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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 77, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/77\hfil Sign-changing solutions]
{Sign-changing solutions of a fourth-order elliptic equation with
supercritical exponent}

\author[K. Ould Bouh \hfil EJDE-2014/77\hfilneg]
{Kamal Ould Bouh}  % in alphabetical order

\address{Kamal Ould Bouh \newline
Department of Mathematics, Taibah University,
P.O. Box: 30002, Almadinah Almunawwarah, Saudi Arabia}
\email{hbouh@taibahu.edu.sa, kamal\_bouh@yahoo.fr}

\thanks{Submitted November 15, 2013. Published March 19, 2014.}
\subjclass[2000]{35J20, 35J60}
\keywords{Sign-changing solutions; critical exponent; bubble-tower solution}

\begin{abstract}
 In this article we study the nonlinear
 elliptic problem involving  nearly critical exponent
 \begin{gather*}
 \Delta^2 u = |u|^{8/(n-4)+\varepsilon}u\quad\text{in } \Omega, \\
 \Delta u=u = 0\quad \text{on } \partial \Omega,
 \end{gather*}
 where $\Omega $ is a smooth bounded domain in $\mathbb{R}^n $ with
 $n \geq 5 $, and $\varepsilon$ is a positive parameter.
 We show that, for $\varepsilon$ small, there is no sign-changing solution
 with low energy which blow up at exactly two points.
 Moreover, we prove that this problem has no bubble-tower
 sign-changing solutions.
\end{abstract}

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


\section{Introduction and statement of results}

 In this article, we consider the  semi-linear
elliptic problem with supercritical nonlinearity
\begin{equation}  \label{ePe}
\begin{gathered}
\Delta^2 u= |u|^{p-1+\varepsilon}u   \quad \text{in } \Omega,  \\
\Delta u= u= 0   \quad \text{on } \partial \Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$,
$n\geq 5$, $\varepsilon$ is a positive real parameter and $p+1= \frac{2n}{n-4}$
is the critical Sobolev exponent for the embedding of
$H^2(\Omega)\cap H_0^1(\Omega)$ into $L^{p+1}(\Omega)$.

Problem \eqref{ePe} is related to the limiting problem (when
$\varepsilon=0$) which exhibits a lack of compactness. In fact,  van Der
Vorst \cite{V1, V2} (see also \cite{Mi}) showed  that
\eqref{ePe} with $\varepsilon=0$ has no positive
solutions if $\Omega$ is a starshaped domain.
Whereas Ebobisse and Ould Ahmedou \cite{EA} proved  that \eqref{ePe}
with $\varepsilon=0$ has a positive
solution provided that some homology group of $\Omega$ is non trivial.
This topological condition is sufficient, but not necessary, as
examples of contractible domains $\Omega$ on which a positive solution
exists as shown in \cite{GGS} (see also \cite{GGSw}).
Note that some problems of type \eqref{ePe}  were studied in case
of Riemannian manifolds, see for example \cite{HRW} and \cite{PV}.

In view of this qualitative change of the situation for \eqref{ePe}
with $\varepsilon=0$,
it is interesting to study the problem \eqref{ePe} with $\varepsilon<0$ and
$\varepsilon>0$ and to understand what happens to the solutions of \eqref{ePe}
(\text{if }they exist) as $\varepsilon \to 0$.

Observe that, when $\varepsilon < 0$, the existence of solutions of
\eqref{ePe} has been proved in \cite{BE,BG}
(see \cite{Bar,BarW,CCN} for the  Laplacian case)
for each $\varepsilon \in (1-p, 0)$. For the positive solutions, Chou and
Geng \cite{CG} made the first study, when $\Omega$ is a convex domain.
They gave the asymptotic behavior of the low energy positive
solution. They used the method of moving planes to show that the
blow-up point is away from the boundary of the domain. The process
is standard if the domain is convex. We note that, for non convex
regions, this method still works in the Laplacian case through the
applications of Kelvin transformations, see \cite{H} (since the
problem is invariant under these transformations). However, the
Navier  boundary conditions are not invariant under the Kelvin
transformation of the biharmonic operator. But the method of moving
planes also works for convex domains, see \cite{CG}. To remove the
convexity assumption, Ben Ayed and El Mehdi \cite{BE} used another
method based on some ideas introduced by  Bahri in \cite{B1}.
This result is the analogous one to the one in \cite{R2} and \cite{H} where
the Laplacian operator was studied.

Concerning the supercritical case, $\varepsilon>0$, the problem \eqref{ePe}
becomes more delicate since we lose the Sobolev embedding which
is an important point to overcome. We recall that, when the
biharmonic operator in \eqref{ePe} is replaced by the Laplacian one,
there are many works devoted to the study of the positive
solutions of the counterpart of \eqref{ePe}. It was proved in
\cite{BEGR} that \eqref{ePe} has no positive solution which blows up
at a single point. This result shows that the situation is
different from the subcritical case. However, Del Pino et al
\cite{DFM} (see also \cite{KR}) gave an existence result for two
blow up points, provided that $\Omega$ satisfies some geometrical
conditions. In sharp contrast to this, very little study has been
made concerning the sign-changing solutions, see \cite{BB1}.

It is well known that problem \eqref{ePe} (with $\varepsilon<0$) has always a
positive least energy solution $u_\varepsilon$ which is obtained by solving
the variational problem
$$
\inf J(u) \quad  \text{where }  J(u):= \frac{\int _\Omega
|\Delta u|^2}{\big(\int _\Omega
    |u|^{p+1+\varepsilon}\big)^{2/(p+1+\varepsilon)}} \quad
u\in H^2(\Omega)\cap H^1_0(\Omega), \quad u\not\equiv    0.
$$



Removing the assumption of the positivity of the solutions, the
study of the asymptotic behavior becomes difficult. The main
difficulty is that the limit problem, after a change of variable,
which is
\begin{equation}\label{i:1}
 \Delta^2 u= |u|^{p-1}u   \quad   \text{in } \mathbb{R}^n,
\end{equation}
has many sign-changing solutions which are unknown. However, an
interesting information about the energy shows that
\cite[Lemma 2]{GGS}
\begin{equation}\label{i:2}
\int_{\mathbb{R}^n} |\Delta w|^2 > 2S^{n/4},
\end{equation}
for each sign-changing solution $w$ of \eqref{i:1},
where $S$ denotes the best minimizers of the Sobolev inequality on
the whole space, that is
\begin{equation*}
 S =\inf\{|\Delta u|^{2}_{L^2(\mathbb{R}^n)}|u|^{-2}_{L^{{2n}/(n-4)}(\mathbb{R}^n)}
: \Delta u\in L^2 ,u\in L^{2n/(n-4)} ,u\not\equiv 0 \}.
\end{equation*}
When we add the positivity assumption, the solutions of
\eqref{i:1} are the family
\begin{equation}\label{i:3}
 \delta _ {(a,\lambda )}(x) = c_0 \frac {\lambda
 ^{(n-4)/2}}{(1+\lambda^2|x-a|^2)^{(n-4)/2}},\quad
 c_0=\big(n(n-4)(n^2-4)\big)^{(n-4)/8},
\end{equation}
with $\lambda >0$ and $a \in \mathbb{R}^n$.

