, given the boundary data
\[
\tilde \Phi  : = (\tilde \varphi^1, \ldots , \tilde \varphi^m ) \in
(\mathbb{C}^{4,\alpha}(S^3))^m \quad \text{and} \quad \tilde \Psi :=
(\tilde \psi^1, \ldots, \tilde \psi^m ) \in (\mathbb{C}^{2,\alpha}(S^3))^m
\]
satisfying \eqref{eq:5.6} and \eqref{eq:7.4}, we use the result of
Proposition~\ref{pr:7.1}, to find $u_{\rm ext}$ a solution of
\[
\Delta^2-\lambda\Delta u-\lambda|\nabla u|^2- \rho^4  e^u  =0
\]
in $\bar \Omega_{r_{\varepsilon,\lambda, \gamma}}  (Y)$, which can be decomposed as
\begin{align*}
&u_{\rm ext}(\varepsilon, \lambda, \gamma, \tilde{\eta}, \tilde \Phi, \tilde \Psi ; x) \\
& =  \sum_{j=1}^m (1+ \tilde{\eta}^j)  G (y^j , x) +
\sum_{j=1}^m \chi_{r_0} (x - y^j)  H^{e} (\tilde{{\varphi}}^j, {
\tilde{\psi}}^j  ;  (x - y^j) / r_{\varepsilon,\lambda, \gamma} ) \\
&\quad +  \tilde {v} (\varepsilon, \lambda, \gamma, \tilde{\eta}, Y, \tilde \Phi,
\tilde \Psi ; x). 
\end{align*}
It remains to determine the parameters and the boundary functions in
such a way that the function which is equal to $u_{int }$ in 
$\cup_j  B_{r_{\varepsilon,\lambda, \gamma}} (y^j)$ and which is equal to 
$u_{\rm ext}$ in $\bar \Omega_{r_{\varepsilon,\lambda, \gamma}}  (Y)$ is a smooth function. 
This amounts to find the boundary data and the parameters so that, for each 
$j=1, \ldots, m$
\begin{equation}
u_{\rm int} = u_{\rm ext} , \quad 
\partial_r u_{\rm int} = \partial_r u_{\rm ext}, \quad 
\Delta u_{\rm int} = \Delta u_{\rm ext} , \quad 
\partial_r \Delta u_{\rm int} = \partial_r \Delta u_{\rm ext},
\label{eq:8.1}
\end{equation}
on $\partial B_{r_{\varepsilon,\lambda, \gamma}} (y^j)$. 
Assuming we have already done so, this
provides for each $\varepsilon, \lambda$  and $\gamma$  are small enough a function 
$w_{\varepsilon, \lambda, \gamma} \in \mathcal{C}^{4, \alpha}(\bar \Omega)$ 
(which is obtained by patching together
the function $u_{\rm int}$ and the function $u_{\rm ext}$) solution of
$\Delta^2  u  -\lambda\Delta u-\lambda|\nabla u|^2- \rho^4  e^u = 0$ and 
elliptic regularity theory
implies that this solution is in fact smooth. This will complete the
proof of our result since, as $\varepsilon, \lambda$  and $\gamma$  tend to $0$, 
the sequence of solutions we have obtained satisfies the required properties,
namely, away from the points $x^j$ the sequence $w_{\varepsilon, \lambda, \gamma}$
 converges to $\sum_j G (x^j ,   \cdot )$.

Before, we proceed, some remarks are due. First 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^2_{\varepsilon, \lambda,\gamma}) \label{eq:8.2}
\end{equation}
 near $\partial B_{r_{\varepsilon,\lambda, \gamma}}$.
Also, the function
\[
\sum_{j=1}^m (1+ \tilde{\eta}^j)  G (y^j  ,x)
\]
which appears in the expression of $u_{\rm ext}$ can be expanded as
\begin{equation} \sum_{\ell=1}^m (1+ \tilde{\eta}^\ell)  G (y^\ell, y^j+x ) 
= - 8  (1+ \tilde{\eta}^j)  \log |x| + E_j (Y ;
y^j) + \nabla E_j (Y ; y^j) \cdot x + \mathcal{O} (r^2_{\varepsilon, \lambda,\gamma})
\label{eq:8.3}
\end{equation} near $\partial B_{r_{\varepsilon,\lambda, \gamma}} (y^j)$. Here, we have defined
\[
E_j(Y ; \cdot ) : = R(y^j,   \cdot  )+ \sum_{\ell \neq j} G
(y^\ell , \cdot).
\]

