# Sign-changing bubble tower solutions for sinh-Poisson type equations on
pierced domains

Pablo Figueroa  Pablo Figueroa
Instituto de Ciencias Físicas y Matemáticas
Facultad de Ciencias
Universidad Austral de Chile
Campus Isla Teja s/n, Valdivia, Chile<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

For asymmetric sinh-Poisson type problems with Dirichlet boundary condition
arising as a mean field equation of equilibrium turbulence vortices with
variable intensities of interest in hydrodynamic turbulence, we address the
existence of sign-changing bubble tower solutions on a pierced domain
$\Omega_{\epsilon}:=\Omega\setminus\displaystyle\overline{B(\xi,\epsilon)}$,
where $\Omega$ is a smooth bounded domain in $\text{I\\!R}^{2}$ and
$B(\xi,\epsilon)$ is a ball centered at $\xi\in\Omega$ with radius
$\epsilon>0$. Precisely, given a small parameter $\rho>0$ and any integer
$m\geq 2$, there exist a radius $\epsilon=\epsilon(\rho)>0$ small enough such
that each sinh-Poisson type equation, either in Liouville form or mean field
form, has a solution $u_{\rho}$ with an asymptotic profile as a sign-changing
tower of $m$ singular Liouville bubbles centered at the same $\xi$ and with
$\epsilon(\rho)\to 0^{+}$ as $\rho$ approaches to zero.

###### Key words and phrases:

sinh-Poisson type equation, pierced domain, tower of bubbles

###### 2020 Mathematics Subject Classification:

35B44, 35J25, 35J60

##  1\. Introduction

Let $\Omega$ be a smooth bounded domain in $\text{I\\!R}^{2}$. Given
$\epsilon>0$ and $\xi\in\Omega$, define
$\Omega_{\epsilon}:=\Omega\setminus\displaystyle\overline{B(\xi,\epsilon)}$, a
pierced domain, where $B(\xi,\epsilon)$ is a ball centered at $\xi$ with
radius $\epsilon$. Inspired by results in [1, 13, 17, 20, 30], we are
interested in constructing sign-changing _bubble tower_ solutions to sinh-
Poisson type equations, either in Liouville form or mean field form, with
variable intensities and Dirichlet boundary conditions on a pierced domain
$\Omega_{\epsilon}$. Precisely, on one hand, we consider the following problem
in Liouville form

$\left\\{\begin{array}[]{ll}-\Delta u=\rho(V_{0}(x)e^{u}-\nu V_{1}(x)e^{-\tau
u})&\text{in $\Omega_{\epsilon}$}\\\ \ \ u=0&\text{on
$\partial\Omega_{\epsilon}$}\end{array}\right.,$ (1.1)

and, on the other hand, we also study the problem in mean field form

$\left\\{\begin{array}[]{ll}\displaystyle-\Delta
u={\lambda_{0}V_{0}(x)e^{u}\over\int_{\Omega_{\epsilon}}V_{0}e^{u}}-{\lambda_{1}\tau
V_{1}(x)e^{-\tau u}\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau u}}&\text{in
$\Omega_{\epsilon}$}\\\ \ \ u=0&\text{on
$\partial\Omega_{\epsilon}$}\end{array}\right.,$ (1.2)

where $\rho>0$ is small, $\lambda_{0},\lambda_{1}>0$, $V_{0},V_{1}>0$ are
smooth potentials in $\Omega$, $\tau>0$, $\epsilon>0$ is a small number and
$\nu\geq 0$. Our aim is to construct for each problem a family of solutions
$u_{\rho}$ for a suitable choice of $\epsilon=\epsilon(\rho)$, with an
asymptotic profile as a sum of positive and negative singular Liouville
bubbles centered at the same point $\xi$ as $\rho\to 0$, on the line of [20,
30].

These equations and related ones have attracted a lot of attention in recent
years due to its relevance in the statistical mechanics description of
2D-turbulence, as initiated by Onsager [28]. Precisely, in this context
Caglioti, Lions, Marchioro, Pulvirenti [6] and Sawada, Suzuki [35] derive the
following equation:

$\displaystyle-\Delta u=\lambda\int\limits_{[-1,1]}{\alpha e^{\alpha
u}\over\int\limits_{\Omega}e^{\alpha u}dx}d\mathcal{P}(\alpha)\ \ \hbox{ in
}\Omega,\ \ \ \ u=0\ \hbox{ on }\partial\Omega,$ (1.3)

where $\Omega$ is a bounded domain in $\mathbb{R}^{2},$ $u$ is the stream
function of the flow, $\lambda>0$ is a constant related to the inverse
temperature and $\mathcal{P}$ is a Borel probability measure in $[-1,1]$
describing the point-vortex intensities distribution. We observe that (1.3) is
obtained under a _deterministic_ assumption on the distribution of the vortex
circulations.

Equation (1.3) includes several well-known problems depending on a suitable
choice of $\mathcal{P}$. For instance, if $\mathcal{P}=\delta_{1}$ is
concentrated at $1$, then (1.3) corresponds to the classical mean field
equation

$-\Delta u=\lambda{e^{u}\over\int\limits_{\Omega}e^{u}dx}\ \ \hbox{ in
}\Omega,\ \ \ \ u=0\ \hbox{ on }\partial\Omega.$ (1.4)

Since there are plenty of results in the literature concerning (1.4), let us
just quote [2, 7, 8, 9, 12, 26]. When
$\mathcal{P}=\sigma\delta_{1}+(1-\sigma)\delta_{-\tau}$ with $\tau\in[-1,1]$
and $\sigma\in[0,1]$, equation (1.3) becomes

$-\Delta
u=\lambda\bigg{(}\sigma{e^{u}\over\int\limits_{\Omega}e^{u}dx}-(1-\sigma)\tau{e^{-\tau
u}\over\int\limits_{\Omega}e^{-\tau u}dx}\bigg{)}\ \ \hbox{ in }\Omega,\ \ \ \
u=0\ \hbox{ on }\ \partial\Omega.$ (1.5)

Setting $\lambda_{0}=\lambda\sigma$, $\lambda_{1}=\lambda(1-\sigma)$ and
$V_{0}=V_{1}=1$ problem (1.5) can be rewritten as (1.2) replacing
$\Omega_{\epsilon}$ by $\Omega$. If $\tau=1$ and $V_{0}=V_{1}\equiv 1$ problem
(1.2) reduces to mean field equation of the equilibrium turbulence, see [5,
21, 24, 27, 31] or its related sinh-Poisson version, see [3, 4, 20, 23, 25],
which have received a considerable interest in recent years. Recently, sign-
changing solutions have been constructed in $\Omega$ for the sinh-Poisson
equation with Robin boundary condition in [19].

Concerning results for $\tau>0$, Pistoia and Ricciardi built in [29] sequences
of blowing-up solutions to (1.2) (in $\Omega$ instead $\Omega_{\epsilon}$)
when $\lambda_{0},\lambda_{1}\tau^{2}$ are close to $8\pi$. Ricciardi and
Takahashi in [32] provided a complete blow-up picture for solution sequences
of (1.2) and successively in [33] Ricciardi et al. constructed min-max
solutions when $\lambda_{0}\to 8\pi^{+}$ and $\lambda_{1}\to 0$ on a multiply
connected domain (in this case the nonlinearity $e^{-\tau u}$ may be treated
as a lower-order term with respect to the main term $e^{u}$). A blow-up
analysis and some existence results are obtained when $\tau>0$ in a compact
Riemann surface in [22, 34]. Bubbling solutions in a compact Riemann surface
has been recently constructed in [18].

On the other hand, on pierced domains, Ahmedou and Pistoia in [1] proved that
on $\Omega_{\epsilon}$, there exists a solution to the classical mean field
equation (1.4) which blows-up at $\xi$ as $\epsilon\to 0$ for any
$\lambda>8\pi$ (extra symmetric conditions are required when $\lambda\in
8\pi\mathbb{N}$). In [13] the authors constructed a family of solutions to the
mean field equation with variable intensities (1.2) blowing-up positively and
negatively at $\xi_{1},\dots,\xi_{m_{1}}$ and $\xi_{m_{1}+1},\dots,\xi_{m}$,
respectively, as $\epsilon_{1},\dots,\epsilon_{m}\to 0$ on a pierced domain
with several holes ($\Omega_{\epsilon}$ is replaced by in
$\Omega\setminus\cup_{i=1}^{m}\overline{B(\xi_{i},\epsilon_{i})}$ ), in the
super-critical regime $\lambda_{0}>8\pi m_{1}$ and
$\lambda_{1}\tau^{2}>8\pi(m-m_{1})$ with $m_{1}\in\\{0,1,\dots,m\\}$.
Recently, in the same spirit of [13], the author in [17] addressed the sinh-
Poisson type equation with variable intensities (1.1) on a pierced domain with
several holes
$\Omega\setminus\cup_{i=1}^{m}\overline{B(\xi_{i},\epsilon_{i})}$. Equation
(1.1) is related, but not equivalent, to problem (1.2) by using the change

$\rho={\lambda_{0}\over\int_{\Omega_{\epsilon}}V_{0}e^{u}}\qquad\text{ and
}\qquad\rho\nu={\lambda_{1}\tau\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau
u}}.$

To the extent of our knowledge, there are by now just few results concerning
non-simple blow-up for sinh-Poisson type problems. Precisely, sign-changing
solutions with non-simple blow-up has been built in [15] for the Neumann sinh-
Poisson equation. Grossi and Pistoia built in [20] a sign-changing bubble
tower solutions for the sinh-Poisson version ($\tau=1$) in a symmetric domain
$\Omega$ with respect to a fixed point $\xi\in\Omega$. After that, Pistoia and
Ricciardi in [30] extend this construction to a sinh-Poisson type equation
with asymmetric exponents ($\tau\neq 1$) under a symmetric assumption on
$\Omega$ depending on either $\tau\in\mathbb{Q}$ or $\tau\notin\mathbb{Q}$. In
both situations [20, 30] the number of bubbles can be arbitrary large.

A matter of interest to us is whether do there exist sign-changing bubble
tower solutions to (1.1) for small values of $\rho$ or to (1.2) for some
values of the parameters $\lambda_{0},\lambda_{1},\tau>0$ on a pierced domain
$\Omega_{\epsilon}$, with bubbles centered at $\xi$. Our first result in this
direction without symmetry assumptions reads as follows.

######  Theorem 1.1.

Let $m\geq 2$ be a positive integer. There exists $\rho_{0}>0$ such that for
all $0<\rho<\rho_{0}$ there is $\epsilon=\epsilon(\rho)$ small enough such
that problem (1.1) has a sign-changing solution $u_{\rho}$ in
$\Omega_{\epsilon}$ blowing-up at $\xi$ in the sense that

$u_{\rho}=(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)G(\cdot,\xi)+o(1)$
(1.6)

locally uniformly in $\bar{\Omega}\setminus\\{\xi\\}$ as $\rho\to 0^{+}$.

Here, for simplicity, we denote

$\nu(i)={1+(-1)^{i}\over 2}=\begin{cases}0&\text{ if $i$ is odd}\\\ 1&\text{
if $i$ is even}\end{cases}\qquad\text{and}\qquad\sigma(i)={1-(-1)^{i}\over
2}=\begin{cases}1&\text{ if $i$ is odd}\\\ 0&\text{ if $i$ is
even}\end{cases},$ (1.7)

$\alpha_{m}$ is given by (2.4) with $i=m$ and
$G(x,y)=-\frac{1}{2\pi}\log|x-y|+H(x,y)$ is the Green’s function of $-\Delta$
in $\Omega$, where the regular part $H$ is a harmonic function in $\Omega$ so
that $H(x,y)=\frac{1}{2\pi}\log|x-y|$ on $\partial\Omega$. Nevertheless, the
latter result may not tell us whether (1.2) has sign-changing bubble tower
solutions for some values of the parameters $\lambda_{0},\lambda_{1},\tau>0$.
Therefore, we perform directly to problem (1.2) a similar procedure.
Conversely, Theorem 1.2 below may not tell us whether (1.1) has sign-changing
bubble tower solutions for _all_ small $\rho>0$. Assume that $\lambda_{0}$ and
$\lambda_{1}$ decompose for some $\alpha_{1}>2$ (see (2.3)), as either

$\lambda_{0}=2\pi
m\left[\alpha_{1}+(m-2)\Big{(}1+{1\over\tau}\Big{)}\right],\quad\lambda_{1}\tau^{2}=2\pi
m\left[\alpha_{1}\tau+m(1+\tau)\right],\quad\text{if }m\text{ even}$ (1.8)

or

$\lambda_{0}=2\pi(m+1)\left[\alpha_{1}+(m-1)\Big{(}1+{1\over\tau}\Big{)}\right],\
\ \lambda_{1}\tau^{2}=2\pi(m-1)\left[\alpha_{1}\tau+(m-1)(1+\tau)\right],\
\text{ if }m\text{ odd}.$ (1.9)

In particular, we choose $\alpha_{1}>2$ and $\alpha_{1}\notin 2\text{I\\!N}$
if $\tau=1$. Our second result (without symmetry assumptions) is the following

######  Theorem 1.2.

If either (1.8) or (1.9) holds for a positive integer $m\geq 2$, then there
exists a radius $\epsilon>0$ small enough such that problem (1.2) has a sign-
changing solution $u_{\epsilon}$ in $\Omega_{\epsilon}$ blowing-up at $\xi$ in
the sense of (1.6) locally uniformly in $\bar{\Omega}\setminus\\{\xi\\}$ as
$\epsilon\to 0^{+}$.

Our solutions correspond to a superposition of highly concentrated vortex
configurations of alternating orientation around the hole $B(\xi,\epsilon)$
and they extend some known results [20, 30] for symmetric domains. We also
point out that a delicate point in the paper concerns the linear theory
developed in Section 5. A bit more complicated analysis is necessary in
comparison with linear theories developed in previous works [1, 11, 13, 14,
16, 20, 30].

Without loss of generality, we shall assume in the rest of the paper that
$\xi=0\in\Omega$, so that $\Omega_{\epsilon}=\Omega\setminus B(0,\epsilon)$
where $\epsilon>0$ is small and $\nu=1$, since we can replace $\nu V_{2}$ by
$V_{2}$. However, we need the presence of $\nu$ when we compare (1.1) with
equation (1.2).