The space $H^2(\Omega)\cap H_0^1(\Omega)$ is equipped with the norm $\|\cdot\|$
and its corresponding inner product $\langle\cdot,\cdot\rangle$ defined by
\begin{equation}
\|u\|=\Big(\int_\Omega |\Delta u|^2\Big)^{1/2}, \quad
\langle u,v\rangle=\int_\Omega \Delta u \Delta
v, \quad  u, v \in H^2(\Omega)\cap H_0^1(\Omega).
\end{equation}
When we study problem \eqref{i:1} in a bounded smooth domain $\Omega$,
we need to introduce the function $P\delta _ {(a,\lambda)}$ which is
the projection of $\delta _ {(a,\lambda)}$  on $H^1_0(\Omega)$. It
satisfies
\begin{equation*}
 \Delta^2 P\delta _{(a, \lambda)} = \Delta^2\delta _{(a,\lambda)} \quad
\text{in } \Omega ; \quad
\Delta P\delta _{(a, \lambda)}=P\delta _{(a, \lambda)}= 0 \quad \text{on }
 \partial \Omega.
\end{equation*}
These functions are almost positive solutions of \eqref{ePe}.
 Our first result deals with the low energy sign-changing solution of
\eqref{ePe} with $\varepsilon >0$. We prove that there is no solution which
blows up at exactly two points. More precisely, we have the following result.

\begin{theorem}\label{t:11}
Let $\Omega$ be any smooth bounded domain in $\mathbb{R}^n$ with $n\geq 5$.
There exists $\varepsilon_0>0$, such that for each
$\varepsilon \in (0,\varepsilon_0)$, problem
\eqref{ePe} has no sign-changing solution  $u_\varepsilon$ which satisfies
\begin{align}\label{n:1}
u_\varepsilon= P\delta_{(a_{\varepsilon,1},\lambda_{\varepsilon,1})}
-P\delta_{(a_{\varepsilon,2},\lambda_{\varepsilon,2})}+v_\varepsilon,
\end{align}
with the $L^{\infty}$-norm of $u_\varepsilon$ at the power
$\varepsilon$ ($|u_\varepsilon|_{\infty}^{\varepsilon}$) begin bounded and
\begin{gather*}
a_{\varepsilon,i}\in \Omega,\quad \lambda_{\varepsilon,i}d(a_{\varepsilon,i},
\partial \Omega) \to \infty  \quad\text{for } i=1,2 \\
 \langle P\delta_{(a_{\varepsilon,1},\lambda_{\varepsilon,1})},
P\delta_{(a_{\varepsilon,2},\lambda_{\varepsilon,2})}\rangle \to 0 \quad \text{and}\quad
 \|v_\varepsilon\|\to 0 \quad \text{as } \varepsilon \to 0 .
\end{gather*}
\end{theorem}


We point out that there are other important phenomena in
sign-changing solutions. Indeed, it is possible to find solutions
having bubble over bubble (bubble-tower solutions). In the case of
the Laplacian operator, Pistoia and Weth \cite{PW}
constructed a family of sign-changing solutions of \eqref{ePe}
($\varepsilon<0$) with $k$ bubbles, $k\geq2$, concentrated at the same
point. This result gives a new phenomenon compared with the
positive case. In their paper, they conjectured that this
phenomenon cannot appear when $\varepsilon>0$. In \cite{BB1}, we gave  an
affirmative answer for the conjecture of Pistoia and Weth. The
following result deals with phenomenon of bubble-tower solutions
for the biharmonic problem \eqref{ePe} with supercritical exponent.

\begin{theorem}\label{t:14}
Let $\Omega$ be any smooth bounded domain in $\mathbb{R}^n$ with $n\geq 5$. There
exists $\varepsilon_0>0$, such that for each $\varepsilon \in (0,\varepsilon_0)$,
problem \eqref{ePe} has no solution  $u_\varepsilon$ of the form
\begin{equation} \label{i:4}
u_\varepsilon = \sum _{i=1}^k (-1)^{i+1}P\delta _{(a_{\varepsilon ,i},
\lambda _{\varepsilon ,i})} + v_\varepsilon,
\end{equation}
with 
$ \lambda_{\varepsilon ,1}\leq \lambda_{\varepsilon ,2}\leq \dots
\leq \lambda_{\varepsilon ,k}$
 and $|u_\varepsilon|_{\infty}^{\varepsilon}$ bounded,
where  $k\geq2$, $a_{\varepsilon ,i}\in\Omega$,
$\min(\lambda _{\varepsilon ,i},\lambda _{\varepsilon,j})
| a_{\varepsilon ,i}-a_{\varepsilon ,j}|$ is bounded,
and $v_\varepsilon \to 0$ in $H^1_0(\Omega)$,
$\lambda _{\varepsilon,i}d(a_{\varepsilon ,i},\partial \Omega) \to +\infty$,
$\langle P\delta _{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})},P\delta
_{(a_{\varepsilon ,j}, \lambda _{\varepsilon ,j})}\rangle \to 0$, for  $i\neq j$,
as $\varepsilon \to 0$.
\end{theorem}

Note that Theorem  \ref{t:14} deals with the bubble-tower solutions
at one point. However Theorem \ref{t:11} says that there are no
solutions which blow up at two points. Combining the ideas of the
proof of Theorems \ref{t:11} and \ref{t:14}, we are able to prove
the following result.

\begin{theorem}\label{t:17}
Let $\Omega$ be any smooth bounded domain in $\mathbb{R}^n$ with $n\geq 5$. There
exists $\varepsilon_0>0$, such that for each $\varepsilon \in (0,\varepsilon_0)$,
 problem \eqref{ePe} has no solution  $u_\varepsilon$ of the form
\begin{eqnarray}\label{ii:4}
u_\varepsilon = \sum _{i=1}^m (-1)^{i+1}P\delta _{(a_{\varepsilon ,i},
\lambda _{\varepsilon ,i})} +
\sum _{i=m+1}^k (-1)^{i-m}P\delta _{(a_{\varepsilon ,i},
\lambda _{\varepsilon ,i})} + v_\varepsilon:=
u_\varepsilon^1+u_\varepsilon^2+v_\varepsilon,
\end{eqnarray}
with  $ |u_\varepsilon|_{\infty}^{\varepsilon}$ bounded,
$\|v_\varepsilon\| \to 0$, $a_{\varepsilon ,i}\to a$  for each $i\leq m$,
$a_{\varepsilon ,i} \to b$ for each $i\geq m+1$ with $a\ne b$, and,
for $j=1,2$, if $u_\varepsilon^j$ contains more
than one bubble then it satisfies the assumptions of Theorem
\ref{t:14}.
\end{theorem}

The proof of our results will be by contradiction. Thus,
throughout this paper we will assume that there exist solutions
$(u_\varepsilon)$ of \eqref{ePe} which satisfy \eqref{n:1}  or  \eqref{i:4}.
In Section 2, we will obtain some information about such $(u_\varepsilon)$
which allow us to develop Sections 3 which deal with some useful
estimates to the proof of our Theorems. Finally, in Section 4, we
combine these estimates to obtain a contradiction. Hence the proof
of our results.