In \eqref{eq:8.1}, all functions are defined on 
$\partial B_{r_{\varepsilon,\lambda, \gamma}} (y^j)$, nevertheless, it will be convenient 
to solve, instead of \eqref{eq:8.1} the following set of equations
\begin{equation}
\begin{gathered}
(u_{\rm int} -u_{\rm ext} )(y^j + r_{\varepsilon,\lambda, \gamma} \cdot ) 
=  0,  \quad 
(\partial_r u_{int } - \partial_r u_{\rm ext}) (y^j 
+ r_{\varepsilon,\lambda, \gamma} \cdot ) =  0, \\
(\Delta  u_{int } - \Delta u_{\rm ext}) (y^j 
+ r_{\varepsilon,\lambda, \gamma}  \cdot ) = 0 ,
 \quad   ( \partial_r   \Delta u_{int } - \partial_r  \Delta
u_{\rm ext} ) (y^j + r_{\varepsilon,\lambda, \gamma}   \cdot )  =  0,
\end{gathered} \label{eq:8.4}
\end{equation}
on $S^3$. Here all functions are considered as functions of 
$z \in S^3$ and we have simply used the change of variables 
$x = y^j + r_{\varepsilon,\lambda, \gamma}z$ to parameterize 
$\partial B_{r_{\varepsilon,\lambda, \gamma}} (y^j)$.

Since the boundary data satisfy \eqref{eq:5.2} and \eqref{eq:5.6},
we decompose
\begin{gather*}
\Phi  = \Phi_0 + \Phi_1 + \Phi^{\perp}, \quad 
\Psi  = 8 \Phi_0 + 12   \Phi_1 + \Psi^{\perp}, \\
\tilde \Phi = \tilde \Phi_0 + \tilde \Phi_1 + \tilde \Phi^{\perp},
\quad  \tilde \Psi = \tilde \Psi_1 + \tilde \Psi^{\perp}
\end{gather*}
where the components of $\Phi_0, \tilde \Phi_0$ are constant
functions on $S^3$, the components of $\Phi_1 , \tilde \Phi_1,
\tilde \Psi_1$ belong to $\ker  (\Delta_{S^3} +3) = \operatorname{span}
\{e_1, \ldots, e_4\}$ and where the components of $\Phi^{\perp} ,
\Psi^{\perp}, \tilde \Phi^{\perp} , \tilde \Psi^{\perp}$ are 
$L^2 (S^3)$ orthogonal to the constant function and the functions $e_1,
\ldots, e_4$. Observe that the components of $\Psi$ over the
constant functions or functions in $\ker  (\Delta_{S^3} +3)$
are determined by the corresponding components of $\Phi$. Moreover,
$\tilde \Psi$ has no component over constant functions.