Finally, we point out some comments about the proofs of the theorems.
Following the ideas presented in [20, 30] about bubble tower solutions for
sinh-Poisson type equations and in [1, 13, 16] about construction of solutions
on pierced domains, we find a solution $u_{\rho}$ using a perturbative
approach, precisely, we look for a solution of (1.1) as

$u_{\rho}=U+\phi,$ (1.10)

where $U$ is a suitable ansatz built using the projection operator
$P_{\epsilon}$ onto $H^{1}_{0}(\Omega_{\epsilon})$(see (2.1)) and $\phi\in
H_{0}^{1}(\Omega_{\epsilon})$ is a small remainder term. Letting
$w_{\delta,\alpha}(x)=\log\frac{2\alpha^{2}\delta^{\alpha}}{(\delta^{\alpha}+|x|^{\alpha})^{2}}$
be a solution to the singular Liouville equation

$\Delta u+|x|^{\alpha-2}e^{u}=0\quad\text{ in
$\text{I\\!R}^{2}$},\qquad\displaystyle\int_{\text{I\\!R}^{2}}|x|^{\alpha-2}e^{u}<+\infty,$

the ansatz $U$ is defined as follows

$U(x)=\sum_{i\text{ odd}}P_{\epsilon}w_{i}(x)-\dfrac{1}{\tau}\sum_{i\text{
even}}P_{\epsilon}w_{i}(x)=\sum_{i=1}^{m}\dfrac{(-1)^{i+1}}{\tau^{\nu(i)}}P_{\epsilon}w_{i}(x),$
(1.11)

where $w_{j}:=w_{\delta_{j},\alpha_{j}}$ for $j=1,\dots,m$. A careful choice
of the parameters $\delta_{j}$’s, $\alpha_{i}$’s and the radius $\epsilon$,
depending on $\rho>0$, is made in section 2 (see (2.2), (2.3)-(2.4) and
(2.21)) in order to make $U$ be a good approximated solution. Indeed, the
error term $R$ for (1.1) given by

$R=\Delta U+\rho(V_{0}(x)e^{U}-V_{1}(x)e^{-\tau U})$ (1.12)

is small in $L^{p}$-norm for $p>1$ close to $1$ (see Lemma 3.1). A
linearization procedure around $U$ leads us to re-formulate (1.1) in terms of
a nonlinear problem for $\phi$ (see equation (3.13)). We will prove the
existence of such a solution $\phi$ to (3.13) by using a fixed point argument,
thanks to some estimates in subsection 3.2 (see (3.20) and (3.21)). The
corresponding solution $u_{\rho}$ in (1.10) behaves as a sign-changing tower
of $m$ singular Liouville bubbles thanks to the asymptotic properties of its
main order term $U$ (see (2.23) in Lemma 2.6). In Section 5 we will prove the
invertibility of the linear operator naturally associated to the problem (see
(3.14)) stated in Proposition 3.1. To conclude Theorem 1.2, the same procedure
with the same ansatz is performed to equation (1.2), assuming (1.8)-(1.9) and
$\epsilon=\epsilon(\rho)$, where the error term is given by

$\mathcal{R}=\Delta
U+\lambda_{0}{V_{0}(x)e^{U}\over\int_{\Omega_{\epsilon}}V_{0}e^{U}}-\lambda_{1}\tau{V_{1}(x)e^{-\tau
U}\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau U}}.$ (1.13)

##  2\. Approximation of the solution

In this section we shall make a choice of the parameters $\alpha_{i}$’s,
$\delta_{j}$’s and $\epsilon=\epsilon(\rho)$ in order to make $U$ a good
approximation. Introduce the function $P_{\epsilon}w$ as the unique solution
of

$\left\\{\begin{array}[]{ll}\Delta P_{\epsilon}w=\Delta w&\text{in
}\Omega_{\epsilon}\\\ P_{\epsilon}w=0,&\text{on
}\partial\Omega_{\epsilon}.\end{array}\right.$ (2.1)

For simplicity, we will denote $h_{0}=H(0,0)$. We have the following
asymptotic expansion of $Pw_{\delta,\alpha}$ as $\delta\to 0$, as shown in [1,
Lemma 2.1] (see also [13, Lemma 2.1]):

######  Lemma 2.1.

The function $P_{\epsilon}w_{\delta,\alpha}$ satisfies

$P_{\epsilon}w_{\delta,\alpha}(x)=w_{\delta,\alpha}(x)-\log(2\alpha^{2}\delta^{\alpha})+4\pi\alpha
H(x,0)-\gamma_{\delta,\epsilon}^{\alpha}G(x,0)+O\Big{(}\delta^{\alpha}+\Big{(}{\epsilon\over\delta}\Big{)}^{\alpha}+\Big{[}1+\Big{|}\frac{\log\delta}{\log\epsilon}\Big{|}\Big{]}\epsilon\Big{)},$

uniformly in $\Omega_{\epsilon}$, where $\gamma_{\delta,\epsilon}^{\alpha}$ is
given by
$\gamma_{\delta,\epsilon}^{\alpha}=\frac{-2\alpha\log\delta+4\pi\alpha
h_{0}}{-{1\over 2\pi}\log\epsilon+h_{0}}.$ In particular, there holds

$P_{\epsilon}w_{\delta,\alpha}(x)=[4\pi\alpha-\gamma_{\delta,\epsilon}^{\alpha}]G(x,0)+O\Big{(}\delta^{\alpha}+\Big{(}{\epsilon\over\delta}\Big{)}^{\alpha}+\Big{[}1+\Big{|}\frac{\log\delta}{\log\epsilon}\Big{|}\Big{]}\epsilon\Big{)}$

as $\epsilon\to 0$ locally uniformly in $\Omega\setminus\\{0\\}$.

Given $m\in\text{I\\!N}$ with $m\geq 2$, consider $\delta_{j}>0$ and
$\alpha_{j}>2$, $j=1,\dots,m$, so that our approximating solution $U$ is
defined by (1.11), parametrized by $\delta_{j}$’s and $\alpha_{i}^{\prime}$s
with $\delta_{j}=\delta_{j}(\rho,\alpha_{1})$ (it also depends on $\tau$,
$h_{0}$, $V_{0}(0)$ and $V_{1}(0)$), where $\nu(i)$ is defined in (1.7) and
$P_{\epsilon}$ the projection operator defined by (2.1) for a suitable choice
of $\epsilon$. In order to have a good approximation, for any $i=1,\dots,m$ we
will assume that

$\delta_{i}^{\alpha_{i}}=d_{i}\rho^{\beta_{i}},$ (2.2)

for small $\rho>0$, where $\alpha_{i}$’s, $\beta_{i}$’s, and $d_{i}$’s will be
specified below. We choose

$\alpha_{1}>2,\quad\text{with}\quad\alpha_{1}\in\begin{cases}\displaystyle\bigcap_{k=0}^{(m-2)/2}\Big{(}2\text{I\\!N}-{4k\over\tau}\Big{)}^{c}\bigcap\Big{[}\bigcap_{k=1}^{m/2}\Big{(}{2\over\tau}\text{I\\!N}-4k+2\Big{)}^{c}\Big{]}&\text{
if $m$ is even}\\\\[17.07182pt]
\displaystyle\bigcap_{k=0}^{(m-1)/2}\Big{(}2\text{I\\!N}-{4k\over\tau}\Big{)}^{c}\bigcap\Big{[}\bigcap_{k=1}^{(m-1)/2}\Big{(}{2\over\tau}\text{I\\!N}-4k+2\Big{)}^{c}\Big{]}&\text{
if $m$ is odd}\end{cases},$ (2.3) $\text{and \ for $\ i\geq
2$}\qquad\alpha_{i}=\begin{cases}\alpha_{1}+2(i-1)+\dfrac{2(i-1)}{\tau}&\text{
if $i$ is odd}\\\\[8.5359pt] \alpha_{1}\tau+2(i-1)\tau+2(i-1)&\text{ if $i$ is
even}\end{cases}.$ (2.4)

Note that

$\alpha_{i}=\left(\alpha_{1}+2i-2+{2i-2\over\tau}\right)\tau^{\nu(i)},\qquad\text{for
$i\geq 2$}$ (2.5)

and $\alpha_{i}>0$ for all $i=1,\dots,m$. Furthermore, several identities and
properties of $\alpha_{i}$’s, $\beta_{i}$’s, $d_{i}$’s and $\epsilon$ will be
proven in order to have a good approximation. From de definition (2.4), it is
readily checked that $\\{\alpha_{2k+1}\\}_{k}$ and $\\{\alpha_{2k}\\}_{k}$ are
increasing in $k$. Since $\alpha_{1}>2$ and
$\alpha_{2}=\alpha_{1}\tau+2\tau+2>2$, it follows that $\alpha_{i}>2$ for all
$i=1,\dots,m$. Notice that all these sets are countable :
$2\text{I\\!N}-{4k\over\tau}$ for $k=0,\dots,\frac{m-2}{2}$ and
${2\over\tau}\text{I\\!N}-4k+2$ for $k=1,\dots,\frac{m}{2}$ with $m$ even;
$2\text{I\\!N}-{4k\over\tau}$ for $k=0,\dots,\frac{m-1}{2}$ and
${2\over\tau}\text{I\\!N}-4k+2$ for $k=1,\dots,\frac{m-1}{2}$ with $m$ odd.
Hence, its union is also countable set and the complement of its union is
dense in IR. Therefore, there exist $\alpha_{1}\in\text{I\\!R}$ satisfying
(2.3).

######  Lemma 2.2.

If $\alpha_{i}$’s are given by (2.4) then

$\alpha_{j+1}=(\alpha_{j}+2)\tau^{(-1)^{j+1}}+2=\begin{cases}(\alpha_{j}+2)\tau+2&\text{
if $j$ is odd}\\\\[8.5359pt] \dfrac{\alpha_{j}+2}{\tau}+2&\text{ if $j$ is
even}\end{cases}$ (2.6)

and

$\sum_{i=1}^{j}{(-1)^{i+1}\over\tau^{\nu(i)}}\alpha_{i}=\begin{cases}-j-{j\over\tau}&\text{if
$j$ is even}\\\ \alpha_{1}+j-1+{j-1\over\tau}&\text{if $j$ is odd}\end{cases}$
(2.7)

holds for all $j\geq 1$. If $\alpha_{1}$ satisfies (2.3) then
$\alpha_{i}\notin 2\text{I\\!N}$ for all $i=1,\dots,m$.

###### Proof:.

Assume first that $i$ is odd. Then, $i+1$ is even, from (2.4) for $i$ odd,
$\alpha_{i}=\alpha_{1}+2i-2+\frac{2i-2}{\tau}$ and direct computations lead us
to obtain that

$(\alpha_{i}+2)\tau+2=\left(\alpha_{1}+2i+{2i-2\over\tau}\right)\tau+2=(\alpha_{1}+2i)\tau+2i.$

On the other hand, from (2.4) for $i+1$ even, we find that
$\alpha_{i+1}=\alpha_{1}\tau+2i\tau+2i$, so that
$\alpha_{i+1}=(\alpha_{i}+2)\tau+2$. Direct computations in case $i$ is even
allows us to conclude (2.6).

From the choice of $\alpha_{i}$’s and (2.5), it follows that

$\begin{split}\sum_{i=1}^{j}{(-1)^{i+1}\over\tau^{\nu(i)}}\alpha_{i}&=\sum_{i=1}^{j}(-1)^{i+1}\bigg{[}\alpha_{1}+2i-2+{2i-2\over\tau}\bigg{]}\\\
&=\alpha_{1}\sum_{i=1}^{j}(-1)^{i+1}+2\sum_{i=1}^{j}(-1)^{i+1}(i-1)+{2\over\tau}\sum_{i=1}^{j}(-1)^{i+1}(i-1)\end{split}$

and we conclude (2.7), in view of

$\sum_{i=1}^{j}(-1)^{i+1}(i-1)=\sum_{i=1}^{j}(-1)^{i+1}i+\sum_{i=1}^{j}(-1)^{i+1}=\begin{cases}-{j\over
2}+0&\text{if $j$ is even}\\\ {j+1\over 2}-1&\text{if $j$ is odd}\end{cases}.$

Finally, assume that $m$ is even. If $\alpha_{1}$ satisfies (2.3) then we have
that $\alpha_{1}\in\big{(}2\text{I\\!N}-{4k\over\tau}\big{)}^{c}$,
$k=0,1,\dots,{m-2\over 2}$ and
$\alpha_{1}\in\big{(}{2\over\tau}\text{I\\!N}-4k+2\big{)}^{c}$,
$k=1,2,\dots,{m\over 2}$. That is to say, $\alpha_{1}+{4k\over\tau}\notin
2\text{I\\!N}$, $k=0,1\dots,{m-2\over 2}$ and $\tau(\alpha_{1}+4k-2)\notin
2\text{I\\!N}$, $k=1,\dots,{m\over 2}$. Therefore,
$\alpha_{2k+1}=\alpha_{1}+4k+{4k\over\tau}\notin 2\text{I\\!N}$,
$k=0,\dots,{m-2\over 2}$ and $\alpha_{2k}=\alpha_{1}\tau+(4k-2)\tau+4k-2\notin
2\text{I\\!N}$, $k=1,\dots,{m\over 2}$. Similar argument lead us to conclude
that $\alpha_{i}\notin 2\text{I\\!N}$ for all $i=1,\dots,m$ if $m$ is odd. ∎

In particular, we have that $\alpha_{1}>2$, $\alpha_{2}=(\alpha_{1}+2)\tau+2$,
$\alpha_{3}={\alpha_{2}+2\over\tau}+2$, $\alpha_{4}=(\alpha_{3}+2)\tau+2$,
$\alpha_{5}={\alpha_{4}+2\over\tau}+2,\ \dots$ Furthermore, we have that

$4\pi\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\alpha_{i}-2\pi(\alpha_{1}-2)=(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2).$
(2.8)

Now, we define $\beta_{i}$ as follows, if $m$ is even then

$\beta_{l}=\tau^{\nu(l)}\bigg{[}m-l+{m-l+1\over\tau}\bigg{]}=\begin{cases}(m-l)\tau+m-l+1,&\text{if
$l$ is even}\\\\[8.5359pt] \displaystyle m-l+{m-l+1\over\tau}&\text{if $l$ is
odd}\end{cases},$ (2.9)

for $l=1,2,\dots,m$. If $m$ is odd then

$\beta_{l}=\tau^{\nu(l)}\left[m-l+1+{m-l\over\tau}\right]=\begin{cases}(m-l+1)\tau+m-l,&\text{if
$l$ is even}\\\\[8.5359pt] \displaystyle m-l+1+{m-l\over\tau}&\text{if $l$ is
odd}\end{cases}$ (2.10)

for $l=1,2,\dots,m$. In any case, either $m$ is odd or even, it is readily
checked that $\beta_{m}=1$ and $\beta_{i}>0$ for all $i=1,\dots,m$. Note that
we can rewrite

$\beta_{l}=\tau^{\nu(l)}\left[{m-l+1\over\tau^{\nu(m)}}+{m-l\over\tau^{\nu(m+1)}}\right].$
(2.11)

Hence, we shall prove the following useful properties from the definition
(2.9) and (2.10). Note that $1-\sigma(i)=\nu(i)=\sigma(i+1)$ and
$\sigma(i)=1-\nu(i)=\nu(i+1)$.

######  Lemma 2.3.

If $\beta_{i}$’s are given by either (2.9) or (2.10) then for any
$l=2,\dots,m$

$\beta_{l}=\begin{cases}\tau\beta_{l-1}-\tau-1,&\text{ if $l$ is
even}\\\\[8.5359pt] \displaystyle{\beta_{l-1}\over\tau}-1-{1\over\tau}&\text{
if $l$ is odd}\end{cases}$ (2.12)

and for any $l=1,\dots,m-1$

$\frac{\beta_{l}-1}{2\tau^{\nu(l)}}=\sum_{j=l+1}^{m}{(-1)^{j+1}\over\tau^{\nu(j)}}\beta_{j}.$
(2.13)

Furthermore, it holds that
$\frac{\beta_{l}}{\alpha_{l}}<\frac{\beta_{l-1}}{\alpha_{l-1}}$ for any
$l=2,\dots,m$, so that, ${\delta_{i}\over\delta_{j}}\to 0$ as $\rho\to 0$ for
any $i<j.$

###### Proof:.

First, assume that $m$ is even. So, $\beta_{l}$’s are given by (2.9). If $l$
is even, then we have that $l-1$ is odd, so that,

$\beta_{l-1}\displaystyle=m-l+1+{m-l+2\over\tau}\qquad\text{ and}$
$\tau\beta_{l-1}-\tau-1=(m-l)\tau+\tau+m-l+2-\tau-1=\beta_{l}.$

Next, assume that $l$ is odd (still $m$ is even). Similarly as above, it
follows that ${\beta_{l-1}\over\tau}-1-{1\over\tau}=\beta_{l}$. Arguing in the
same way for $\beta_{l}$’s given by (2.10) when $m$ is odd, we conclude
(2.12).