\section{Preliminary results}

In this Section, we assume that there exist solutions $(u_\varepsilon)$ of
\eqref{ePe} which satisfy
\begin{equation}\label{e:20}
u_\varepsilon = \sum _{i=1}^k (-1)^{i+1}P\delta _{(a_{\varepsilon ,i},
 \lambda _{\varepsilon ,i})} + v_\varepsilon,
\end{equation}
with  $|u_\varepsilon|_{\infty}^{\varepsilon}$  bounded,
$k\geq 2$,   $a_{\varepsilon,i}\in\Omega$,  and as $\varepsilon \to 0$,
$\|v_\varepsilon\| \to 0$,
$\lambda_{\varepsilon,i}d(a_{\varepsilon ,i},\partial \Omega) \to +\infty$,
$\langle P\delta_{(a_{\varepsilon ,i}, \lambda _{\varepsilon ,i})},
P\delta _{(a_{\varepsilon ,j}, \lambda _{\varepsilon ,j})}\rangle
\to 0$ for $i\neq j$.
We will collect some useful information
used in the next sections. First, from \eqref{e:20}, it is easy to
see that the following remark holds.

\begin{remark}\label{l:21} \rm
Let $(u_\varepsilon)$ be a family of sign-changing
solutions of \eqref{ePe} satisfying \eqref{e:20}. Then
\begin{itemize}
\item[(i)] $ u_\varepsilon \rightharpoonup 0$  as  $\varepsilon \to 0$,
\item[(ii)] $\int_{\Omega}|u_\varepsilon|^{p+1+\varepsilon}
=\int_{\Omega}|\Delta u_\varepsilon|^2=kS^{n/4}+o(1)$,
\item[(iii)] $M_{\varepsilon,+}:=\max_\Omega u_\varepsilon  \to +\infty$,
$M_{\varepsilon,-}:=-\min_\Omega u_\varepsilon  \to +\infty$  as
$\varepsilon \to 0$.
\end{itemize}
\end{remark}

Secondly, arguing as in \cite{BC} and \cite{R}, we see that for
$u_\varepsilon$ satisfying \eqref{e:20}, there is a unique way to choose
$\alpha_i$, $a_i$, $\lambda _i$ and $v$ such that
\begin{equation}\label{e:11}
u_\varepsilon = \sum_{i=1}^k  (-1)^{i+1}\alpha _iP\delta _{(a_i, \lambda _i)} + v,
\end{equation}
with
\begin{equation} \label{e:12}
\begin{gathered}
\alpha_i \in \mathbb{R}, \quad \alpha_i \to 1,\\
a_i \in \Omega, \quad \lambda_i \in \mathbb{R}^*_+,\quad \lambda_i d(a_i ,
\partial\Omega )\to +\infty,\\
v \to 0 \quad\text{in } H^2(\Omega)\cap H^1_0(\Omega), \quad v \in E,
\end{gathered}
\end{equation}
where $E$ denotes the subspace of $H^1_0(\Omega)$  defined by
\begin{equation}\label{E}
E:=\big\{w: \langle w,\varphi\rangle=0,\; \forall\varphi\in
\operatorname{span}\{P\delta _i, \partial P\delta_i/\partial \lambda_i,
\partial P\delta _i/\partial a^j_i,\, i\leq k; j\leq n \}\big\}.
\end{equation}
Here, $a_i^j$ denotes the $j$-th
component of $a_i$ and in the sequel, in order to simplify the
notations, we set
 \begin{equation}
\delta _{(a_i , \lambda _i) }=\delta _i, \quad
P\delta _{(a_i , \lambda _i) }=P\delta _i.
\end{equation}
 In the following, we always assume that $u_\varepsilon$
(which satisfies \eqref{e:20}) is written as in \eqref{e:11} and
\eqref{e:12} holds.


\begin{lemma}\label{l:23}
Let $u_\varepsilon$ satisfying the assumption of above theorems.
Then $\lambda _i$ occurring in \eqref{e:11} satisfies
\begin{equation}\label{e:28}
\lambda _i ^\varepsilon  \to 1  \quad \text{as $\varepsilon \to 0$, for each }
 i\leq k.
\end{equation}
\end{lemma}