We first consider the $L^2(S^3)$-orthogonal projection of
\eqref{eq:8.4} onto the space of functions which are orthogonal to
the constant function and the functions $e_1, \ldots, e_4$. This
yields the system
\begin{equation}
\begin{gathered}
\varphi^{j,\perp} -\tilde \varphi^{j,\perp} 
 =   M_0^{(j)} (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y, 
\Phi, \tilde \Phi, \Psi, \tilde \Psi) \\
\partial_r  H^i (\varphi^{j,\perp} , \psi^{j,\perp}   ; \cdot) -
\partial_r  H^e (\tilde \varphi^{j,\perp} , \tilde \psi^{j,\perp}  ; \cdot) 
 =  M_1^{(j)} (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y, \Phi, 
\tilde \Phi, \Psi, \tilde \Psi )   \\
\psi^{j,\perp} - \tilde \psi^{j,\perp} =  M_2^{(j)} 
(\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y, \Phi, \tilde \Phi, \Psi, \tilde \Psi ) 
\\
\begin{aligned}
&\partial_r  \Delta  H^i (\varphi^{j,\perp}, \psi^{j,\perp}  ;
\cdot) - \partial_r  \Delta  H^e ( \tilde \varphi^{j,\perp} , 
\tilde \psi^{j,\perp}  ; \cdot) ) \\
&=  M_3^{(j)} (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y, \Phi, 
\tilde \Phi, \Psi, \tilde \Psi ) 
\end{aligned}
\end{gathered} \label{eq:8.5}
\end{equation}
where the functions $M_k^{(j)}$ are nonlinear functions of the
parameters $\varepsilon$, $\tilde{\eta}$, $Y$, $T$ and the boundary data $\Phi$,
$\tilde \Phi$, $\Psi$ and $\tilde \Psi$. Moreover, using
\eqref{eq:8.2} and \eqref{eq:8.3} and also \eqref{eq:6.11} (keeping
in mind that $\mu \in (1,2)$) and \eqref{eq:7.10} (keeping in mind
that $\nu \in (-1,0)$), we conclude that, for each $j=1, \ldots, m$
and $k = 0, 1, 2, 3$
\begin{equation}
\|M_k^{(j)}\|_{\mathcal{C}^{4-k, \alpha} (S^3)} \leq c r^2_{\varepsilon, \lambda,\gamma}
\label{eq:8.55}
\end{equation}
for some constant $c >0$ independent of $\kappa$ 
(provided $\varepsilon \in (0, \varepsilon_\kappa), \lambda \in (0, \lambda_\kappa)$ and 
$\gamma \in (0, \gamma_\kappa))$.

Thanks to the result of Lemma \ref{le:5.3} and \eqref{eq:8.55}, this
last system can be re-written as
\[
(\Phi^{\perp} , \tilde \Phi^{\perp} , \Psi^{\perp}, \tilde
\Psi^{\perp} ) = M(\varepsilon, \lambda,\gamma, \tilde{\eta}, T, Y, \Phi, \tilde \Phi, \Psi,
\tilde \Psi )
\]
where
\[
\|M \|_{(\mathcal{C}^{4, \alpha} (S^3) )^{2m} \times 
(\mathcal{C}^{2, \alpha} (S^3) )^{2m}} \leq c r^2_{\varepsilon, \lambda,\gamma}
\]for some
constant $c >0$ independent of $\kappa$ 
(provided $(\varepsilon_\kappa, \lambda_\kappa, \gamma_\kappa) \in (0,
\varepsilon_\kappa)^3$). Moreover, \eqref{eq:6.7} and \eqref{eq:7.7} imply
(reducing $\varepsilon_\kappa, \lambda_\kappa$ and $\gamma_\kappa$ if necessary) that, 
the mapping $M$ is a
contraction from the ball of radius $\kappa  r^2_{\varepsilon, \lambda,\gamma}$ in 
$(\mathcal{C}^{4, \alpha} (S^3) )^{2m} \times (\mathcal{C}^{2, \alpha} (S^3))^{2m}$ 
into itself and as such has a unique fixed point in this
set. Observe that this fixed point depends continuously on $\varepsilon, \lambda, \gamma$,
$\tilde{\eta}$, $T$, $Y$ and also on $\Phi_0$, $\tilde \Phi_0$, $\Phi_1$,
$\tilde \Phi_1$ and $\tilde \Psi_1$.

We insert this  fixed point in \eqref{eq:8.4} and now project the
corresponding system over the set of functions spanned by 
$e_1, \ldots, e_4$ and finally over the set of constant functions.