Now, for any $m$, we shall prove that

$\beta_{l}=\begin{cases}\displaystyle
1+2\sum_{i=1}^{m-l}(-1)^{i+1}\tau^{\sigma(i)}\beta_{l+i},&\text{if $l$ is
even}\\\\[11.38092pt] \displaystyle
1+2\sum_{i=1}^{m-l}{(-1)^{i+1}\over\tau^{\sigma(i)}}\beta_{l+i}&\text{if $l$
is odd}\end{cases}.$ (2.14)

Assume that $m$ is even. If $l$ is even then $\nu(i+l)=1-\sigma(i)$ and from
(2.11) we get that

$\begin{split}\sum_{i=1}^{m-l}(-1)^{i+1}\tau^{\sigma(i)}\beta_{l+i}&=\sum_{i=1}^{m-l}(-1)^{i+1}\tau\left[{\beta_{l}\over\tau}-\Big{(}1+{1\over\tau}\Big{)}i\right]=\beta_{l}\sum_{i=1}^{m-l}(-1)^{i+1}-(\tau+1)\sum_{i=1}^{m-l}(-1)^{i+1}i\end{split}$

and (2.14) follows, in view of $m-l$ is even, $\sum_{i=1}^{m-l}(-1)^{i+1}=0$
and $\sum_{i=1}^{m-l}(-1)^{i+1}i=-{m-l\over 2}.$ Next, assume that $l$ is odd
(still $m$ is even) so that $\nu(i+l)=\sigma(i)$ and similarly as above

$\begin{split}\sum_{i=1}^{m-l}{(-1)^{i+1}\over\tau^{\sigma(i)}}\beta_{l+i}&=\sum_{i=1}^{m-l}(-1)^{i+1}\left[\beta_{l}-\Big{(}1+{1\over\tau}\Big{)}i\right]=\beta_{l}\sum_{i=1}^{m-l}(-1)^{i+1}-\Big{(}1+{1\over\tau}\Big{)}\sum_{i=1}^{m-l}(-1)^{i+1}i=\frac{\beta_{l}-1}{2},\end{split}$

in view of $m-l$ is odd, $\sum_{i=1}^{m-l}(-1)^{i+1}=1$ and
$\sum_{i=1}^{m-l}(-1)^{i+1}i={m-l+1\over 2}.$ Arguing in the same way for
$\beta_{l}$’s given by (2.10) when $m$ is odd we conclude (2.14). Now, from
(2.14) taking $j=l+i$ we have that if $l$ is even then

$\beta_{l}=1+2\sum_{i=1}^{m-l}(-1)^{i+1}\tau^{\sigma(i)}\beta_{l+i}=1+2\sum_{j=l+1}^{m}(-1)^{j+1}{\tau\over\tau^{\nu(j)}}\beta_{i}$

in view of $\sigma(j-l)=\sigma(j)=1-\nu(j)$, and if $l$ is odd then

$\beta_{l}=1+2\sum_{i=1}^{m-l}\frac{(-1)^{i+1}}{\tau^{\sigma(i)}}\beta_{l+i}=1+2\sum_{j=l+1}^{m}{(-1)^{j}\over\tau^{\nu(j)}}\beta_{i},$

in view of $\sigma(j-l)=\nu(j)$. Thus, we deduce (2.13).

Now, we know that
$0<\alpha_{l-1}+\alpha_{l-1}\tau+2\beta_{l-1}+2\beta_{l-1}\tau$ for any $l$.
Hence, for $l$ even we have that $\beta_{l}=\tau\beta_{l-1}-\tau-1$ and
$\alpha_{l}=(\alpha_{l-1}+2)\tau+2$ by using (2.12) and (2.6) respectively, so
that

$\begin{split}\alpha_{l-1}[\tau\beta_{l-1}-\tau-1]&<[(\alpha_{l-1}+2)\tau+2]\beta_{l-1}\iff\alpha_{l-1}\beta_{l}<\alpha_{l}\beta_{l-1}\end{split}$

By using (2.6) and (2.12), for $l$ odd we have that
$\beta_{l}=\frac{\beta_{l-1}}{\tau}-\frac{1}{\tau}-1$ and
$\alpha_{l}=\frac{\alpha_{l-1}+2}{\tau}+2$, and it is readily checked that
$\alpha_{l-1}\beta_{l}<\alpha_{l}\beta_{l-1}$. Thus, we deduce that
${\delta_{i}\over\delta_{j}}\to 0$ as $\rho\to 0$ for any $i<j$, in view of

${\delta_{i}\over\delta_{j}}={d_{i}^{1/\alpha_{i}}\rho^{\beta_{i}/\alpha_{i}}\over
d_{j}^{1/\alpha_{j}}\rho^{\beta_{j}/\alpha_{j}}}={d_{i}^{1/\alpha_{i}}\over
d_{j}^{1/\alpha_{j}}}\rho^{{\beta_{i}\over\alpha_{i}}-{\beta_{j}\over\alpha_{j}}}\qquad\text{and}\qquad{\beta_{j}\over\alpha_{j}}<{\beta_{i}\over\alpha_{i}}.$

This finishes the proof. ∎

Now, we define $d_{i}$’s by

$\log d_{m}=a_{m}\quad\text{and}\quad\log
d_{l}=a_{l}+2\tau^{\nu(l)}\sum_{i=l+1}^{m}{a_{i}\over\tau^{\nu(i)}}=\begin{cases}\displaystyle
a_{l}+2\sum_{i=l+1}^{m}a_{i}\tau^{\sigma(i)},&\text{if $l$ is
even}\\\\[11.38092pt] \displaystyle
a_{l}+2\sum_{i=l+1}^{m}{a_{i}\over\tau^{\nu(i)}}&\text{if $l$ is
odd}\end{cases}$ (2.15)

for $l=1,2,\dots,m-1$, where

$a_{l}=\log\Big{(}{\tau^{\nu(l)}V_{\nu(l)}(0)\over
2\alpha_{l}^{2}}\Big{)}+{(-1)^{l}\over\tau^{\sigma(l)}}2\pi(\alpha_{m}+2)h_{0}=\begin{cases}\displaystyle\log\Big{(}{\tau
V_{1}(0)\over 2\alpha_{l}^{2}}\Big{)}+2\pi(\alpha_{m}+2)h_{0},&\text{if $l$ is
even}\\\\[8.5359pt] \displaystyle\log\Big{(}{V_{0}(0)\over
2\alpha_{l}^{2}}\Big{)}-{2\pi\over\tau}(\alpha_{m}+2)h_{0}&\text{if $l$ is
odd}\end{cases}$ (2.16)

when $m$ even, while when $m$ is odd

$\begin{split}a_{l}&=\log\Big{(}{\tau^{\nu(l)}V_{\nu(l)}(0)\over
2\alpha_{l}^{2}}\Big{)}+(-1)^{l+1}\tau^{\nu(l)}2\pi(\alpha_{m}+2)h_{0}\\\
&=\begin{cases}\displaystyle\log\Big{(}{\tau V_{1}(0)\over
2\alpha_{l}^{2}}\Big{)}-2\pi(\alpha_{m}+2)\tau h_{0},&\text{if $l$ is
even}\\\\[8.5359pt] \displaystyle\log\Big{(}{V_{0}(0)\over
2\alpha_{l}^{2}}\Big{)}+2\pi(\alpha_{m}+2)h_{0}&\text{if $l$ is
odd}\end{cases}.\end{split}$ (2.17)

######  Lemma 2.4.

If $d_{i}$’s are given by (2.15) then for any $l=1,\dots,m-1$

$\log d_{l}=\begin{cases}\displaystyle
a_{l}+2\sum_{i=1}^{m-l}(-1)^{i+1}\tau^{\sigma(i)}\log d_{l+i},&\text{if $l$ is
even}\\\\[14.22636pt] \displaystyle
a_{l}+2\sum_{i=1}^{m-l}{(-1)^{i+1}\over\tau^{\sigma(i)}}\log d_{l+i}&\text{if
$l$ is odd}\end{cases}$ (2.18)

and consequently,

$\frac{\log
d_{l}-a_{l}}{2\tau^{\nu(l)}}=\sum_{j=l+1}^{m}{(-1)^{j+1}\over\tau^{\nu(j)}}\log
d_{j}.$ (2.19)

###### Proof:.

First, assume that $m$ is even. So, $a_{l}$’s are given by (2.16). If $l$ is
even, then we have that $\nu(i+l)=1-\sigma(i)$,
$\frac{\tau}{\tau^{\nu(j)}}=\tau^{\sigma(j)}$ and $m-l$ is even, so that,
$\sigma(m-l)=0$ and

$\begin{split}&\sum_{i=1}^{m-l}(-1)^{i+1}\tau^{\sigma(i)}\log
d_{l+i}=\sum_{i=1}^{m-l-1}(-1)^{i+1}\tau^{\sigma(i)}\bigg{[}a_{l+i}+2\tau^{\nu(l+i)}\sum_{j=l+i+1}^{m}{a_{j}\over\tau^{\nu(j)}}\bigg{]}-a_{m}\\\
&=\sum_{i=1}^{m-l-1}(-1)^{i+1}\tau^{\sigma(i)}a_{l+i}+2\tau\sum_{i=1}^{m-l-1}\sum_{j=l+i+1}^{m}(-1)^{i+1}{a_{j}\over\tau^{\nu(j)}}-a_{m}\\\
&=\sum_{j=l+1}^{m-1}(-1)^{j+1}\tau^{\sigma(j)}a_{j}+\sum_{j=l+2}^{m}[1+(-1)^{j}]a_{j}\tau^{\sigma(j)}-a_{m}=\sum_{j=l+1}^{m}a_{j}\tau^{\sigma(j)}\end{split}$

and (2.18) follows. Similarly as above, if $l$ is odd, then we have that
$\nu(i+l)=\sigma(i)$, $\frac{\tau}{\tau^{\nu(j)}}=\tau^{\sigma(j)}$ and
$m-l+1$ is even, so that, $\sigma(m-l+1)=0$ and

$\begin{split}&\sum_{i=1}^{m-l}{(-1)^{i+1}\over\tau^{\sigma(i)}}\log
d_{l+i}=\sum_{i=1}^{m-l-1}{(-1)^{i+1}\over\tau^{\sigma(i)}}a_{l+i}+2\sum_{i=1}^{m-l-1}\sum_{j=l+i+1}^{m}(-1)^{i+1}{a_{j}\over\tau^{\nu(j)}}+{a_{m}\over\tau}\\\
&=\sum_{j=l+1}^{m-1}{(-1)^{j}\over\tau^{\nu(j)}}a_{j}+\sum_{j=l+2}^{m}[1+(-1)^{j+1}]{a_{j}\over\tau^{\nu(j)}}+{a_{m}\over\tau}=\sum_{j=l+1}^{m}{a_{j}\over\tau^{\nu(j)}}\end{split}$

and we conclude (2.18). Similar arguments work out in case $m$ is odd. We
deduce (2.19) by using the change $j=l+i$. This conclude the proof. ∎

Now, $\epsilon=\epsilon(\rho)$ is chosen so that

$\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\gamma_{i}=2\pi(\alpha_{1}-2),\qquad\text{where
}\ \ \gamma_{j}=\gamma_{\delta_{j},\epsilon}^{\alpha_{j}},\ \ j=1,\dots,m$
(2.20)

and $\gamma_{\delta,\epsilon}^{\alpha}$ is given in Lemma 2.1. Precisely,
$\epsilon=\epsilon(\rho)$ is given by

$\epsilon^{\alpha_{1}-2}=\begin{cases}\displaystyle{[V_{0}(0)]^{m}[\tau
V_{1}(0)]^{m\over\tau}\over\prod_{i=1}^{m}\alpha_{i}^{4/\tau^{\nu(i)}}}e^{{2\pi\over\tau}(\alpha_{m}+2)h_{0}}\Big{(}{\rho\over
2}\Big{)}^{m+{m\over\tau}}&\text{if $m$ is even}\\\\[17.07182pt]
\displaystyle{[V_{0}(0)]^{m+1}[\tau
V_{1}(0)]^{m-1\over\tau}\over\prod_{i=1}^{m}\alpha_{i}^{4/\tau^{\nu(i)}}}e^{2\pi(\alpha_{m}+2)h_{0}}\Big{(}{\rho\over
2}\Big{)}^{m+1+{m-1\over\tau}}&\text{if $m$ is odd}\end{cases}$ (2.21)

It is readily checked that $\epsilon(\rho)\to 0^{+}$ as $\rho\to 0^{+}$.
Moreover, $\epsilon^{\alpha_{1}-2}\sim\rho^{\beta_{1}+1}$, in view of

$\beta_{1}=\begin{cases}m-1+{m\over\tau}&\text{if $m$ is even}\\\
m+{m-1\over\tau}&\text{if $m$ is odd}\end{cases}$ (2.22)

######  Lemma 2.5.

If $\epsilon$ is given by (2.21) then it holds (2.20) and
$\frac{\epsilon}{\delta_{1}}\to 0$ as $\rho\to 0$.