\begin{proof}
By Remark \ref{l:21}, we know that
\begin{equation}\label{e:29}
\int _{\Omega}|u_\varepsilon| ^{p+1+\varepsilon} = kS^{n/4} + o(1) \quad
\text{as } \varepsilon \to 0.
\end{equation}
Furthermore,
\begin{equation}\label{e:210}
\int _{\Omega}|u_\varepsilon| ^{p+1+\varepsilon}
=\int_{\Omega} -\Delta u_\varepsilon u_\varepsilon
= \int_{\Omega}|u_\varepsilon |^{p-1+\varepsilon}u_\varepsilon
\Big(\sum  (-1)^{i+1}\alpha  _iP\delta_i\Big)+O(\|v\|^2).
\end{equation}
Observe that
\begin{equation} \label{e:211}
\begin{aligned}
&\int _{\Omega}|u_\varepsilon |^{p-1+\varepsilon}u_\varepsilon
\Big(\sum  (-1)^{i+1}\alpha  _iP\delta_i\Big) \\
&=\sum\alpha  _i ^{p+\varepsilon +1}\int _{\Omega}P\delta _i^{p+\varepsilon +1}
+O\Big(\sum_{j\neq i}\int _{\Omega} P\delta _i^{p+\varepsilon}P\delta
_j\Big) \\
&\quad +O\Big(\sum\int _{\Omega}\alpha  _i P\delta _i^{p+\varepsilon}|v|+\sum\int _{\Omega}\alpha
_i P\delta _i
|v|^{p+\varepsilon}\Big)
 :=\sum A_i,
\end{aligned}
\end{equation}
where
\begin{equation*}
A_i:= \alpha  _i ^{p+\varepsilon +1}\int _{\Omega}P\delta _i ^{p+\varepsilon
+1}+O\Big(\sum_{j\neq i}\int _{\Omega} P\delta _i^{p+\varepsilon}P\delta _j+\int
_{\Omega}P\delta _i^{p+\varepsilon}|v|+\int _{\Omega}P\delta _i|v|^{p+\varepsilon}\Big).
\end{equation*}
 Easy computations show that
\begin{gather*}
\int _{\Omega}P\delta _i ^{p+1+\varepsilon}= \lambda _i ^{\varepsilon(n-4)/2}\left(S^{n/4} +o(1)\right),\\
\int _{\Omega}P\delta _i^{p+\varepsilon}|v|=\lambda _i ^{\varepsilon(n-4)/2}O(|v |_{L^{p+1}}),\\
\int _{\Omega}P\delta _i|v|^{p+\varepsilon}=\lambda _i
^{\varepsilon(n-4)/2}O(|v|_{L^{p+1}}^{p+\varepsilon}).
\end{gather*}
Recall that for $i\neq j$ (see \cite{B1})
\begin{equation*}
\int _{\mathbb{R}^n} \delta _i^p\delta _j
=\int _{\mathbb{R}^n} \delta _j^p\delta
_i=c\varepsilon_{ij}+ O\Big(\varepsilon_{ij}^{n/(n-4)}\log\varepsilon_{ij}^{-1}\Big),
\end{equation*}
where $c$ is a positive constant and, for $i\neq j$,  $\varepsilon_{ij}$ is
defined by
\begin{equation}\label{e:215}
\varepsilon_{ij}=\Big(\frac{\lambda_i}{\lambda_j}+\frac{\lambda_j}{\lambda_i}
+\lambda_i\lambda_j|a_i-a_j|^2\Big)^{(4-n)/2}.
\end{equation}
Hence, we obtain
\begin{equation*}
\int _{\Omega} P\delta _i^{p+\varepsilon}P\delta _j
=O\Big(\lambda _i ^{\varepsilon(n-4)/2}\int
_{\Omega} P\delta _i^pP\delta _j\Big)
=\lambda _i ^{\varepsilon(n-4)/2}O(\varepsilon_{ij}), \quad
\text{for } i\neq j.
\end{equation*}
Thus
\begin{equation}\label{e:212}
A_i= \alpha  _i ^{p+\varepsilon +1}\lambda _i ^{\varepsilon(n-4)/2}\left(S^{n/4} +
o(1)\right).
\end{equation}
Therefore \eqref{e:210}, \eqref{e:211}, and \eqref{e:212}  provide
us with
\begin{equation}\label{e:213}
\int _{\Omega}|u_\varepsilon| ^{p+1+\varepsilon}
=\Big(\sum \alpha  _i ^{p+1+\varepsilon} \lambda _i
^{\varepsilon(n-4)/2}\Big)\Big(S^{n/4}+o(1)\Big) +o(1).
\end{equation}
Combining \eqref{e:29}, \eqref{e:213}, and the fact that $\alpha _i$
satisfies \eqref{e:12}, the lemma follows.
\end{proof}

\begin{remark}[\cite{BEGR,R2} ]\label{r:24} \rm
We recall the  estimate
\begin{equation}\label{e:214}
\delta  _i^\varepsilon (x) - c_0^\varepsilon \lambda _i ^{\varepsilon (n-4)/2}
= O\left(\varepsilon \log (1+\lambda
_i ^2  |x-a_i|^2)\right) \quad \text{in } \Omega ,
\end{equation}
which will be very useful in the next section.
\end{remark}

\section{Some useful estimates}

As usual in this type of problems, we first deal with the $v$-part
of $u_\varepsilon$, in order to show that it is negligible with respect to
the concentration phenomenon.

\begin{lemma}\label{l:41}
The function $v$ defined in  \eqref{e:11}, satisfies the
estimate
\begin{align*}
&\|v\| \\
&\leq c\varepsilon+ c
\begin{cases}
\sum_i \frac{1}{(\lambda_id_i)^{n-4}}+
\sum_{i\neq j}\varepsilon_{ij}(\log\varepsilon_{ij}^{-1})^{(n-4)/n}
&\text{if } n < 12,\\[4pt]
 \sum_i \frac{1}{(\lambda_id_i)^{(n+4)/2-\varepsilon(n-4)}}
 +\sum_{i\neq j}\varepsilon_{ij}^{(n+4)/2(n-4)}
(\log \varepsilon_{ij}^{-1})^{(n+4)/2n} &{if } n \geq 12,
\end{cases}
\end{align*}
where $\varepsilon_{ij}$ is defined in \eqref{e:215} and
$d_i:=d(a_i,\partial\Omega)$ for $i\leq k$.
\end{lemma}

\begin{proof}
Since $u_\varepsilon = \sum  (-1)^{i+1}\alpha  _iP\delta _i+v$ is a solution of
\eqref{ePe} and $v \in E$ (see \eqref{E}), we obtain
\begin{align*}
\int_{\Omega}-\Delta u_\varepsilon v
& =\|v\|^2=\int_{\Omega}| u_\varepsilon |^{p-1+\varepsilon}u_\varepsilon v\\
&= \int_{\Omega}| \sum  (-1)^{i+1}\alpha  _iP\delta
_i|^{p-1+\varepsilon}(\sum (-1)^{i+1}\alpha  _iP\delta _i)v\\
&\quad +p\int_{\Omega}| \sum (-1)^{i+1}\alpha  _iP\delta _i
|^{p-1+\varepsilon}v^2 +o(\|v\|^2).
\end{align*}
Hence,  we have
\begin{equation}\label{e:41}
Q(v,v)= f(v)+ o(\|v\|^2),
\end{equation}
where
\begin{align*}
 Q(v,v)=&\|v\|^2-p\int_{\Omega}| \sum  (-1)^{i+1}\alpha  _iP\delta _i|^{p-1+\varepsilon}v^2,\\
f(v)=&\int_{\Omega}| \sum  (-1)^{i+1}\alpha  _iP\delta
_i|^{p-1+\varepsilon}(\sum (-1)^{i+1}\alpha  _iP\delta _i)v .
\end{align*}
 Using Remark \ref{r:24} and according to \cite{B1}, it is easy to see
that
\begin{equation*}
Q(v,v)=\|v\|^2-p\sum_{i=1}^k\int_{\Omega}(P\delta _i)^{p-1+\varepsilon}v^2
+o(\|v\|^2)
\end{equation*}
is positive definite; that is, there exists $c> 0$ independent of
$\varepsilon$, satisfying $ Q(v,v)\geq c \|v\|^2 $, for each $ v  \in E $.
Then, from \eqref{e:41} we get
\begin{equation*}
\|v\|^2= O(\|f(v)\|).
\end{equation*}
Now, using Lemma \ref{l:23}, we obtain
\begin{equation} \label{e:42}
\begin{aligned}
f(v)&=\sum(-1)^{i+1}\int_{\Omega}( \alpha_iP\delta _i)^{p+\varepsilon}v\\
 &\quad + O\Big(\sum_{i\neq j} \int_{\Omega}(\delta _i\delta
_j)^{p/2}| v| +\sum_{i\neq j} \int_{\Omega}\delta _i^{p-1}\delta _j|
v| (\text{if }n<12)\Big).
\end{aligned}
\end{equation}
Using Remark \ref{r:24} and the fact that $v \in E $, we obtain
\begin{align*}
&\big| \int_{\Omega}P\delta _i^{p+\varepsilon}v\big| \\
&= |\int \delta_i^{p+\varepsilon}v|
+O\Big(\int \delta _i^{p-1+\varepsilon}\theta_i| v|\Big)  \\
& \leq c\varepsilon \int \log(1+\lambda_i^2| x-a_i|^2)\delta _i^p| v| + c| \theta_i|_{L^{\infty}}\int \delta _i^{p-1+\varepsilon}|
 v|  \\
& \leq c\|v\|\Big(\varepsilon+ \frac{1}{(\lambda_id_i)^{n-4}}
(\text{if }n<12) +\frac{1}{(\lambda_id_i)^{\frac{n+4}{2}+\varepsilon(n-4)}}(\text{if }
 n\geq12)\Big),
\end{align*}
where $\theta _i:=\theta _{a_i ,\lambda _i}:=\delta _i-P\delta _i$.