The first projection yields the system of equations
\begin{equation}
\begin{gathered}
\Phi_1  =  \bar M_{1}(\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y,\Phi_0, 
 \tilde \Phi_0,  \Phi_1, \tilde \Phi_1, \tilde \Psi_1 )  \\
\tilde \Phi_1  =  \bar M_2 (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y ,
  \Phi_0, \tilde \Phi_0,  \Phi_1, \tilde \Phi_1, \tilde \Psi_1)  \\
\Psi_1   =  \bar M_3 (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y,\Phi_0, 
\tilde \Phi_0,  \Phi_1, \tilde \Phi_1, \tilde \Psi_1 )  \\
r_{\varepsilon,\lambda, \gamma}   \nabla E_j (Y ; y^j)  
 =  \bar M_4^{(j)}  (\varepsilon, \lambda, \gamma, \tilde{\eta},
T, Y ,\Phi_0, \tilde \Phi_0,  \Phi_1, \tilde \Phi_1, \tilde \Psi_1)
\end{gathered}
 \label{eq:8.6}
\end{equation}
where the functions $\bar M_k$ (and also $\bar M_4^{(j)}$) are
nonlinear functions depending continuously on the parameters 
$\varepsilon, \lambda, \gamma$, $\tilde{\eta}$, $T$, $Y$ and the components of the 
boundary data $\Phi_0$, $\tilde \Phi_0$, $\Phi_1$, $\tilde \Phi_1$ and $\tilde
\Psi_1$. Moreover,
\[
|\bar M_k | \leq c r^2_{\varepsilon, \lambda,\gamma}
\]
for some constant $c >0$ independent of $\kappa$ (provided provided 
$\varepsilon \in (0, \varepsilon_\kappa), \lambda \in (0, \lambda_\kappa)$ and 
$\gamma \in (0, \gamma_\kappa))$.

Let us comment briefly on how these equations are obtained. These
equations simply come from \eqref{eq:8.1} when expansions
\eqref{eq:8.2} and \eqref{eq:8.3} are taken into account, together
with the expression of $H^i (\varphi^j, \psi^j   ; \cdot)$ and
$H^{e} (\tilde \varphi^j , \tilde \psi^j  ; \cdot)$ given in
Lemma \ref{le:5.1} and Lemma \ref{le:5.2}, and also the estimates
\eqref{eq:6.11} and \eqref{eq:7.10}. Observe that the projection of
the term $x \to \nabla E_j (Y ; y^j) \cdot x$ which
arises in \eqref{eq:8.3}, as well as the projection of its partial
derivative with respect to $r$, over the set of constant function is
equal to $0$. Moreover, this term projects identically over the set
of functions spanned by $e_1, \ldots, e_4$ as well as its derivative
with respect to $r$. Finally, its Laplacian vanishes identically.

Recall that we have define in the introduction the function
\[
W (Y) : =  \sum_{j=1}^m R  (y^j, y^j )+ \sum_{j_1\neq j_2} G
(y^{j_1}, y^{j_2})
\]
Using the symmetries of the functions $G$ and $R$, namely the fact
that
\[
G(x,y) = G(y,x) \quad \text{and} \quad R(x,y) = R(y,x)
\]
we obtain
\[
\nabla W|_Y = 2  (\nabla E_1 (Y ; y^1), \ldots, \nabla E_m (Y, y^m)).
\]
Now, we have assumed that the point  $X= (x^1, \ldots, x^m)$ is a
nondegenerate critical point of the functional $W$ and hence
$\nabla W |_X =0$,
and
\[
(\mathbb{R}^4)^m \ni Z \mapsto D (\nabla W)|_{X}  (Z) \in (
\mathbb{R}^4)^m
\]
is invertible. Therefore, the last equation can be rewritten as
\[
r_{\varepsilon, \lambda,\gamma}  (Y-X) = \bar M_5  (\varepsilon, \lambda, \gamma, 
\tilde{\eta}, T, Y ,\Phi_0, \tilde
\Phi_0, \Phi_1, \tilde \Phi_1, \tilde \Psi_1)
\]