###### Proof:.

First, assume that $m$ is even so that, from (2.21) it follows that

$\begin{split}(\alpha_{1}-2)\log\epsilon=m\log V_{0}(0)+{m\over\tau}\log[\tau
V_{1}(0)]+{2\pi\over\tau}(\alpha_{m}+2)h_{0}-4\sum_{i=1}^{m}{\log\alpha_{i}\over\tau^{\nu(i)}}+\Big{(}m+{m\over\tau}\Big{)}\log\Big{(}{\rho\over
2}\Big{)}.\end{split}$

On the other hand, from the definition of $\delta_{i}$ and $\gamma_{i}$ it
follows that

$\Big{[}-{1\over 2\pi}\log\epsilon+h_{0}\Big{]}\gamma_{i}=-2\log
d_{i}-2\beta_{i}\log\rho+4\pi\alpha_{i}h_{0}$

so that,

$\Big{[}-{1\over
2\pi}\log\epsilon+h_{0}\Big{]}\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\gamma_{i}=-2\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\log
d_{i}-2\log\rho\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\beta_{i}+4\pi
h_{0}\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\alpha_{i}.$

Hence, we compute the sums involved as follows. From Lemma 2.4 we have that
for any $m$, either odd or even,

$\log d_{1}=a_{1}+2\sum_{j=2}^{m}{(-1)^{j}\over\tau^{\nu(j)}}\log d_{j}.$

Then by using (2.15) and $m=2k$ for some $k\in\text{I\\!N}$ we get that

$\begin{split}&\sum_{j=1}^{m}{(-1)^{j}\over\tau^{\nu(j)}}\log
d_{j}={a_{1}+\log d_{1}\over
2}=\sum_{i=1}^{m}{a_{i}\over\tau^{\nu(i)}}=\sum_{i=1}^{k}{1\over\tau}\left[\log[\tau
V_{1}(0)]+2\pi(\alpha_{m}+2)h_{0}-\log(2\alpha_{2i}^{2})\right]\\\
&\quad+\sum_{i=1}^{k}\left[\log
V_{0}(0)-{2\pi\over\tau}(\alpha_{m}+2)h_{0}-\log(2\alpha_{2i-1}^{2})\right]\\\
&={m\over 2}\log V_{0}(0)+{m\over 2\tau}\log[\tau
V_{1}(0)]-\Big{(}m+{m\over\tau}\Big{)}{\log 2\over
2}-2\sum_{i=1}^{m}{\log\alpha_{i}\over\tau^{\nu(i)}}.\end{split}$

Also, from the choice of $\beta_{i}$’s in (2.9), it is readily checked that

$\begin{split}\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\beta_{i}&=\sum_{i=1}^{m}(-1)^{i+1}\left[m+{m+1\over\tau}-\Big{(}1+{1\over\tau}\Big{)}i\right]={m\over
2}+{m\over 2\tau}.\end{split}$

Therefore, by using (2.7) we obtain that

$\begin{split}&\Big{[}-{1\over
2\pi}\log\epsilon+h_{0}\Big{]}\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\gamma_{i}=-m\log
V_{0}(0)-{m\over\tau}\log[\tau V_{1}(0)]+\Big{(}m+{m\over\tau}\Big{)}\log
2+4\sum_{i=1}^{m}{\log\alpha_{i}\over\tau^{\nu(i)}}\\\
&\quad-\Big{(}m+{m\over\tau}\Big{)}\log\rho+4\pi
h_{0}\Big{(}-m-{m\over\tau}\Big{)}=-(\alpha_{1}-2)\log\epsilon+{2\pi\over\tau}(\alpha_{m}+2)h_{0}-4\pi
h_{0}\Big{(}m+{m\over\tau}\Big{)}\\\
&=-(\alpha_{1}-2)\log\epsilon+2\pi(\alpha_{1}-2)h_{0},\end{split}$

in view of ${\alpha_{m}+2\over\tau}-2m-{2m\over\tau}=\alpha_{1}-2$ and (2.20)
follows. Similarly, (2.20) is proven if $m$ is odd.

Finally, taking into account (2.22), (2.21) and
$\delta_{1}^{\alpha_{1}}=d_{1}\rho^{\beta_{1}}$, we deduce that
${\epsilon\over\delta_{1}}\to
0\iff{\beta_{1}+1\over\alpha_{1}-2}>{\beta_{1}\over\alpha_{1}},$ in view of
$\epsilon^{\alpha_{1}-2}\sim\rho^{\beta_{1}+1}$. Indeed, it is readily checked
that
${\beta_{1}+1\over\alpha_{1}-2}>{\beta_{1}+1\over\alpha_{1}}>{\beta_{1}\over\alpha_{1}}.$
This completes the proof. ∎

Let us stress that the behavior of $\epsilon=\epsilon(\rho)$ and
$\delta_{j}=\delta_{j}(\rho)$’s with respect to $\rho$ is given by

${\epsilon\over\delta_{j}},\ \ {\delta_{i}\over\delta_{j}}\to 0\quad\text{as
}\ \ \rho\to 0\ \ \text{for }\ i<j\ \ \text{and }\ \ \epsilon\to 0\ \ \text{as
}\ \ \rho\to 0.$

Assume that $\delta_{i}$’s are given by (2.2) with $\alpha_{i}$’s,
$\beta_{i}$’s and $d_{i}$’s defined in (2.4), (2.9)-(2.10) and (2.15),
respectively, and $\epsilon$ is given by (2.21). Notice that
$\frac{\log\delta_{j}}{\log\epsilon}=O(1)$ as $\rho\to 0$, for all
$j=1,\dots,m$. Now, define the _shrinking annuli_

$A_{i}=\\{x\in\Omega\mid\sqrt{\delta_{i-1}\delta_{i}}<|x|\leq\sqrt{\delta_{i}\delta_{i+1}}\\},\qquad
j=1,\dots,m,$

where for simplicity we denote $\delta_{0}=\frac{\epsilon^{2}}{\delta_{1}}$
and $\delta_{m+1}=\frac{M_{0}^{2}}{\delta_{m}}$ with
$M_{0}=\sup\\{|x|:x\in\Omega\\}$, so that,
$\Omega_{\epsilon}=\cup_{j=1}^{m}A_{j}$, $\cap_{j=1}^{m}A_{j}=\emptyset$ and
$\frac{A_{j}}{\delta_{j}}$ approaches to $\text{I\\!R}^{2}$ as $\rho\to 0$ for
each $j=1,\dots,m$.

######  Lemma 2.6.

There exist $\eta>0$ such that the following expansions hold

$\begin{split}U(\delta_{j}y)=&\,{(-1)^{j+1}\over\tau^{\nu(j)}}\left[-2\log\delta_{j}-a_{j}-\log\rho+\log{|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\right]\\\
&\,+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{j}y,0)+O\left(\rho^{\eta}\right)\end{split}$
(2.23)

for any $j=1,\dots,m$, uniformly for $\delta_{j}y\in A_{j}$, $j=1,\dots,m$.

###### Proof:.

From the expansions of $P_{\epsilon}w_{j}$, $j=1,\dots,m$ and the definition
of $\gamma_{j}$ in Lemma 2.1, (2.8) and (2.20) we obtain that

$\begin{split}U(x)=&\,\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\left[w_{i}-\log(2\alpha_{i}^{2}\delta_{i}^{\alpha_{i}})\right]+{1\over
2\pi}\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\gamma_{i}\log|x|\\\
&\,+\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\left[4\pi\alpha_{i}-\gamma_{i}\right]H(x,0)+O\left(\sum_{j=1}^{m}\left[\delta_{j}^{\alpha_{j}}+\Big{(}{\epsilon\over\delta_{j}}\Big{)}^{\alpha_{j}}\right]+\epsilon\right)\\\
=&\,\sum_{i=1}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\log{1\over(\delta_{i}^{\alpha_{i}}+|x|^{\alpha_{i}})^{2}}+(\alpha_{1}-2)\log|x|+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(x,0)\\\
&\,+O\left(\rho^{\eta}\right),\end{split}$ (2.24)

where we choose

$0<\eta\leq\min\bigg{\\{}1,{\beta_{1}+1\over\alpha_{1}-2},\min\Big{\\{}\alpha_{j}\Big{(}{\beta_{1}+1\over\alpha_{1}-2}-{\beta_{j}\over\alpha_{j}}\Big{)}\
:\ j=1,\dots,m\Big{\\}}\bigg{\\}},$ (2.25)

in view $\beta_{j}$ are decreasing so that $\delta_{j}^{\alpha_{j}}=O(\rho)$
for all $j=1,\dots,m$,
$\big{(}{\epsilon\over\delta_{j}}\big{)}^{\alpha_{j}}=O\big{(}\rho^{\alpha_{j}({\beta_{1}+1\over\alpha_{1}-2}-{\beta_{j}\over\alpha_{j}})}\big{)}$
and $\epsilon=O\big{(}\rho^{\beta_{1}+1\over\alpha_{1}-2}\big{)}$. Now, from
the choice of $\delta_{i}$’s in (2.2), (2.13) and (2.19) we have that for $j$
odd

$\begin{split}2\sum_{i=j+1}^{m}{(-1)^{i}\over\tau^{(\nu(i)}}\alpha_{i}\log\delta_{i}&=2\sum_{i=j+1}^{m}{(-1)^{i}\over\tau^{(\nu(i)}}\log
d_{i}+2\log\rho\sum_{i=j+1}^{m}{(-1)^{i}\over\tau^{(\nu(i)}}\beta_{i}\\\
&=2\frac{\log
d_{j}-a_{j}}{2}+2\frac{\beta_{j}-1}{2}\log\rho=\alpha_{j}\log\delta_{j}-a_{j}-\log\rho\end{split}$

and similarly, for $j$ even
$2\sum_{i=j+1}^{m}{(-1)^{i}\over\tau^{(\nu(i)}}\alpha_{i}\log\delta_{i}=-{1\over\tau}\big{[}\alpha_{j}\log\delta_{j}-a_{j}-\log\rho\big{]}.$
Thus, we rewrite as

$2\sum_{i=j+1}^{m}{(-1)^{i}\over\tau^{(\nu(i)}}\alpha_{i}\log\delta_{i}={(-1)^{j+1}\over\tau^{\nu(j)}}\big{[}\alpha_{j}\log\delta_{j}-a_{j}-\log\rho\big{]}.$
(2.26)

On the other hand, for any $y\in\frac{A_{1}}{\delta_{1}}$ it holds that

$\begin{split}U(\delta_{1}y)=&\,-2\alpha_{1}\log\delta_{1}+\log{1\over(1+|y|^{\alpha_{1}})^{2}}+(\alpha_{1}-2)\log(\delta_{1}|y|)\\\
&\,+\sum_{i=2}^{m}{(-1)^{i+1}\over\tau^{\nu(i)}}\log\frac{1}{(\delta_{i}^{\alpha_{i}}+\delta_{1}^{\alpha_{i}}|y|^{\alpha_{i}})^{2}}+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{1}y,0)+O\left(\rho^{\eta}\right)\\\
=&\,-(\alpha_{1}+2)\log\delta_{1}+\sum_{i=2}^{m}(-1)^{i}{2\alpha_{i}\over\tau^{\nu(i)}}\log\delta_{i}+\log{|y|^{\alpha_{1}-2}\over(1+|y|^{\alpha_{1}})^{2}}\\\
&\,+\sum_{i=2}^{m}{(-1)^{i}\over\tau^{\nu(i)}}2\log\left(1+\Big{(}{\delta_{1}|y|\over\delta_{i}}\Big{)}^{\alpha_{i}}\right)+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{1}y,0)+O\left(\rho^{\eta}\right)\\\
=&\,-2\log\delta_{1}-a_{1}-\log\rho+\log{|y|^{\alpha_{1}-2}\over(1+|y|^{\alpha_{1}})^{2}}+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{1}y,0)\\\
&\,+O\left(\sum_{j=2}^{m}\Big{(}{\delta_{1}\over\delta_{j}}\Big{)}^{\alpha_{j}\over
2}+\rho^{\eta}\right),\end{split}$ (2.27)

in view of (2.26) and

$\log\left(1+\Big{(}{\delta_{1}|y|\over\delta_{i}}\Big{)}^{\alpha_{i}}\right)=O\left(\Big{(}{\delta_{1}|y|\over\delta_{i}}\Big{)}^{\alpha_{i}}\right)=O\left(\Big{(}{\delta_{1}\over\delta_{i}}\Big{)}^{\alpha_{i}\over
2}\right),\quad i=2,\dots,m.$

Choosing $\eta>0$ satisfying (2.25) and

$0<\eta\leq{1\over
2}\min\Big{\\{}\alpha_{i}\Big{(}{\beta_{1}\over\alpha_{1}}-{\beta_{i}\over\alpha_{i}}\Big{)}\
: \ 1<i\Big{\\}}$ (2.28)

we get that $\big{(}{\delta_{1}\over\delta_{i}}\big{)}^{\alpha_{i}\over
2}=O\left(\rho^{\eta}\right)$, $i=2,\dots,m.$ Similarly, for
$y\in\frac{A_{j}}{\delta_{j}}$, $1<j<m$ we find that

$\begin{split}&U(\delta_{j}y)=\sum_{i=1}^{j-1}\frac{(-1)^{i+1}}{\tau^{\nu(i)}}\bigg{[}-2\alpha_{i}\log(\delta_{j}|y|)-2\log\left(1+\Big{(}{\delta_{i}\over\delta_{j}|y|}\Big{)}^{\alpha_{i}}\right)\bigg{]}+(\alpha_{1}-2)\log(\delta_{j}|y|)\\\
&\,+{(-1)^{j+1}\over\tau^{\nu(j)}}\bigg{[}-2\alpha_{j}\log\delta_{j}+\log{1\over(1+|y|^{\alpha_{j}})^{2}}\bigg{]}+\sum_{i=j+1}^{m}\frac{(-1)^{i}}{\tau^{\nu(i)}}2\alpha_{i}\log\delta_{i}\\\
&\,+\sum_{i=j+1}^{m}\frac{(-1)^{i}}{\tau^{\nu(i)}}2\log\bigg{(}1+\Big{(}{\delta_{j}|y|\over\delta_{i}}\Big{)}^{\alpha_{i}}\bigg{)}+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{j}y,0)+O\left(\rho^{\eta}\right)\\\
&=-2\log\delta_{j}\sum_{i=1}^{j}\frac{(-1)^{i+1}}{\tau^{\nu(i)}}\alpha_{i}+2\sum_{i=j+1}^{m}{(-1)^{i}\over\tau^{\nu(i)}}\alpha_{i}\log\delta_{i}+(\alpha_{1}-2)\log\delta_{j}+(\alpha_{1}-2)\log|y|\\\
&\,-2\log|y|\sum_{i=1}^{j-1}\frac{(-1)^{i+1}}{\tau^{\nu(i)}}\alpha_{i}+{(-1)^{j+1}\over\tau^{\nu(j)}}\log{1\over(1+|y|^{\alpha_{j}})^{2}}\\\
&\,+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{j}y,0)+O\bigg{(}\sum_{i=1}^{j-1}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}\over
2}+\sum_{i=j+1}^{m}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}\over
2}+\rho^{\eta}\bigg{)},\end{split}$ (2.29)

in view of

$\log\left(1+\Big{(}{\delta_{i}\over\delta_{j}|y|}\Big{)}^{\alpha_{i}}\right)=O\left(\Big{(}{\delta_{i}\over\delta_{j}|y|}\Big{)}^{\alpha_{i}}\right)=O\left(\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}\over
2}\right),\quad i<j,$
$\log\left(1+\Big{(}{\delta_{j}|y|\over\delta_{i}}\Big{)}^{\alpha_{i}}\right)=O\left(\Big{(}{\delta_{j}|y|\over\delta_{i}}\Big{)}^{\alpha_{i}}\right)=O\left(\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}\over
2}\right),\quad j<i,$

and by using again (2.20). Now, we choose $\eta$ satisfying (2.25), (2.28) and
smaller, if necessary, so that

$0<\eta\leq{1\over
2}\min\Big{\\{}\alpha_{i}\Big{(}{\beta_{i}\over\alpha_{i}}-{\beta_{j}\over\alpha_{j}}\Big{)}\
: \ i<j\Big{\\}},\ \ j=2,\dots,m$

and

$0<\eta\leq{1\over
2}\min\Big{\\{}\alpha_{i}\Big{(}{\beta_{j}\over\alpha_{j}}-{\beta_{i}\over\alpha_{i}}\Big{)}\
: \ j<i\Big{\\}},\ \ j=1,\dots,m-1$

thus, $\big{(}{\delta_{i}\over\delta_{j}}\big{)}^{\alpha_{i}\over
2}=O\left(\rho^{\eta}\right)$, $i<j$ and
$\big{(}{\delta_{j}\over\delta_{i}}\big{)}^{\alpha_{i}\over
2}=O\left(\rho^{\eta}\right)$, $j<i$. Hence, by using (2.7) and (2.26) for $j$
even we get that