For the other integrals of \eqref{e:42}, we use Holder's
inequality and we obtain for $i\neq j$
\begin{align*}
\int_{\Omega}(\delta _i\delta _j)^{p/2}| v|
&\leq c\|v\| \Big( \int_{\Omega}(\delta _i\delta_j)^{n/(n-4)}\Big)^{(n+4)/2n}\\
&\leq c\|v\|\varepsilon_{ij}^{(n+4)/2(n-4)}
(\log\varepsilon_{ij}^{-1})^{(n+4)/2n}
\end{align*}
and
if $n<12$, we have $p-1 = 8/(n-4)>1$; therefore
\begin{align}\label{e:45}
\int_{\Omega}\delta _i^{p-1}\delta _j| v| \leq c\|v\| \Big(
\int_{\Omega}(\delta _i\delta _j)^{n/(n-4)}\Big)^{(n-4)/n}\leq
c\|v\|\varepsilon_{ij}(\log\varepsilon_{ij}^{-1})^{(n-4)/n}.
\end{align}
Combining \eqref{e:42}--\eqref{e:45}, the proof follows.
\end{proof}

Now,  we need to introduce some notations before to state the
crucial point in the proof of our Theorems. We denote by $G$ the
Green's function  defined by : $\forall x \in \Omega$
\begin{gather*}
\Delta^2 G(x,.)=c_n\delta_x\quad\text{in } \Omega,\\
\Delta G(x,.)=G(x,.)=0\quad\text{on } \partial\Omega,
\end{gather*}
where $\delta_x$ is the Dirac mass at $x$ and $c_n=(n-4)(n-2) \omega_n$, 
with $\omega_n$ is the area of the unit sphere of $\mathbb{R}^n$. 
We denote by $H$ the regular part of $G$, that is, 
$$
 H(x_1,x_2)=|x_1-x_2|^{4-n} -G(x_1,x_2) \quad \text{for } 
(x_1,x_2)\in \Omega^2\setminus \Gamma
$$
with $\Gamma= \{(y,y): y\in\Omega\}$.

\begin{proposition} \label{p:42} 
Assume that $n\geq 5$ and let $\alpha _i$, $a_i$ and $\lambda_i$ be the 
variables defined in \eqref{e:11} with $k=2$.  We have
\begin{equation}
 \begin{aligned}
&\Big|\alpha _i c_1\frac{n-4}{2}\frac{H(a_i,a_i)}{\lambda_i^{n-4}}
-\alpha _j c_1\Big(\lambda_i\frac{\partial \varepsilon_{12}}{\partial \lambda_i} +
\frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}}
\Big)+\alpha _i \frac{n-4}{2}c_2 \varepsilon\Big|  \\
&\leq c\varepsilon^2
+c  \begin{cases}
 \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^{n-2}}+ \varepsilon_{12}^{\frac{n}{n-4}}
\log\varepsilon_{12}^{-1}+ \varepsilon_{12}^{2}
(\log\varepsilon_{12}^{-1})^{\frac{2(n-4)}{n}}& \text{if } n\geq 6),\\
 \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^2}+ \varepsilon_{12}^{2}
(\log\varepsilon_{12}^{-1})^{2/5}\quad
&\text{if } n=5,
\end{cases}
\end{aligned} \label{D2}
\end{equation}
where $i, j\in \{1,2\}$ with $i\ne j$  and $c_1$, $c_2$ are
positive constants.
\end{proposition}