The projection of \eqref{eq:8.4} over the constant function, leads
to the system
\begin{equation}
\begin{gathered} 
(\log 1/ r^2_{\varepsilon, \lambda,\gamma})^{-1}  \log (\tau^j/ \tau^j_*)
  =  \bar M_6 (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y ,\Phi_0, \tilde \Phi_0,  
\Phi_1, \tilde \Phi_1, \Psi_1, \tilde \Psi_1) \\
\tilde \Phi_0  =  \bar M_7 (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y ,\Phi_0, 
\tilde \Phi_0,  \Phi_1, \tilde \Phi_1, \Psi_1, \tilde \Psi_1)  \\
\Phi_0  =  \bar M_8 (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y ,\Phi_0, 
 \tilde \Phi_0,  \Phi_1, \tilde \Phi_1, \Psi_1, \tilde \Psi_1) \\
\tilde{\eta}  =  \bar M_9 (\varepsilon, \lambda, \gamma, \tilde{\eta}, T, Y ,\Phi_0, 
\tilde \Phi_0, \Phi_1, \tilde \Phi_1, \Psi_1, \tilde \Psi_1)
\end{gathered} \label{eq:8.7}
\end{equation}
where the function $\bar M_k$ satisfy the usual properties. We are
now in a position to define $\tau_-$ and $\tau^+$ since, according
to the above, as $\varepsilon$, $\lambda$ and $\gamma$ tend to $0$ we expect that $y_i$ will
converge to $x_i$ and that $\tau_i$ will converge to $\tau_i^*$
satisfying \eqref{eq:8.0} and hence it is enough to choose $\tau_-$ and 
$\tau^+$ in such a way that
\[
4 \log(\tau_-)  <  -\sup_{i} E_{i}(Y, x_i) \leq  - \inf_i E_{i}(Y, x_i)  < 4 \log(\tau^+).
\]
So, if we define the parameters $U : = (u^1, \ldots, u^m)$ where
\[
u^j = \frac{1}{\log 1/{r^2_{\varepsilon,\lambda, \gamma}}}   \log (\tau^j/\tau^j_*),
\quad 
Z = r_{\varepsilon,\lambda, \gamma}  (Y-X)
\]
so that the system we have to solve reads
\begin{equation}
( \varepsilon, \lambda, \gamma, U , \tilde{\eta} , Z , \Phi_0 ,\tilde \Phi_0
, \Phi_1 , \tilde \Phi_1 , \tilde \Psi_1  ) = \bar M (\varepsilon, \lambda,
\gamma, U , \tilde{\eta} , Z , \Phi_0 ,\tilde \Phi_0 , \Phi_1 ,
\tilde \Phi_1 , \tilde \Psi_1 ). \label{eq:8.8}
\end{equation}
where as usual, the nonlinear function $\bar M$ depends continuously
on the parameters $T  , \tilde{\eta} , Z$ and the functions $\Phi_0 ,
\tilde \Phi_0 , \Phi_1 , \tilde \Psi_1 $ and is bounded (in the
appropriate norm) by a constant (independent of $\varepsilon, \lambda, \gamma$ and $\kappa$)
time $r^2_{\varepsilon, \lambda,\gamma}$, provided 
$\varepsilon \in (0, \varepsilon_\kappa), \lambda\in (0, \lambda_\kappa)$ and
 $ \gamma\in (0, \gamma_\kappa)$. Observe that
\begin{gather*}
U , \tilde{\eta} \in  \mathbb{R}^m , \quad  
Z \in (\mathbb{R}^4)^m , \quad 
\Phi_0 , \tilde \Phi_0 \in \mathbb{R}^m \\
\Phi_1 ,  \tilde \Phi_1 , \tilde \Psi_1  \in (\ker   (\Delta_{S^3}+3))^m .
\end{gather*}
In addition, reducing $\varepsilon_\kappa, \lambda_\kappa$ and
$\gamma_\kappa$ if necessary, this nonlinear mapping sends the ball
of radius $\kappa  r^2_{\varepsilon, \lambda,\gamma}$ (for the natural
product norm) into itself, provided $\kappa$ is fixed large enough,
$\varepsilon \in (0, \varepsilon_\kappa)$, $\lambda \in (0,\lambda_\kappa)$ and
$\gamma \in (0, \gamma_\kappa)$. Applying Schauder's fixed point
Theorem in the ball of radius $\kappa  r^2_{ \varepsilon, \lambda, \gamma
}$ in the product space where the entries live yields the existence
of a solution of \eqref{eq:8.8} and this completes the proof of
Theorem~\ref{th:1.1}.

\subsection*{Acknowledgments}
This project was founded by the national plan of sciences,
technology and innovation (MAAR-IFAH), King Abdulaziz city for
sciences and Technology, Kingdom of Saudi Arabia, Award number
(12-MAT2912-02).


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\end{document}