$\begin{split}&U(\delta_{j}y)=-2\log\delta_{j}\Big{[}-j-{j\over\tau}\Big{]}-{1\over\tau}\left[\alpha_{j}\log\delta_{j}-a_{j}-\log\rho\right]+(\alpha_{1}-2)\log\delta_{j}+(\alpha_{1}-2)\log|y|\\\
&\,-2\log|y|\left[\alpha_{1}+j-2+\frac{j-2}{\tau}\right]-{1\over\tau}\log{1\over(1+|y|^{\alpha_{j}})^{2}}+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{j}y,0)+O\left(\rho^{\eta}\right)\\\
\end{split}$

and we conclude (2.23). Similarly, for $j$ odd we get that

$\begin{split}&U(\delta_{j}y)=-2\log\delta_{j}\Big{[}\alpha_{1}+j-1+{j-1\over\tau}\Big{]}+\alpha_{j}\log\delta_{j}-a_{j}-\log\rho+(\alpha_{1}-2)\log\delta_{j}+(\alpha_{1}-2)\log|y|\\\
&\,-2\log|y|\left[-j+1-\frac{j-1}{\tau}\right]+\log{1\over(1+|y|^{\alpha_{j}})^{2}}+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{j}y,0)+O(\rho^{\eta})\\\
\end{split}$

and we obtain (2.23). Also, we have for any $y\in{A_{m}\over\delta_{m}}$

$\begin{split}&U(\delta_{m}y)=\sum_{i=1}^{m-1}\frac{(-1)^{i+1}}{\tau^{\nu(i)}}\left[-2\alpha_{i}\log(\delta_{m}|y|)-2\log\bigg{(}1+\Big{(}{\delta_{i}\over\delta_{m}|y|}\Big{)}^{\alpha_{i}}\bigg{)}\right]+(\alpha_{1}-2)\log(\delta_{m}|y|)\\\
&\,+{(-1)^{m+1}\over\tau^{\nu(m)}}\left[-2\alpha_{m}\log\delta_{m}+\log{1\over(1+|y|^{\alpha_{m}})^{2}}\right]+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{j}y,0)+O(\rho^{\eta})\\\
&={(-1)^{m+1}\over\tau^{\nu(m)}}\left[-2\log\delta_{m}-a_{m}-\log\rho+\log{|y|^{\alpha_{m}-2}\over(1+|y|^{\alpha_{m}})^{2}}+2\pi(\alpha_{m}+2)H(\delta_{m}y,0)\right]+O(\rho^{\eta}).\end{split}$
(2.30)

This completes the proof. ∎

##  3\. Problem in Liouville form (1.1)

###  3.1. Error estimate

In order to evaluate how well the approximation $U$ satisfies the equation
(1.1) (and get the invertibility of the linearized operator), we will use the
norms

$\|h\|_{p}=\left(\int_{\Omega_{\epsilon}}|h(x)|^{p}\,dx\right)^{1/p}\qquad\text{and}\qquad\|h\|=\left(\int_{\Omega_{\epsilon}}|\nabla
h(x)|^{2}\,dx\right)^{1/2},$

the usual norms in the Banach spaces $L^{p}(\Omega_{\epsilon})$ and
$H_{0}^{1}(\Omega_{\epsilon})$, respectively. Assume that $\delta_{i}$’s are
given by (2.2) with $\alpha_{i}$’s, $\beta_{i}$’s and $d_{i}$’s defined in
(2.4), (2.9)-(2.10) and (2.15), respectively, and $\epsilon$ is given by
(2.21). Let us evaluate the approximation rate of $U$ in $\|\cdot\|_{p}$,
encoded in (1.12)

######  Lemma 3.1.

There exist $\rho_{0}>0$, a constant $C>0$ and $1<p_{0}<2$, such that for any
$\rho\in(0,\rho_{0})$ and $p\in(1,p_{0})$ it holds

$\|R\|_{p}\leq C\rho^{\eta_{p}}$ (3.1)

for some $\eta_{p}>0$.

###### Proof:.

First, note that $\Delta
U=\sum_{i=1}^{m}{(-1)^{i}\over\tau^{\nu(i)}}|x|^{\alpha_{i}-2}e^{w_{i}}$ so
that, for any $1<j<m$ and $y\in\frac{A_{j}}{\delta_{j}}$ we have that

$\Delta
U(\delta_{j}y)={(-1)^{j}\over\tau^{\nu(j)}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}+O\bigg{(}\sum_{i=1}^{j-1}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}}{1\over\delta_{j}^{2}|y|^{\alpha_{i}+2}}+\sum_{i=j+1}^{m}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{j}^{2}}\bigg{)}.$
(3.2)

Similarly, we obtain that

$\Delta
U(\delta_{1}y)=-{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}+O\bigg{(}\sum_{i=2}^{m}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{j}^{2}}\bigg{)},\qquad
y\in{A_{1}\over\delta_{1}}$ (3.3)

and

$\Delta
U(\delta_{m}y)={(-1)^{m}\over\tau^{\nu(m)}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}+O\bigg{(}\sum_{i=1}^{m-1}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}}{1\over\delta_{j}^{2}|y|^{\alpha_{i}+2}}\bigg{)},\qquad
y\in{A_{m}\over\delta_{m}}$ (3.4)

By using (2.16) and (2.23) (from Lemma 2.6) for $y\in\frac{A_{j}}{\delta_{j}}$
and any $j$ odd we have that

$\begin{split}\rho
V_{0}(\delta_{j}y)e^{U(\delta_{j}y)}&={2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}{V_{0}(\delta_{j}y)\over
V_{0}(0)}\exp\bigg{(}(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)[H(\delta_{j}y,0)-h_{0}]+O(\rho^{\eta})\bigg{)}\\\
&={2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}\left[1+O(\delta_{j}|y|+\rho^{\eta})\right]\end{split}$
(3.5)

and

$\begin{split}&\rho V_{1}(\delta_{j}y)e^{-\tau U(\delta_{j}y)}=\rho
V_{1}(\delta_{j}y)\exp\bigg{(}-\tau\Big{[}-2\log\delta_{j}-a_{j}-\log\rho+\log{|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}\\\
&\,+(-1)^{m+1}{2\pi\over\tau^{\nu(m)}}(\alpha_{m}+2)H(\delta_{j}y,0)\Big{]}+O(\rho^{\eta})\bigg{)}=O\bigg{(}\rho^{1+\tau}\delta_{j}^{2\tau}{(1+|y|^{\alpha_{j}})^{2\tau}\over|y|^{(\alpha_{j}-2)\tau}}\bigg{)},\end{split}$
(3.6)

Similarly, by using (2.17) and $1-\nu(m)=\sigma(m)$ for
$y\in\frac{A_{j}}{\delta_{j}}$ and $j$ even

$\begin{split}\rho V_{1}(\delta_{j}y)e^{-\tau
U(\delta_{j}y)}&={2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}\tau(1+|y|^{\alpha_{j}})^{2}}{V_{1}(\delta_{j}y)\over
V_{1}(0)}\exp\bigg{(}(-1)^{m}2\pi\tau^{\sigma(m)}(\alpha_{m}+2)[H(\delta_{j}y,0)-h_{0}]+O(\rho^{\eta})\bigg{)}\\\
&={2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}\tau(1+|y|^{\alpha_{j}})^{2}}\left[1+O(\delta_{j}|y|+\rho^{\eta})\right]\end{split}$
(3.7)

and

$\rho
V_{0}(\delta_{j}y)e^{U(\delta_{j}y)}=O\bigg{(}\rho^{1+1/\tau}\delta_{j}^{2/\tau}{(1+|y|^{\alpha_{j}})^{2/\tau}\over|y|^{(\alpha_{j}-2)/\tau}}\bigg{)}.$
(3.8)

Hence, by using (3.3), (3.5) and (3.6) for $\delta_{1}y\in A_{1}$ we get that

$\begin{split}R(\delta_{1}y)=&\,{1\over\delta_{1}^{2}}{2\alpha_{1}^{2}|y|^{\alpha_{1}-2}\over(1+|y|^{\alpha_{1}})^{2}}O(\delta_{1}|y|+\rho^{\eta})+O\bigg{(}\rho^{1+\tau}\delta_{1}^{2\tau}{(1+|y|^{\alpha_{1}})^{2\tau}\over|y|^{(\alpha_{1}-2)\tau}}+\sum_{i=2}^{m}\Big{(}{\delta_{1}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{1}^{2}}\bigg{)},\end{split}$
(3.9)

by using (3.2), (3.5) and (3.6) for $1<j<m$, $j$ odd and $\delta_{j}y\in
A_{j}$

$\begin{split}R(\delta_{j}y)=&\,{1\over\delta_{j}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}O(\delta_{j}|y|+\rho^{\eta})\\\
&\,+O\bigg{(}\rho^{1+\tau}\delta_{j}^{2\tau}{(1+|y|^{\alpha_{j}})^{2\tau}\over|y|^{(\alpha_{j}-2)\tau}}+\sum_{i=1}^{j-1}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}}{1\over\delta_{j}^{2}|y|^{\alpha_{i}+2}}+\sum_{i=j+1}^{m}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{j}^{2}}\bigg{)},\end{split}$
(3.10)

by using (3.2), (3.7) and (3.8) for $1<j<m$, $j$ even and $\delta_{j}y\in
A_{j}$

$\begin{split}R(\delta_{j}y)=&\,{1\over\delta_{j}^{2}\tau}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}O(\delta_{j}|y|+\rho^{\eta})\\\
&\,+O\bigg{(}\rho^{1+1/\tau}\delta_{j}^{2/\tau}{(1+|y|^{\alpha_{j}})^{2/\tau}\over|y|^{(\alpha_{j}-2)/\tau}}+\sum_{i=1}^{j-1}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}}{1\over\delta_{j}^{2}|y|^{\alpha_{i}+2}}+\sum_{i=j+1}^{m}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{j}^{2}}\bigg{)}\end{split}$
(3.11)

and by (3.4) for $\delta_{m}y\in A_{m}$

$\begin{split}R(\delta_{m}y)=&\,{1\over\delta_{m}^{2}\tau^{\nu(m)}}{2\alpha_{m}^{2}|y|^{\alpha_{m}-2}\over(1+|y|^{\alpha_{m}})^{2}}O(\delta_{m}|y|+\rho^{\eta})\\\
&\,+O\bigg{(}\rho^{1+\tau^{(-1)^{m+1}}}\delta_{m}^{2\tau^{(-1)^{m+1}}}{(1+|y|^{\alpha_{m}})^{2\tau^{(-1)^{m+1}}}\over|y|^{(\alpha_{m}-2)\tau^{(-1)^{m+1}}}}+\sum_{i=1}^{m-1}\Big{(}{\delta_{i}\over\delta_{m}}\Big{)}^{\alpha_{i}}{1\over\delta_{m}^{2}|y|^{\alpha_{i}+2}}\bigg{)}\end{split}$
(3.12)

By (3.9)-(3.12), we finally get that there exist $\rho_{0}>0$ small enough and
$p_{0}>1$ close to 1, so that for all $0<\rho\leq\rho_{0}$ and $1<p\leq p_{0}$

$\begin{split}\int_{\Omega_{\epsilon}}|R(x)|^{p}\,dx=&\,\sum_{j=1}^{m}\int_{A_{j}}|R(x)|^{p}\,dx=\sum_{j=1}^{m}\delta_{j}^{2}\int_{A_{j}}|R(\delta_{j}y)|^{p}\,dy=O\left(\rho^{p\eta_{p}}\right),\end{split}$

for some $\eta_{p}>0$, see Appendix, Lemma A.1 for more details. This
completes the proof. ∎

###  3.2. The nonlinear problem and proof of Theorem 1.1.

In this subsection, we will look for a solution $u$ of (1.1) in the form
$u=U+\phi$, for some small remainder term $\phi$. In terms of $\phi$, the
problem (1.1) is equivalent to find $\phi\in H_{0}^{1}(\Omega_{\epsilon})$ so
that

$\left\\{\begin{array}[]{ll}L(\phi)=-[R+\Lambda(\phi)+N(\phi)],&\text{ in
$\Omega_{\epsilon}$},\\\ \phi=0,&\text{ on
$\partial\Omega_{\epsilon}$}.\end{array}\right.$ (3.13)

where the linear operators $L$ and $\Lambda$ are defined as

$L(\phi)=\Delta\phi+K(x)\phi,\quad
K(x)=\sum_{i=1}^{m}|x|^{\alpha_{i}-2}e^{w_{i}}$ (3.14)

and

$\Lambda(\phi)=\rho[V_{0}(x)e^{U}+\tau V_{1}(x)e^{-\tau U}]\phi-K\phi$ (3.15)

The nonlinear part $N$ is given by

$\begin{split}N(\phi)=&\,\rho V_{0}(x)e^{U}\big{(}e^{\phi}-\phi-1\big{)}-\rho
V_{1}(x)e^{-\tau U}\big{(}e^{-\tau\phi}+\tau\phi-1\big{)}.\end{split}$ (3.16)

######  Proposition 3.1.

For any $p>1$ there exists $\rho_{0}>0$ so that for all $0<\rho\leq\rho_{0}$,
$h\in L^{p}(\Omega_{\epsilon})$ there is a unique solution $\phi\in
H_{0}^{1}(\Omega_{\epsilon})$ of

$\left\\{\begin{array}[]{ll}L(\phi)=h&\text{in }\Omega_{\epsilon}\\\
\phi=0&\text{on }\partial\Omega_{\epsilon}.\end{array}\right.$ (3.17)

Moreover, there is a constant $C>0$ independent of $\rho$ such that

$\|\phi\|\leq C|\log\rho|\|h\|_{p}.$ (3.18)

The latter proposition implies that the unique solution $\phi=T(h)$ of (3.17)
defines a continuous linear map from $L^{p}(\Omega_{\epsilon})$ into
$H_{0}^{1}(\Omega_{\epsilon})$, with norm bounded by $C|\log\rho|$. We are now
in position to study the nonlinear problem.

######  Proposition 3.2.

There exist $p_{0}>1$ and $\rho_{0}>0$ so that for any $1<p<p_{0}$ and all
$0<\rho\leq\rho_{0}$, the problem

$\left\\{\begin{array}[]{ll}L(\phi)=-[R+\Lambda(\phi)+N(\phi)],&\text{in
}\Omega_{\epsilon}\\\ \phi=0,&\text{on
}\partial\Omega_{\epsilon}\end{array}\right.$ (3.19)

admits a unique solution $\phi(\rho)\in H_{0}^{1}(\Omega_{\epsilon})$, where
$N$, $\Lambda$ and $R$ are given by (3.16), (3.15) and (1.12), respectively.
Moreover, there is a constant $C>0$ such that for some $\eta_{p}>0$

$\|\phi\|_{\infty}\leq C\rho^{\eta_{p}}|\log\rho|$

Here, $\eta_{p}$ is the same as in (3.1). We shall use the following
estimates.

######  Lemma 3.2.

There exist $p_{0}>1$ and $\rho_{0}>0$ so that for any $1<p<p_{0}$ and all
$0<\rho\leq\rho_{0}$ it holds

$\|\Lambda(\phi)\|_{p}\leq C\rho^{\eta_{p}^{\prime}}\|\phi\|,$ (3.20)

for all $\phi\in H_{0}^{1}(\Omega_{\epsilon})$ with $\|\phi\|\leq
M\rho^{\eta_{p}}|\log\rho|$, for some $\eta_{p}^{\prime}>0$.