\begin{proof} 
Let
$$
c_1=c_0^\frac{2n}{n-4}\int_{\mathbb{R}^n}\frac{dx}{(1+|x|^2)^{(n+4)/2}}, \quad 
c_2=\frac{n-4}{2}c_0^\frac{2n}{n-4}\int_{\mathbb{R}^n}
\log(1+|x|^2)\frac{|x|^2-1}{(1+|x|^2)^{n+1}}dx.
$$
It suffices to prove the proposition for $i=1$.  Multiplying
\eqref{ePe} by $\lambda_1\partial P \delta_1 / \partial \lambda_1$ and integrating
on $\Omega$, we obtain
\begin{equation}\label{e:47}
 \alpha_1 \int_{\Omega} \delta_1^p\lambda_1\frac{\partial P \delta_1}
 {\partial \lambda_1}
 -\alpha_2 \int_{\Omega} \delta_2^p\lambda_1\frac{\partial P \delta_2}
 {\partial \lambda_2}
 =\int_{\Omega}|u_\varepsilon|^{p-1+\varepsilon}u_\varepsilon 
 \lambda_1\frac{\partial P \delta_1}{\partial\lambda_1}.
\end{equation}
Using \cite{B1}, we derive
\begin{gather*}
 \int_{\Omega} \delta_1^p\lambda_1\frac{\partial P \delta_1}{\partial
 \lambda_1}  =\frac{n-4}{2}c_1\frac{H(a_1,a_1)}{\lambda_1^{n-4}}+
 O\Big(\frac{\log(\lambda_1  d_1)}{(\lambda_1d_1)^{n-1}}\Big),\\
 \int_{\Omega} \delta_2^p\lambda_1\frac{\partial P \delta_1}{\partial
 \lambda_1}  =c_1\Big(\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1}
 + \frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+ R,
\end{gather*}
where $R$ satisfies
\begin{equation}\label{e:410}
 R = O\Big(\sum_{k=1,2} \frac{\log(\lambda_k
d_k)}{(\lambda_kd_k)^{n-1}}+ \varepsilon_{12}^{\frac{n}{n-4}}
\log\varepsilon_{12}^{-1}\Big).
\end{equation}
For the other term of \eqref{e:47}, we have
\begin{equation} \label{e:411}
\begin{aligned}
&\int_{\Omega}|u_\varepsilon|^{p-1+\varepsilon}u_\varepsilon
\lambda_1\frac{\partial P \delta_1}{\partial\lambda_1}\\
&= \int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta
_2|^{p-1+\varepsilon}(\alpha_1P\delta _1-\alpha_2P\delta
_2)\lambda_1\frac{\partial P \delta_1}{\partial\lambda_1}
\\
&\quad +(p+\varepsilon)\int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta
_2|^{p-1+\varepsilon}v\lambda_1\frac{\partial P \delta_1}{\partial
 \lambda_1}+ O\Big(\|v\|^2+\varepsilon_{12}^{\frac{n}{n-4}}
\log\varepsilon_{12}^{-1}\Big).
\end{aligned}
\end{equation}
The above integral can be written as
\begin{equation} \label{e:412}
\begin{aligned}
&\int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta_2|^{p-1+\varepsilon}
v\lambda_1\frac{\partial P \delta_1}{\partial  \lambda_1}\\
&=\int_{\Omega}(\alpha_1P\delta _1)^{p-1+\varepsilon}v\lambda_1
\frac{\partial P \delta_1}{\partial  \lambda_1}
+ O\Big(\int_{\Omega\setminus A}P\delta
_2^{p-1}P\delta _1|v| +\int_{A}P\delta _1^{p-1}P\delta _2|v|\Big),
\end{aligned}
\end{equation}
where $A = \{x : 2\alpha_2P\delta_2 \leq \alpha_1P\delta_1 \}$.
Observe that, for $n \geq 12$, we have   $p-1 = 8/(n-4)\leq 1$,
thus
\begin{align*}
\int_{\Omega\setminus A}P\delta _2^{p-1}P\delta _1|v|
+\int_{A}P\delta _1^{p-1}P\delta _2|v|
&\leq c  \int_{\Omega}|v|(\delta _1\delta _2)^{\frac{n+4}{2(n-4)}}
\\
&\leq c\|v\|\varepsilon_{12}^{(n+4)/2(n-4)}(\log\varepsilon_{12}^{-1})^{(n+4)/2n}.
\end{align*}
For $n < 12$, we have
\begin{equation}
\int_{\Omega\setminus A}P\delta _2^{p-1}P\delta _1|v| +\int_{A}P\delta _1^{p-1}P\delta
_2|v| \leq c \varepsilon_{12}(\log\varepsilon_{12}^{-1})^{(n-4)/n}\|v\|.
\end{equation}
For the other integral in \eqref{e:412}, using \cite{B1},
\cite{R2} and Remark \ref{r:24}, we obtain
\begin{align*}
&\int_{\Omega}P\delta_1^{p-1+\varepsilon}v\lambda_1\frac{\partial P
\delta_1}{\partial \lambda_1}  \\
&= O\Big( \|v\|\Big[\varepsilon +\Big(
\frac{1}{(\lambda_1d_1)^{\inf(n-4,(n+4)/2)}}(\text{if }n\neq12)
+\frac{\log(\lambda_1d_1)}{(\lambda_1d_1)^4}(\text{if }
 n=12)\Big)\Big]\Big).
\end{align*}
It remains to estimate the second integral of \eqref{e:411}. We
have
\begin{align*}
&\int_{\Omega}| \alpha_1P\delta _1-\alpha_2P\delta
_2|^{p-1+\varepsilon}(\alpha_1P\delta _1-\alpha_2P\delta
_2)\lambda_1\frac{\partial P
\delta_1}{\partial\lambda_1} \\
&=\int_{\Omega}(\alpha_1P\delta
_1)^{p+\varepsilon}\lambda_1\frac{\partial P \delta_1}{\partial
\lambda_1}-\int_{\Omega}(\alpha_2P\delta _2)^{p+\varepsilon}\lambda_1\frac{\partial
P \delta_1}{\partial \lambda_1}\\
&\quad -(p+\varepsilon)\int_{\Omega}\alpha_2P\delta
_2(\alpha_1P\delta _1)^{p-1+\varepsilon}\lambda_1\frac{\partial P
\delta_1}{\partial \lambda_1}+
O\Big(\varepsilon_{12}^{\frac{n}{n-4}}\log\varepsilon_{12}^{-1}\Big) .
\end{align*}
Now, using Remark \ref{r:24} and \cite{B1}, we have
\begin{align*}
 \int_{\Omega}P\delta _1^{p+\varepsilon}\lambda_1\frac{\partial P
\delta_1}{\partial \lambda_1}
&= \frac{n-4}{2}\Big(c_2\varepsilon
+2c_1\frac{H(a_1,a_1)}{\lambda_1^{n-4}}\Big) \\
&\quad + O\Big(\varepsilon^2+ \frac{\log(\lambda_1 d_1)}
{(\lambda_1d_1)^{n-1}}+\frac{1}{(\lambda_1d_1)^2}(\text{if }n=5)\Big) ,
\end{align*}
\[
 \int_{\Omega}P\delta _2^{p+\varepsilon}\lambda_1\frac{\partial P
\delta_1}{\partial \lambda_1}=c_1\Big(\lambda_1\frac{\partial
\varepsilon_{12}}{\partial \lambda_1} +
\frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+ R_2,
\]
\begin{equation} \label{e:419}
p\int_{\Omega}P\delta _2P\delta _1^{p-1+\varepsilon}\lambda_1\frac{\partial P
\delta_1}{\partial \lambda_1} =c_1\Big(\lambda_1\frac{\partial
\varepsilon_{12}}{\partial \lambda_1} +
\frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big)+ R_1,
\end{equation}
where for $i=1,2$,
\begin{align*}
R_i &=  O\Big(\varepsilon
\varepsilon_{12}(\log\varepsilon_{12}^{-1})^{\frac{n-4}{n}}\Big)
+\Big(\varepsilon_{12}^{\frac{n}{n-4}}(\log\varepsilon_{12}^{-1})
+\frac{\log(\lambda_id_i)}{(\lambda_id_i)^{n}}\text{if }n\geq8\Big)\\
&\quad +\Big(\frac{\varepsilon_{12}(\log\varepsilon_{12}
^{-1})^{\frac{n-4}{n}}}{(\lambda_id_i)^{n-4}}\text{if }n<8\Big)
\end{align*}
Therefore, combining \eqref{e:47}--\eqref{e:419}, and Lemma \ref{l:41},
the proof of Proposition \ref{p:42} follows.
\end{proof}