###### Proof:.

Arguing in the same way as in [13, Lemma 3.3], for simplicity, denote
$W=\rho[V_{0}e^{U}+\tau V_{1}e^{-\tau U}]$, so that the linear operator
$\Lambda$ is re-written as $\Lambda(\phi)=(W-K)\phi$. By using (3.5)-(3.6) and
(3.7)-(3.8), we find that for $i$ odd

$\delta_{i}^{2}W(\delta_{i}y)={2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha}_{i})^{2}}\left[1+O(|\delta_{i}y|+\rho^{\eta})\right]+O\Big{(}\rho^{1+\tau}\delta_{i}^{2+2\tau}{(1+|y|^{\alpha_{i}})^{2\tau}\over|y|^{(\alpha_{i}-2)\tau}}\Big{)}$

uniformly for $\delta_{i}y\in A_{i}$ and for $i$ even

$\delta_{i}^{2}W(\delta_{i}y)={2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha}_{i})^{2}}\left[1+O(|\delta_{i}y|+\rho^{\eta})\right]+O\Big{(}\rho^{1+1/\tau}\delta_{i}^{2+2/\tau}{(1+|y|^{\alpha_{i}})^{2/\tau}\over|y|^{(\alpha_{i}-2)/\tau}}\Big{)}$

uniformly for $x\in A_{i}$ and $i$ even. Hence, for any $q\geq 1$ and for $i$
odd there holds

$\begin{split}\left\|W-K\right\|_{L^{q}(A_{i})}^{q}\leq&\,C\bigg{[}\delta_{i}^{2-q}\int_{A_{i}\over\delta_{i}}\bigg{|}{2\alpha_{i}^{2}|y|^{\alpha_{i}-1}\over(1+|y|^{\alpha_{i}})^{2}}\bigg{|}^{q}\,dy+\rho^{\eta
q}\delta_{i}^{2-2q}\int_{A_{i}\over\delta_{i}}\bigg{|}{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}\bigg{|}^{q}\,dy\\\
&\,+\rho^{(1+\tau)q}\delta_{i}^{2+2\tau
q}\int_{A_{i}\over\delta_{i}}\bigg{|}{(1+|y|^{\alpha_{i}})^{2\tau}\over|y|^{(\alpha_{i}-2)\tau}}\bigg{|}^{q}dy+\sum_{j<i}\delta_{i}^{2-2q}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{j}q}\int_{A_{i}\over\delta_{i}}\bigg{|}{1\over|y|^{\alpha_{j}+2}}\bigg{|}^{q}dy\\\
&+\sum_{j>i}\delta_{i}^{2-2q}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{j}q}\int_{A_{i}\over\delta_{i}}|y|^{(\alpha_{j}-2)q}dy\bigg{]}\leq
C\rho^{q\eta^{\prime}_{1,q}}\end{split}$

for some $\eta^{\prime}_{1,q}>0$, see Lemma A.1. Similarly, there holds
$\left\|W-K\right\|_{L^{q}(A_{i})}^{q}\leq C\rho^{q\eta^{\prime}_{2,q}}$ for
any $q\geq 1$ and for $i$ even for some $\eta^{\prime}_{2,q}>0$. Hence, we get
that

$\begin{split}\|\Lambda(\phi)\|_{p}\leq&\,\left\|\left(W-K\right)\phi\right\|_{p}\leq\left\|W-K\right\|_{pr}\|\phi\|_{ps}\leq
C\rho^{\eta^{\prime}_{p}}\|\phi\|,\end{split}$

where
$\eta^{\prime}_{p}=\min\left\\{\eta^{\prime}_{1,pr},\eta^{\prime}_{2,ps}\right\\}$
with $r$, $s$ satisfying $\frac{1}{r}+\frac{1}{s}=1$. Furthermore, we have
used the Hölder’s inequality $\|uv\|_{q}\leq\|u\|_{qr}\|v\|_{qs}$ with
${1\over r}+{1\over s}=1$ and the inclusions
$L^{p}(\Omega_{\epsilon})\hookrightarrow L^{pr}(\Omega_{\epsilon})$ for any
$r>1$ and $H^{1}_{0}(\Omega_{\epsilon})\hookrightarrow
L^{q}(\Omega_{\epsilon})$ for any $q>1$. Let us stress that we can choose $p$,
$r$ and $s$ close enough to 1 such that $\eta_{p}^{\prime}>0$. ∎

######  Lemma 3.3.

There exist $p_{0}>1$ and $\rho_{0}>0$ so that for any $1<p<p_{0}$ and all
$0<\rho\leq\rho_{0}$ it holds

$\|N(\phi_{1})-N(\phi_{2})\|_{p}\leq
C\rho^{\eta_{p}^{\prime\prime}}\|\phi_{1}-\phi_{2}\|$ (3.21)

for all $\phi_{i}\in H_{0}^{1}(\Omega_{\epsilon})$ with $\|\phi_{i}\|\leq
M\rho^{\eta_{p}}|\log\rho|$, $i=1,2$, and for some
$\eta_{p}^{\prime\prime}>0$. In particular, we have that

$\|N(\phi)\|_{p}\leq C\rho^{\eta_{p}^{\prime\prime}}\ \|\phi\|$ (3.22)

for all $\phi\in H_{0}^{1}(\Omega_{\epsilon})$ with $\|\phi\|\leq
M\rho^{\eta_{p}}|\log\rho|$.

###### Proof:.

We will argue in the same way as in [1, Lemma 5.1], see also [13, Lemma 3.4].
First, denoting $f_{i}(\phi)=V_{i}(x)e^{(-\tau)^{i}(U+\phi)}$ we point out
that

$N(\phi)=\sum_{i=0}^{1}\rho\left\\{f_{i}(\phi)-f_{i}(0)-f^{\prime}_{i}(0)[\phi]\right\\}\
\ \ \text{and}\ \ \ N(\phi_{1})-N(\phi_{2})=\sum_{i=0}^{1}\rho
f_{i}^{\prime\prime}(\tilde{\phi}_{\mu_{i}})[\phi_{\theta_{i}},\phi_{1}-\phi_{2}],$
(3.23)

by the mean value theorem, where
$\phi_{\theta_{i}}=\theta_{i}\phi_{1}+(1-\theta_{i})\phi_{2}$,
$\tilde{\phi}_{\mu_{i}}=\mu_{i}\phi_{\theta_{i}}$ for some
$\theta_{i},\mu_{i}\in[0,1]$, $i=0,1$, and
$f_{i}^{\prime\prime}(\phi)[\psi,v]=\tau^{2i}V_{i}(x)e^{(-\tau)^{i}(U+\phi)}\psi
v$. Using Hölder’s inequalities we get that

$\left\|f_{i}^{\prime\prime}(\phi)[\psi,v]\right\|_{p}\leq|\tau|^{2i}\|V_{i}e^{(-\tau)^{i}(U+\phi)}\|_{pr_{i}}\|\psi\|_{ps_{i}}\|v\|_{pt_{i}}$
(3.24)

with ${1\over r_{i}}+{1\over s_{i}}+{1\over t_{i}}=1$. We have used the
Hölder’s inequality and $\|uvw\|_{q}\leq\|u\|_{qr}\|v\|_{qs}\|w\|_{qt}$ with
${1\over r}+{1\over s}+{1\over t}=1$. Now, let us estimate
$\|V_{i}e^{(-\tau)^{i}(U+\phi)}\|_{pr_{i}}$ with
$\phi=\tilde{\phi}_{\mu_{i}}$, $i=0,1$. By (3.5) and the change of variable
$x=\delta_{i}y$ let us estimate

$\int_{A_{i}}\left|V_{0}e^{U}\right|^{q}dx=O\left(\rho^{-q}\delta_{i}^{2-2q}\int_{\frac{A_{i}}{\delta_{i}}}\left|\frac{|y|^{\alpha_{i}-2}}{(1+|y|^{\alpha_{i}})^{2}}\right|^{q}\left[1+O\big{(}\rho^{\eta}+\delta_{i}|y|\big{)}\right]^{q}\right)=O\left(\rho^{{\beta_{i}\over\alpha_{i}}(2-2q)-q}\right)$
(3.25)

for any $i$ odd and similarly, by (3.7) we get that

$\displaystyle\int_{A_{i}}\left|V_{1}e^{-\tau
U}\right|^{q}dx=O\left(\rho^{{\beta_{i}\over\alpha_{i}}(2-2q)-q}\right)$
(3.26)

for any $i$ even, in view of (2.2). By (3.6) we get the estimate

$\displaystyle\int_{A_{i}}\left|V_{0}e^{U}\right|^{q}dx$ $\displaystyle=$
$\displaystyle
O\bigg{(}\rho^{q\over\tau}\delta_{i}^{2+{2q\over\tau}}\int\limits_{\frac{A_{i}}{\delta_{i}}}\Big{[}\frac{|y|^{\alpha_{i}-2}}{(1+|y|^{\alpha_{i}})^{2}}\Big{]}^{-{q\over\tau}}dy\bigg{)}=O\big{(}\rho^{-q+q\eta_{q}}\big{)}$

for all $i$ even, in view of Lemma A.1. Similarly, by (3.8) we deduce that

$\displaystyle\int_{A_{i}}\left|V_{1}e^{-\tau
U}\right|^{q}dx=O(\rho^{-q+q\eta_{q}})$ (3.27)

for $i$ odd, again in view of Lemma A.1. Therefore, by using (3.25) and (3.2)
and that $\frac{\beta_{i}}{\alpha_{i}}$ are decreasing, we deduce that

$\left\|V_{0}e^{U}\right\|_{q}^{q}=\sum_{i=1\atop i\text{
odd}}^{m}O\left(\rho^{{\beta_{i}\over\alpha_{i}}(2-2q)-q}\right)+O\big{(}\rho^{-q+q\eta_{q}}\big{)}=O\left(\rho^{{\beta_{1}\over\alpha_{1}}(2-2q)-q}\right)\quad\text{for
any }q\geq 1$ (3.28)

and, by using (3.26) and (3.27) we obtain that

$\left\|V_{1}e^{-\tau
U}\right\|_{q}^{q}=O\left(\rho^{{\beta_{1}\over\alpha_{1}}(2-2q)-q}\right)\quad\text{for
any }q\geq 1.$ (3.29)

On the other hand, using the estimate $|e^{a}-1|\leq C|a|$ uniformly for any
$a$ in compact subsets of IR, Hölder’s inequality with ${1\over
s_{i}^{\prime}}+{1\over t_{i}^{\prime}}=1$, $i=0,1$,
$\|\tilde{\phi}_{\mu_{i}}\|\leq M\rho^{\eta_{p}}|\log\rho|\leq C$, $i=0,1$ and
triangle inequality we find that

$\big{\|}V_{i}e^{(-\tau)^{i}(U+\tilde{\phi}_{\mu_{i}})}\big{\|}_{q}=O\big{(}\rho^{\eta_{0,qs_{i}^{\prime}}-1+\eta_{p}}|\log\rho|+\rho^{\eta_{0,q}-1}\big{)},$
(3.30)

where for $q>1$ we denote $\eta_{0,q}={\beta_{1}(2-2q)\over\alpha_{1}q}$ (on
the line of [17, eq.(3.14) proof of Lemma 3.3]). Note that $\eta_{0,q}<0$ for
any $q>1$. Also, choosing $q$ and $s_{i}^{\prime}$, $i=0,1$, close enough to
1, we get that $0<\eta_{p}+\eta_{0,qs_{i}^{\prime}}$. Now, we can conclude the
estimate by using (3.23)-(3.30) to get

$\begin{split}\|N(\phi_{1})-N(\phi_{2})\|_{p}&\,\leq\,C\sum_{i=0}^{1}\rho\|V_{i}e^{(-\tau)^{i}(U+\tilde{\phi}_{\mu_{i}})}\|_{pr_{i}}\|\phi_{\theta_{i}}\|\|\phi_{1}-\phi_{2}\|\leq\,C\sum_{i=0}^{1}\rho^{\eta_{p}+\eta_{0,pr_{i}}}|\log\rho|\|\phi_{1}-\phi_{2}\|\end{split}$

and (3.21) follows, where $\eta^{\prime\prime}_{p}={1\over
2}\min\\{\eta_{p}+\eta_{0,pr_{i}}\ :\ i=0,1\\}>0$ choosing $r_{i}$ close to 1
so that $\eta_{p}+\eta_{0,pr_{i}}>0$ for $i=0,1$. Let us stress that $p>1$ is
chosen so that $\eta_{p}>0$ and $\eta_{p}^{\prime}>0$. ∎

###### Proof of the Proposition 3.2.

Taking into account Proposition 3.1, estimates (3.1), (3.20), (3.21), (3.22)
and standard arguments it turns out that for all $\rho>0$ sufficiently small
$\mathcal{A}$ is a contraction mapping of $\mathcal{F}_{M}$ (for $M$ large
enough), and therefore a unique fixed point of $\mathcal{A}$ exists in
$\mathcal{F}_{M}$, where $\mathcal{A}(\phi):=-T(R+\Lambda(\phi)+N(\phi))$ and
$\mathcal{F}_{M}=\\{\phi\in H_{0}^{1}(\Omega_{\epsilon}):\|\phi\|\leq
M\rho^{\eta_{p}}|\log\rho|\\}$. See the proof of Proposition 3.2 in [13] for
more details. ∎

###### Proof of the Theorem 1.1.

Taking into account (1.10) and the definition of $U$, the existence of a
solution to equation (1.1) follows directly by Proposition 3.2. The asymptotic
behavior of $u_{\rho}$ as $\rho\to 0^{+}$ follows from (2.23) in Lemma 2.6 and
estimate for $\phi$ in Proposition 3.2. Furthermore, we have that $u_{\rho}$
has the desired concentration properties (1.6) as $\rho\to 0^{+}$ locally
uniformly in $\bar{\Omega}\setminus\\{0\\}$. ∎

##  4\. Problem in mean field form (1.2)

###  4.1. Error estimate

Assume that $\delta_{i}$’s are given by (2.2) with $\alpha_{i}$’s,
$\beta_{i}$’s and $d_{i}$’s defined in (2.4), (2.9)-(2.10) and (2.15),
respectively, and $\epsilon$ is given by (2.21). Let us evaluate the error
term in $\|\cdot\|_{p}$, encoded in (1.13).

######  Lemma 4.1.

There exist $\rho_{0}>0$, a constant $C>0$ and $1<p_{0}<2$, such that for any
$\rho\in(0,\rho_{0})$ and $p\in(1,p_{0})$ it holds

$\|\mathcal{R}\|_{p}\leq C\rho^{\eta_{p}}$ (4.1)

for some $\eta_{p}>0$.

###### Proof:.

By using (3.5) and (3.8) we have that

$\begin{split}\int_{\Omega_{\epsilon}}&\,V_{0}(x)e^{U}=\sum_{j=1}^{m}\int_{A_{j}\over\delta_{j}}\delta_{j}^{2}V_{0}(\delta_{j}y)e^{U(\delta_{j}y)}dy=\sum_{j=1\atop
j\text{
odd}}^{m}{1\over\rho}\int_{A_{j}\over\delta_{j}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}\left[1+O(\delta_{j}|y|+\rho^{\eta})\right]dy\\\
&+\sum_{j=1\atop j\text{
even}}^{m}O\bigg{(}\rho^{1/\tau}\delta_{j}^{2+2/\tau}\int_{A_{j}\over\delta_{j}}{(1+|y|^{\alpha_{j}})^{2/\tau}\over|y|^{(\alpha_{j}-2)/\tau}}dy\bigg{)}={1\over\rho}\bigg{[}4\pi\sum_{j=1\atop
j\text{ odd}}^{m}\alpha_{j}+O(\rho^{\eta})\bigg{]}\end{split}$ (4.2)

and similarly, from (3.6) and (3.7) we get that

$\begin{split}\int_{\Omega_{\epsilon}}V_{1}(x)e^{-\tau
U}=&\,{1\over\rho\tau}\bigg{[}4\pi\sum_{j=1\atop j\text{
even}}^{m}\alpha_{j}+O(\rho^{\eta})\bigg{]},\end{split}$ (4.3)

in view of
$\int\limits_{\mathbb{R}^{2}}{|y|^{\alpha_{i}-2}\over\left(1+|y|^{\alpha_{i}}\right)^{2}}dy={2\pi\over\alpha_{i}}.$
From assumptions (1.8)-(1.9) it follows that $\lambda_{0}=4\pi\sum_{j=1\atop
j\text{ odd}}^{m}\alpha_{j}$ and $\lambda_{1}\tau^{2}=4\pi\sum_{j=1\atop
j\text{ even}}^{m}\alpha_{j}$ so that, by using (4.2)-(4.3) we find that for
$y\in\frac{A_{j}}{\delta_{j}}$ and any $j$ odd

$\lambda_{0}{V_{0}(\delta_{j}y)e^{U(\delta_{j}y)}\over\int_{\Omega_{\epsilon}}V_{0}(x)e^{U}}={2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}(1+|y|^{\alpha_{j}})^{2}}\left[1+O(\delta_{j}|y|+\rho^{\eta})\right]$
(4.4)

and

$\lambda_{1}\tau{V_{1}(\delta_{j}y)e^{-\tau
U(\delta_{j}y)}\over\int_{\Omega_{\epsilon}}V_{1}(x)e^{-\tau
U}}=O\bigg{(}\rho^{1+\tau}\delta_{j}^{2\tau}{(1+|y|^{\alpha_{j}})^{2\tau}\over|y|^{(\alpha_{j}-2)\tau}}\bigg{)}.$
(4.5)

Similarly, for any $y\in\frac{A_{j}}{\delta_{j}}$ and $j$ even

$\lambda_{0}{V_{0}(\delta_{j}y)e^{U(\delta_{j}y)}\over\int_{\Omega_{\epsilon}}V_{0}(x)e^{U}}=O\bigg{(}\rho^{1+1/\tau}\delta_{j}^{2/\tau}{(1+|y|^{\alpha_{j}})^{2/\tau}\over|y|^{(\alpha_{j}-2)/\tau}}\bigg{)}$
(4.6)

and

$\lambda_{1}\tau{V_{1}(\delta_{j}y)e^{-\tau
U(\delta_{j}y)}\over\int_{\Omega_{\epsilon}}V_{1}(x)e^{-\tau
U}}={2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over\delta_{j}^{2}\tau(1+|y|^{\alpha_{j}})^{2}}\left[1+O(\delta_{j}|y|+\rho^{\eta})\right].$
(4.7)

Hence, similar to the proof of Lemma 3.1 by using (3.2)-(3.4) and (4.4)-(4.7)
for $\delta_{1}y\in A_{1}$ we get that

$\begin{split}\mathcal{R}(\delta_{1}y)=&\,{1\over\delta_{1}^{2}}{2\alpha_{1}^{2}|y|^{\alpha_{1}-2}\over(1+|y|^{\alpha_{1}})^{2}}O(\delta_{1}|y|+\rho^{\eta})+O\bigg{(}\rho^{1+\tau}\delta_{1}^{2\tau}{(1+|y|^{\alpha_{1}})^{2\tau}\over|y|^{(\alpha_{1}-2)\tau}}+\sum_{i=2}^{m}\Big{(}{\delta_{1}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{1}^{2}}\bigg{)},\end{split}$
(4.8)

for $1<j<m$, $j$ odd and $\delta_{j}y\in A_{j}$

$\begin{split}\mathcal{R}(\delta_{j}y)=&\,{1\over\delta_{j}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}O(\delta_{j}|y|+\rho^{\eta})\\\
&\,+O\bigg{(}\rho^{1+\tau}\delta_{j}^{2\tau}{(1+|y|^{\alpha_{j}})^{2\tau}\over|y|^{(\alpha_{j}-2)\tau}}+\sum_{i=1}^{j-1}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}}{1\over\delta_{j}^{2}|y|^{\alpha_{i}+2}}+\sum_{i=j+1}^{m}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{j}^{2}}\bigg{)}\end{split}$
(4.9)

for $1<j<m$, $j$ even and $\delta_{j}y\in A_{j}$

$\begin{split}\mathcal{R}(\delta_{j}y)=&\,{1\over\delta_{j}^{2}\tau}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}O(\delta_{j}|y|+\rho^{\eta})\\\
&\,+O\bigg{(}\rho^{1+1/\tau}\delta_{j}^{2/\tau}{(1+|y|^{\alpha_{j}})^{2/\tau}\over|y|^{(\alpha_{j}-2)/\tau}}+\sum_{i=1}^{j-1}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{i}}{1\over\delta_{j}^{2}|y|^{\alpha_{i}+2}}+\sum_{i=j+1}^{m}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{i}}{|y|^{\alpha_{i}-2}\over\delta_{j}^{2}}\bigg{)}\end{split}$
(4.10)

and for $\delta_{m}y\in A_{m}$

$\begin{split}\mathcal{R}(\delta_{m}y)=&\,{1\over\delta_{m}^{2}\tau^{\nu(m)}}{2\alpha_{m}^{2}|y|^{\alpha_{m}-2}\over(1+|y|^{\alpha_{m}})^{2}}O(\delta_{m}|y|+\rho^{\eta})\\\
&\,+O\bigg{(}\rho^{1+\tau^{(-1)^{m+1}}}\delta_{m}^{2\tau^{(-1)^{m+1}}}{(1+|y|^{\alpha_{m}})^{2\tau^{(-1)^{m+1}}}\over|y|^{(\alpha_{m}-2)\tau^{(-1)^{m+1}}}}+\sum_{i=1}^{m-1}\Big{(}{\delta_{i}\over\delta_{m}}\Big{)}^{\alpha_{i}}{1\over\delta_{m}^{2}|y|^{\alpha_{i}+2}}\bigg{)}\end{split}$
(4.11)

By (4.8)-(4.11) and similar ideas to get (3.1) we conclude the proof. ∎

###  4.2. The nonlinear problem and proof of Theorem 1.2.

In this section we shall study the following nonlinear problem:

$\left\\{\begin{array}[]{ll}\mathcal{L}(\phi)=-[\mathcal{R}+\Lambda_{0}(\phi)+\mathcal{N}(\phi)]&\text{in
}\Omega_{\epsilon}\\\ \phi=0,&\text{on
}\partial\Omega_{\epsilon},\end{array}\right.$ (4.12)

where the linear operators $\mathcal{L},\Lambda_{0}$ are defined as

$\mathcal{L}(\phi)=\Delta\phi+K_{0}\left(\phi-{1\over\lambda_{0}}\int_{\Omega_{\epsilon}}K_{0}\phi
dx\right)+K_{1}\left(\phi-{1\over\lambda_{1}\tau^{2}}\int_{\Omega_{\epsilon}}K_{1}\phi
dx\right)$ (4.13)

and

$\begin{split}\Lambda_{0}(\phi)=&\,\lambda_{0}{V_{0}e^{U}\over\int_{\Omega_{\epsilon}}V_{0}e^{U}dx}\left(\phi-{\int_{\Omega_{\epsilon}}V_{0}e^{U}\phi
dx\over\int_{\Omega_{\epsilon}}V_{0}e^{U}dx}\right)+\lambda_{1}\tau^{2}{V_{1}e^{-\tau
U}\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau
U}dx}\left(\phi-{\int_{\Omega_{\epsilon}}V_{1}e^{-\tau U}\phi
dx\over\int_{\Omega_{\epsilon}}V_{1}(x)e^{-\tau U}dx}\right)\\\
&\,-K_{0}\left(\phi-{1\over\lambda_{0}}\int_{\Omega_{\epsilon}}K_{0}\phi
dx\right)-K_{1}\left(\phi-{1\over\lambda_{1}\tau^{2}}\int_{\Omega_{\epsilon}}K_{1}\phi
dx\right)\end{split}$ (4.14)

with

$K_{0}=\sum_{k=1\atop k\text{ odd}}^{m}|x|^{\alpha_{k}-2}e^{w_{k}},\quad
K_{1}=\sum_{k=1\atop k\text{ even}}^{m}|x|^{\alpha_{k}-2}e^{w_{k}}.$ (4.15)

The nonlinear term $\mathcal{N}(\phi)$ is given by

$\displaystyle\mathcal{N}(\phi)=$
$\displaystyle\lambda_{0}\left[{V_{0}e^{U+\phi}\over\int_{\Omega_{\epsilon}}V_{0}e^{U+\phi}dx}-{V_{0}e^{U}\over\int_{\Omega_{\epsilon}}V_{0}e^{U}dx}-{V_{0}e^{U}\over\int_{\Omega_{\epsilon}}V_{0}e^{U}dx}\left(\phi-{\int_{\Omega_{\epsilon}}V_{0}e^{U}\phi
dx\over\int_{\Omega_{\epsilon}}V_{0}e^{U}dx}\right)\right]$ (4.16)
$\displaystyle-\lambda_{1}\tau\left[{V_{1}e^{-\tau(U+\phi)}dx\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau(U+\phi)}dx}-{V_{1}e^{-\tau
U}\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau U}dx}+\tau{V_{1}e^{-\tau
U}\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau
U}dx}\left(\phi-{\int_{\Omega_{\epsilon}}V_{1}e^{-\tau U}\phi
dx\over\int_{\Omega_{\epsilon}}V_{1}e^{-\tau U}dx}\right)\right].$

It is readily checked that $\phi$ is a solution to (3.13) if and only if
$u_{\epsilon}$ given by (1.11) is a solution to (LABEL:p22). In Section 5 we
will prove the following result.

######  Proposition 4.1.

For any $p>1,$ there exists $\rho_{0}>0$ and $C>0$ such that for any
$\rho\in(0,\rho_{0})$ and $h\in L^{p}(\Omega_{\epsilon})$ there exists a
unique $\phi\in H^{1}_{0}(\Omega_{\epsilon})$ solution of

$\mathcal{L}(\phi)=h\ \hbox{ in }\ \Omega_{\epsilon},\ \ \ \phi=0\ \hbox{ on
}\ \partial\Omega_{\epsilon},$ (4.17)

which satisfies

$\|\phi\|\leq C|\log\rho|\,\|h\|_{p}.$ (4.18)

######  Lemma 4.2.

There exist $p_{0}>1$ and $\rho_{0}>0$ so that for any $1<p<p_{0}$ and all
$0<\rho\leq\rho_{0}$ it holds

$\|\Lambda_{0}(\phi)\|_{p}\leq C\rho^{\sigma_{p}^{\prime}}\|\phi\|,$ (4.19)

for all $\phi\in H_{0}^{1}(\Omega_{\epsilon})$ with $\|\phi\|\leq
M\rho^{\sigma_{p}}|\log\rho|$, for some $\sigma_{p}^{\prime}>0$.

###### Proof:.

Arguing in the same way as in [13, Lemma 3.3], denote
$W_{i}={\lambda_{i}\tau^{2i}V_{i}e^{(-\tau)^{i}U}\over\int_{\Omega_{\epsilon}}V_{i}e^{(-\tau)^{i}U}dx}$
for $i=0,1$. By using (4.4), (4.6), Lemma A.1 and similar computations to
prove (3.20), we find that $\left\|W_{0}-K_{0}\right\|_{L^{q}(A_{i})}^{q}\leq
C\rho^{q\sigma^{\prime}_{0,q}}$ for any $i$ for some $\sigma^{\prime}_{0,q}$.
Similarly, we find that $\left\|W_{1}-K_{1}\right\|_{L^{q}(A_{i})}^{q}\leq
C\rho^{q\sigma^{\prime}_{1,q}}$ for any $i$ for some $\sigma^{\prime}_{1,q}$.
It is possible to see that taking $q>1$ close enough to 1, we get that
$\sigma^{\prime}_{i,q}>0$ for $i=0,1$.

Notice that $\Lambda_{0}$ is a linear operator and we re-write
$\Lambda_{0}(\phi)$ as

$\begin{split}\Lambda_{0}(\phi)=&\,\sum_{i=0}^{1}\bigg{[}\left(W_{i}-K_{i}\right)\phi-{1\over\lambda_{i}\tau^{2i}}\left(W_{i}-K_{i}\right)\int_{\Omega_{\epsilon}}W_{i}\phi+{1\over\lambda_{i}\tau^{2i}}K_{i}\int_{\Omega_{\epsilon}}\left(K_{i}-W_{i}\right)\phi\bigg{]}.\end{split}$

Hence, we get that

$\begin{split}\|\Lambda_{0}(\phi)\|_{p}\leq&\,\sum_{i=0}^{1}\bigg{[}\left\|W_{i}-K_{i}\right\|_{pr_{i0}}\|\phi\|_{ps_{i0}}+{\|W_{i}\|_{r_{i1}}\over\lambda_{i}\tau^{2i}}\left\|W_{i}-K_{i}\right\|_{p}\|\phi\|_{s_{i1}}+{\|K_{i}\|_{p}\over\lambda_{i}\tau^{2(i-1)}}\left\|K_{i}-W_{i}\right\|_{r_{i2}}\|\phi\|_{s_{i2}}\bigg{]}\\\
\leq&\,C\sum_{i=0}^{1}\bigg{[}\rho^{\sigma^{\prime}_{i,pr_{i0}}}\|\phi\|+\rho^{\sigma^{\prime}_{i,p}+\sigma_{3,r_{i1}}}\|\phi\|+\rho^{\sigma^{\prime}_{i,r_{i2}}+\sigma_{3,p}}\|\phi\|\bigg{]}\end{split}$

and (4.19) follows, where
$\sigma^{\prime}_{p}=\min\left\\{\sigma^{\prime}_{i,pr_{i0}};\sigma^{\prime}_{i,p}+\sigma_{3,r_{i1}};\sigma^{\prime}_{i,r_{i2}}+\sigma_{3,p}\mid
i=0,1\right\\}$ with $r_{ij}$, $s_{ij}$, $i=0,1$, $j=0,1,2$ satisfying
$\frac{1}{r_{ij}}+\frac{1}{s_{ij}}=1$. We have used that