\section{Proof of main theorems}

\begin{proof}[Proof of Theorem \ref{t:11}]
Arguing by contradiction, let us suppose that the problem
\eqref{ePe} has a solution $u_{\varepsilon}$ as stated in Theorem
\ref{t:11}. This solution has to satisfy \eqref{e:11} and  from
Proposition \ref{p:42}, we have
\begin{equation} \label{e:51}
\begin{aligned}
&c_1\frac{n-4}{2}\frac{H(a_i,a_i)}{\lambda_i^{n-4}}
-c_1\Big(\lambda_i\frac{\partial \varepsilon_{12}}{\partial \lambda_i} +
\frac{n-4}{2}\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}}
\Big)+\frac{n-4}{2}c_2 \varepsilon  \\
 & = o\Big(\varepsilon+ \sum_{k=1,2}
\frac{1}{(\lambda_kd_k)^{n-4}}+ \varepsilon_{12}\Big), \quad \text{for }
i=1,2.
\end{aligned}
\end{equation}
Furthermore, an easy computation shows that
\begin{equation}\label{e:111}
\lambda_i\frac{\partial \varepsilon_{12}}{\partial
\lambda_i}=-\frac{n-4}{2}\varepsilon_{12}
\big(1-2\frac{\lambda_j}{\lambda_i}\varepsilon_{12}^{2/n-4}\big),
\quad \text{for } i,j=1,2;\;  j\ne i.
\end{equation}
Without loss of generality, we can assume that $\lambda_2 \geq \lambda_1$.
We distinguish two cases and in each one, we will find a contradiction
which implies our theorem.
\smallskip

\noindent\textbf{Case 1.}  $\frac{\lambda_1\lambda_2|
a_1-a_2|^2}{{\lambda_2}/{\lambda_1}}\to +\infty$. In this
case, it is easy to obtain 
\begin{equation}
\varepsilon_{12}=\frac{1}{(\lambda_1\lambda_2|a_1-a_2|^2)^{(n-4)/2}}
+ o(\varepsilon_{12}), 
\end{equation}
which implies that
\begin{equation}\label{e:54}
 \lambda_i\frac{\partial \varepsilon_{12}}{\partial
\lambda_i}=-\frac{n-4}{2}\frac{1}{(\lambda_1\lambda_2|a_1-a_2|^2)^{(n-4)/2}}
+o(\varepsilon_{12})\quad \text{for }  i=1,2.
\end{equation}
Then from \eqref{e:51} and \eqref{e:54}, we obtain
\begin{align*}
&\frac{c_1}{2}\Big(\frac{H(a_1,a_1)}{\lambda_1^{n-4}}
 +\frac{H(a_2,a_2)}{\lambda_2^{n-4}}\Big) 
 +\frac{c_1}{(\lambda_1\lambda_2)^{(n-4)/2}}
 \Big(\frac{1}{|a_1-a_2|^{n-4}} -H(a_1,a_2)\Big)+c_2\varepsilon \\
& = o\Big(\varepsilon+ \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^{n-4}}
 + \varepsilon_{12}\Big).
\end{align*}
Using the fact that  
\begin{gather*}
G(a_1,a_2):=\frac{1}{|a_1-a_2|^{n-4}} -H(a_1,a_2) >0, \\ 
\varepsilon_{12}=O\Big(\frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}}+
\frac{G(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}} \Big),
\end{gather*}
we derive a contradiction in this case. 
\smallskip

\noindent\textbf{Case 2.}
$\frac{\lambda_1\lambda_2| a_1-a_2|^2}{{\lambda_2}/{\lambda_1}}\to c \geq 0$.
 In this case, we remark that $\lambda_2/\lambda_1 \to +\infty$ (since
$\varepsilon_{12} \to 0$). Multiplying \eqref{e:51} by 2 for $i=2$ and
adding to \eqref{e:51} for $i=1$, we obtain:
\begin{equation} \label{e:57}
\begin{aligned}
 & c_1\Big(\frac{H(a_1,a_1)}{\lambda_1^{n-4}}+2\frac{H(a_2,a_2)}{\lambda_2^{n-4}}\Big)
 -\frac{2c_1}{n-4}\Big(\lambda_1\frac{\partial \varepsilon_{12}}{\partial
 \lambda_1}+
 2\lambda_2\frac{\partial \varepsilon_{12}}{\partial \lambda_2}\Big)\\
&- \frac{3H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}}+3c_2\varepsilon   \\
 & = o\Big(\varepsilon+  \sum_{k=1,2} \frac{1}{(\lambda_kd_k)^{n-4}}
+ \varepsilon_{12}\Big).
\end{aligned}
\end{equation}
Now, using \eqref{e:111} and the fact that $\lambda_2\geq \lambda_1$, an
easy computation shows that
\begin{equation} \label{e123}
-\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1}
-2\lambda_2\frac{\partial \varepsilon_{12}}{\partial \lambda_2}
 \geq \frac{n-4}{4} \varepsilon_{12}.
\end{equation}
Furthermore, since
$H(a_1,a_2)\leq cd_1^{4-n}$ and $\lambda_2/\lambda_1 \to \infty$, we obtain
 \begin{equation}\label{e147}
 \frac{H(a_1,a_2)}{(\lambda_1\lambda_2)^{(n-4)/2}}
=o\Big( \frac{1}{(\lambda_1d_1)^{n-4}}\Big).
\end{equation}
Then we derive a contradiction from \eqref{e:57}, \eqref{e123} and
\eqref{e147}. Our proof is thereby complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{t:14}]
Arguing by contradiction, let us assume that problem \eqref{ePe} has
solutions $(u_\varepsilon)$ as stated in Theorem \ref{t:14}. 
From Section 2, these solutions have to satisfy \eqref{e:11} and \eqref{e:12}.
As in the proof of Proposition \ref{p:42},  we have for each $i=1,\dots, k$
\begin{equation} \label{Ei}
\begin{aligned}
&c_1\frac{n-4}{2}\frac{H(a_i,a_i)}{\lambda_i^{n-4}}
 +c_1 \sum_{j\neq i}(-1)^{j+1}\Big(\lambda_i\frac{\partial
\varepsilon_{ij}}{\partial \lambda_i} +
\frac{n-4}{2}\frac{H(a_i,a_j)}{(\lambda_i\lambda_j)^{(n-4)/2}}
\Big)+\frac{n-4}{2}c_2 \varepsilon  \\
 & = o\Big(\varepsilon+  \sum_{j=1}^k
\frac{1}{(\lambda_jd_j)^{n-4}}+ \sum_{r\neq j}\varepsilon_{rj}\Big).
\end{aligned}
\end{equation}
Observe that,  if $j<i$, we have $\lambda_j|a_i-a_j|$ is bounded (by
the assumption) which implies that
 \begin{equation} \label{rome}
|a_i-a_j| =o(d_j) ,\quad d_i/d_j=1+o(1) \,  \forall  i,j,
\quad
 \varepsilon_{ij} \geq c (\lambda_j/\lambda_i)^{(n-4)/2}\, \forall j < i,
\end{equation}
where $c$ is a positive constant. Using \eqref{rome}, easy
computations show that
 \begin{equation}\label{rome2}
\begin{gathered}
\varepsilon_{(i-1)j}+  \varepsilon_{i(j+1)}=o(\varepsilon_{ij})\quad \forall i < j, \\
\frac{H(a_i,a_j)}{(\lambda_i\lambda_j)^{(n-4)/2}}=o(\frac{1}{(\lambda_1d_1)^{n-4}})
\text{ if } (i,j) \ne (1,1).
\end{gathered}
\end{equation}
Thus, using \eqref{rome2}, \eqref{Ei} can be written as
\begin{align} \label{E1'}
 c_1\frac{n-4}{2}\frac{H(a_1,a_1)}{\lambda_1^{n-4}}
-c_1\lambda_1\frac{\partial \varepsilon_{12}}{\partial \lambda_1}
+\frac{n-4}{2}c_2 \varepsilon
= o\Big(\varepsilon+
\frac{1}{(\lambda_1d_1)^{n-4}}+ \sum_{r\neq j}\varepsilon_{rj}\Big),\\
\label{Ek'}
-c_1\frac{\partial \varepsilon_{(k-1)k}}{\partial \lambda_k}
+\frac{n-4}{2}c_2 \varepsilon
= o\Big(\varepsilon+ \frac{1}{(\lambda_1d_1)^{n-4}}
 + \sum_{r\neq j}\varepsilon_{rj}\Big),
\end{align}
and for $1<i<k$,
\begin{equation} \label{Ei'}
-c_1\lambda_i\frac{\partial \varepsilon_{(i-1)i}}{\partial
\lambda_i}-c_1\lambda_i\frac{\partial \varepsilon_{i(i+1)}}{\partial
\lambda_i}+\frac{n-4}{2}c_2 \varepsilon
 = o\Big(\varepsilon+ \frac{1}{(\lambda_1d_1)^{n-4}}+ \sum_{r\neq j}
\varepsilon_{rj}\Big).
\end{equation}
Using \eqref{e:111} and \eqref{Ek'}, we derive that
 \begin{equation}\label{fin1}
\varepsilon = o\Big(\frac{1}{(\lambda_1d_1)^{n-4}} +
\sum_{r\neq j}\varepsilon_{ij}\Big),\quad
\varepsilon_{(k-1)k} =
o\Big(\frac{1}{(\lambda_1d_1)^{n-4}}+ \sum_{r\neq j}\varepsilon_{rj}\Big).
\end{equation}
Now, using \eqref{fin1} and \eqref{Ek'} with $k-1$ instead of $k$,
we derive the estimate of $\varepsilon_{(k-2)(k-1)}$ and by induction we get
 \begin{equation}\label{fin2}
\varepsilon_{(i-1)i} = o\Big( \frac{1}{(\lambda_1d_1)^{n-4}} + \sum_{r\neq
j}\varepsilon_{rj}\Big) \quad \text{for } i=2, \dots, k.
\end{equation}
Finally, using \eqref{rome2}, \eqref{fin1}, \eqref{fin2} and \eqref{E1'},
we obtain
\begin{equation*}
\frac{H(a_1,a_1)}{\lambda_1^{n-4}}
  = o(\frac{1}{(\lambda_1d_1)^{n-4}}),
\end{equation*}
which gives a contradiction. Hence, our theorem is proved.
\end{proof}


\begin{proof}[Proof of Theorem \ref{t:17}]
Arguing by contradiction, let us assume that problem \eqref{ePe} has
solutions $(u_\varepsilon)$ as stated in Theorem \ref{t:17}. From Section
2, these solutions have to satisfy \eqref{e:11} and \eqref{e:12}.
Without loss of generality, in the sequel, we will assume that
$\lambda_1d_1\leq \lambda_{m+1}d_{m+1}$. As in the proof of Theorem
\ref{t:14},  \eqref{Ei} is satisfied for each $i=1, \dots , k$.
Furthermore, \eqref{rome2} holds if $i,j \leq m$ or  $i,j >m$
(in the last case, we require that $(i,j)\ne (m+1,m+1)$). 

Observe that since $a\ne b$, it is easy to obtain that $|a_i-a_j|
\geq c>0$ for each $i\leq m$ and $j\geq m+1$. Hence for $i\leq m$
and $j\geq m+1$ we have
 \begin{gather}
\lambda_r\frac{\partial \varepsilon_{ij}}{\partial \lambda_r}
= -\frac{n-4}{2}\frac{1}{(\lambda_i\lambda_j|a_i-a_j|^2)^{(n-4)/2}}
+o(\varepsilon_{ij}),
\quad  \text{for } r=i,j,\label{wx1}
\\
\varepsilon_{ij} + \frac{H(a_i,a_j)}{(\lambda_i \lambda_j)^{(n-4)/2}}
 =o\Big(\varepsilon_{1(m+1)}+\frac{1}{(\lambda_1d_1)^{n-4}}\Big)
\quad \text{for } (i,j)\ne (1,m+1).\label{wx2}
\end{gather} 
Now using \eqref{rome2}, \eqref{wx1} and \eqref{wx2}, we derive that
\eqref{Ei'}  holds for each $i\not \in \{1,m+1\}$. However, since 
the first bubble  in the second bubble tower $u_{\varepsilon}^{2}$ 
has negative sign, for $i=1,m+1$, we have
\[
c_1\frac{H(a_i,a_i)}{\lambda_i^{n-4}}
-\frac{2c_1}{n-4}\lambda_i\frac{\partial \varepsilon_{i(i+1)}}{\partial
\lambda_i}+c_1 \varepsilon_{1(m+1)}+c_2 \varepsilon  = o\Big(\varepsilon+
\frac{1}{(\lambda_1d_1)^{n-4}}+ \sum_{r\neq j}\varepsilon_{rj}\Big).
\]
Finally, arguing as in Theorem \ref{t:14}, we derive a
contradiction. Hence our result is proved.
\end{proof}

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