$\begin{split}\|W_{0}\|_{r_{01}}^{r_{01}}&\leq C\bigg{[}\sum_{j=1\atop j\text{
odd}}^{m}\delta_{j}^{2-2r_{01}}\int_{A_{j}\over\delta_{j}}\left|{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{i}})^{2}}\right|^{r_{01}}+\sum_{j=1\atop
j\text{
even}}^{m}\rho^{(1+1/\tau)r_{01}}\delta_{j}^{2+2{r_{01}\over\tau}}\int_{A_{j}\over\delta_{j}}\bigg{|}{(1+|y|^{\alpha_{j}})^{2/\tau}\over|y|^{(\alpha_{j}-2)/\tau}}\bigg{|}^{r_{01}}\bigg{]}\\\
&\leq C\rho^{\sigma_{3,r_{01}}}\end{split}$

and similarly, $\|W_{1}\|_{r_{11}}^{r_{11}}\leq C\rho^{\sigma_{3,r_{11}}}$,
where $\sigma_{3,q}={\beta_{1}\over\alpha_{1}}(2-2q)$ and similarly that
$\|K_{i}\|_{p}^{p}\leq C\rho^{\sigma_{3,p}}$, $i=0,1$. Note that
${2-2q\over\alpha_{j}q}<1$ for any $j=1,\dots,m$. Furthermore, we have used
the Hölder’s inequality $\|uv\|_{q}\leq\|u\|_{qr}\|v\|_{qs}$ with ${1\over
r}+{1\over s}=1$ and the inclusions $L^{p}(\Omega_{\epsilon})\hookrightarrow
L^{pr}(\Omega_{\epsilon})$ for any $r>1$ and
$H^{1}_{0}(\Omega_{\epsilon})\hookrightarrow L^{q}(\Omega_{\epsilon})$ for any
$q>1$. Let us stress that we can choose $p$, $r_{ij}$ and $s_{ij}$, $i=1,2$,
$j=0,1,2$, close enough to 1 such that $\sigma_{p}^{\prime}>0$. ∎

######  Lemma 4.3.

There exist $p_{0}>1$ and $\rho_{0}>0$ so that for any $1<p<p_{0}$ and all
$0<\rho\leq\rho_{0}$ it holds

$\|\mathcal{N}(\phi_{1})-\mathcal{N}(\phi_{2})\|_{p}\leq
C\rho^{\sigma_{p}^{\prime\prime}}\|\phi_{1}-\phi_{2}\|$ (4.20)

for all $\phi_{i}\in H_{0}^{1}(\Omega_{\epsilon})$ with $\|\phi_{i}\|\leq
M\rho^{\sigma_{p}}|\log\rho|$, $i=1,2$, and for some
$\sigma_{p}^{\prime\prime}>0$. In particular, we have that

$\|\mathcal{N}(\phi)\|_{p}\leq C\rho^{\sigma_{p}^{\prime\prime}}\ \|\phi\|$
(4.21)

for all $\phi\in H_{0}^{1}(\Omega_{\epsilon})$ with $\|\phi\|\leq
M\rho^{\sigma_{p}}|\log\rho|$.

###### Proof:.

Arguing in the same way as in [1, Lemma 5.1], we denote
$g_{i}(\phi)={V_{i}(x)e^{(-\tau)^{i}(U+\phi)}\over\int_{\Omega_{\epsilon}}V_{i}(x)e^{(-\tau)^{i}(U+\phi)}}$
and point out that

$\mathcal{N}(\phi)=\sum_{i=0}^{1}\lambda_{i}(-\tau)^{i}\left\\{g_{i}(\phi)-g_{i}(0)-g^{\prime}_{i}(0)[\phi]\right\\}\
\ \ \text{and by the mean value theorem we get that}$
$\mathcal{N}(\phi_{1})-\mathcal{N}(\phi_{2})=\sum_{i=1}^{2}\lambda_{i}(-\tau)^{i}g_{i}^{\prime\prime}(\tilde{\phi}_{\mu_{i}})[\phi_{\theta_{i}},\phi_{1}-\phi_{2}],$
(4.22)

where $\phi_{\theta_{i}}=\theta_{i}\phi_{1}+(1-\theta_{i})\phi_{2}$,
$\tilde{\phi}_{\mu_{i}}=\mu_{i}\phi_{\theta_{i}}$ for some
$\theta_{i},\mu_{i}\in[0,1]$, $i=0,1$, and

$\begin{split}g_{i}^{\prime\prime}(\phi)[\psi,v]=&\,\tau^{2i}\bigg{[}{V_{i}(x)e^{u_{i}}\psi
v\over\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}}-{V_{i}(x)e^{u_{i}}v\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}\psi\over\big{(}\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}\big{)}^{2}}-{V_{i}(x)e^{u_{i}}\psi\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}v\over\big{(}\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}\big{)}^{2}}\\\
&\,-{V_{i}(x)e^{u_{i}}\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}\psi
v\over\big{(}\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}\big{)}^{2}}+2{V_{i}(x)e^{u_{i}}\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}v\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}\psi\over\big{(}\int_{\Omega_{\epsilon}}V_{i}(x)e^{u_{i}}\big{)}^{3}}\bigg{]},\end{split}$

where for simplicity we denote $u_{i}=(-\tau)^{i}(U+\phi)$. Using Hölder’s
inequalities we get that

$\begin{split}\left\|g_{i}^{\prime\prime}(\phi)[\psi,v]\right\|_{p}\leq&\,C\left[{\|V_{i}e^{u_{i}}\|_{pr_{i}}\over\|V_{i}e^{u_{i}}\|_{1}}+{\|V_{i}e^{u_{i}}\|_{pr_{i}}^{2}\over\|V_{i}e^{u_{i}}\|_{1}^{2}}+{\|V_{i}e^{u_{i}}\|_{pr_{i}}^{3}\over\|V_{i}e^{u_{i}}\|_{1}^{3}}\right]\|\psi\|\|v\|,\end{split}$
(4.23)

with ${1\over r_{i}}+{1\over s_{i}}+{1\over t_{i}}=1$, ${1\over r_{i}}+{1\over
q_{i}}=1$, ${1\over pr_{i}}+{1\over\tilde{r}_{i}}=1$. We have used the
Hölder’s inequality, the inclusions presented in the previous Lemma and
$\|uvw\|_{q}\leq\|u\|_{qr}\|v\|_{qs}\|w\|_{qt}$ with ${1\over r}+{1\over
s}+{1\over t}=1$. Now, let us estimate
${\|V_{i}e^{u_{i}}\|_{pr_{i}}\over\|V_{i}e^{u_{i}}\|_{1}}$ with
$\phi=\tilde{\phi}_{\mu_{i}}$, $i=0,1$. Taking into account (3.28)-(3.29) and
(4.2)-(4.3), we obtain that for $i=0,1$

$\left\|V_{i}e^{(-\tau)^{i}U}\right\|_{q}^{q}=O\left(\rho^{{\beta_{1}\over\alpha_{1}}(2-2q)-q}\right)\quad\text{for
any }q\geq
1\quad\text{and}\quad\left\|V_{i}e^{(-\tau)^{i}U}\right\|_{1}\geq{C_{i}\over\rho}\quad\text{for
some }C_{i}>0.$

Note that $\sigma_{3,q}-1\leq{2-(\alpha_{i}+2)q\over\alpha_{i}q}\quad\text{
for any}\quad i=1,\dots,m$. By the previous estimates we find that
$\big{\|}V_{i}e^{(-\tau)^{i}U+\tilde{\phi}_{\mu_{i}}}\big{\|}_{q}=O\big{(}\rho^{\eta_{0,qs_{i}^{\prime}}-1+\eta_{p}}|\log\rho|+\rho^{\eta_{0,q}-1}\big{)}$
(on the line of [17, eq. (3.14) proof of Lemma 3.3]. Also, choosing $s_{i}$,
$i=0,1$, close enough to 1, we get that $\sigma_{p}+\eta_{0,s_{i}}>0$ and

$\left\|V_{i}e^{(-\tau)^{i}(U+\tilde{\phi}_{\mu_{i}})}\right\|_{1}\geq{C_{i}\over\rho}-C\rho^{\eta_{0,s_{i}}-1+\sigma_{p}}|\log\rho|\geq{1\over\rho}\left(C_{i}-C\rho^{\eta_{0,s_{i}}+\sigma_{p}}\
|\log\rho|\right)\geq{C_{i}\over 2\rho}.$

Taking $q=pr_{i}$, we obtain the estimate for $i=0,1$

$\begin{split}{\|V_{i}e^{(-\tau)^{i}(U+\tilde{\phi}_{\mu_{i}})}\|_{pr_{i}}\over\|V_{i}e^{(-\tau)^{i}(U+\tilde{\phi}_{\mu_{i}})}\|_{1}}&=O\left(\rho^{\sigma_{3,pr_{i}}}\Big{[}\rho^{\sigma_{p}+\sigma_{3,pr_{i}s_{i}}-\sigma_{pr_{i}}}|\log\rho|+1\Big{]}\right)=O\left(\rho^{\sigma_{3,pr_{i}}}\right)\end{split}$
(4.24)

choosing $s_{i}>1$ close enough to 1 so that
$\sigma_{p}+\sigma_{3,pr_{i}s_{i}}-\sigma_{pr_{i}}>0$, $i=1,2$. Now, we can
conclude the estimate by using (4.22)-(4.24) to get

$\begin{split}\|\mathcal{N}(\phi_{1})-\mathcal{N}(\phi_{2})\|_{p}&\,\leq\,C\sum_{i=0}^{1}\lambda_{i}\rho^{\sigma_{p}+3\sigma_{3,pr_{i}}}|\log\rho|\|\phi_{1}-\phi_{2}\|\leq\,C\rho^{\sigma_{p}^{\prime\prime}}\|\phi_{1}-\phi_{2}\|,\end{split}$

where $\sigma^{\prime\prime}_{p}={1\over
2}\min\\{\sigma_{p}+3\sigma_{3,pr_{i}}\ :\ i=0,1\\}>0$ choosing $r_{i}$ close
to 1 so that $\sigma_{p}+3\sigma_{3,pr_{i}}>0$ for $i=0,1$. Let us stress that
$p>1$ is chosen so that $\sigma_{p}>0$. ∎

Taking into account previous estimates (4.1), (4.19), (4.20) and (4.21), and
by using the same argument as in the proof of Proposition 3.2 we conclude the
following

######  Proposition 4.2.

There exist $p_{0}>1$ and $\rho_{0}>0$ so that for any $1<p<p_{0}$ and all
$0<\rho\leq\rho_{0}$, the problem (4.12) admits a unique solution
$\phi(\rho)\in H_{0}^{1}(\Omega_{\epsilon})$, where $\mathcal{L}$,
$\mathcal{R}$, $\Lambda_{0}(\phi)$ and $\mathcal{N}$ are given by (4.13),
(1.13), (4.14) and (4.16), respectively. Moreover, there is a constant $C>0$
such that

$\|\phi\|_{\infty}\leq C\rho^{\sigma_{p}}|\log\rho|,$

for some $\sigma_{p}>0$.

###### Proof of the Theorem 1.2.

Arguing in the same way as in the proof of Theorem 1.1, the existence of a
solution (1.10) to equation (1.2) follows directly by Proposition 4.2 and the
definition of $U$ in (1.11). ∎

##  5\. The linear theory

In this section we present the invertibility of the linear operators $L$ and
$\mathcal{L}$ defined in (3.14) and (4.13) respectively. Roughly speaking in
the scale annulus $\frac{A_{j}}{\delta_{j}}$ the operator $L$ apporaches to
the following linear operator in $\text{I\\!R}^{2}$

$L_{j}(\phi)=\Delta\phi+{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\phi,\qquad
j=1,\dots,m.$

It is well known that, in case $\alpha_{j}\in 2\text{I\\!N}$, the bounded
solutions of $L_{j}(\phi)=0$ in $\text{I\\!R}^{2}$ are precisely linear
combinations of the functions

$Y_{1j}(y)={|y|^{\alpha_{j}\over 2}\over
1+|y|^{\alpha_{j}}}\cos\Big{(}{\alpha_{j}\over 2}\theta\Big{)},\quad
Y_{2j}(y)={|y|^{\alpha_{j}\over 2}\over
1+|y|^{\alpha_{j}}}\sin\Big{(}{\alpha_{j}\over
2}\theta\Big{)}\quad\text{and}\quad Y_{0j}(y)=\,{1-|y|^{\alpha_{j}}\over
1+|y|^{\alpha_{j}}},$

which are written in polar coordinates for $j=1,\dots,m$. See [10] for a
proof. In our case, we will consider solutions of $L_{j}(\phi)=0$ with
$\alpha_{j}\notin 2\text{I\\!N}$ for all $j=1,\dots,m$ such that
$\int_{\text{I\\!R}^{2}}|\nabla\phi(y)|^{2}\,dy<+\infty$, which are multiples
of $Y_{0j}$, see [1, Theorem A.1] for a proof. Another key element in the
study of $\mathcal{L}$, which shows technical details, is to get rid of the
presence of

$\tilde{c}_{j}(\phi)=-{1\over\lambda_{j}\tau^{2j}}\int_{\Omega_{\epsilon}}K_{j}\phi\qquad
j=0,1.$ (5.1)

Let us introduce the following Banach spaces for $j=1,2,\dots,m$

$L_{\alpha_{j}}(\text{I\\!R}^{2})=\left\\{u\in
W_{\text{loc}}^{1,2}(\text{I\\!R}^{2})\ :\
\int_{\text{I\\!R}^{2}}{|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}|u(y)|^{2}\,dy<+\infty\right\\}$
(5.2)

and

$H_{\alpha_{j}}(\text{I\\!R}^{2})=\left\\{u\in
W_{\text{loc}}^{1,2}(\text{I\\!R}^{2})\ :\ \int_{\text{I\\!R}^{2}}|\nabla
u(y)|^{2}\,dy+\int_{\text{I\\!R}^{2}}{|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}|u(y)|^{2}\,dy<+\infty\right\\}$
(5.3)

endowed with the norms

$\|u\|_{L_{\alpha_{j}}}:=\left(\int_{\text{I\\!R}^{2}}{|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}|u(y)|^{2}\,dy\right)^{1/2}$

and

$\|u\|_{H_{\alpha_{j}}}:=\left(\int_{\text{I\\!R}^{2}}|\nabla
u(y)|^{2}\,dy+\int_{\text{I\\!R}^{2}}{|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}|u(y)|^{2}\,dy\right)^{1/2}.$

We point out the compactness of the embedding
$i_{\alpha_{j}}:H_{\alpha_{j}}(\text{I\\!R}^{2})\to
L_{\alpha_{j}}(\text{I\\!R}^{2})$, (see for example [20]).

###### Proof of Proposition 3.1.

Let us assume the existence of $p>1$, sequences $\rho_{n}\to 0$,
$\epsilon_{n}=\epsilon(\rho_{n})\to 0$, functions $h_{n}\in
L^{p}(\Omega_{\epsilon_{n}})$, $\phi_{n}\in H_{0}^{1}(\Omega_{\epsilon_{n}})$
such that

$L(\phi_{n})=h_{n}\ \ \text{in}\ \ \Omega_{\epsilon_{n}},\ \ \phi_{n}=0\ \
\text{on}\ \ \partial\Omega_{\epsilon_{n}}$

$\|\phi_{n}\|=1$ and $|\log\rho_{n}|\,\|h_{n}\|_{p}=o(1)$ as $n\to+\infty$. We
will omit the subscript $n$ in $\delta_{i,n}=\delta_{i}$. Recall that
$\delta_{i}^{\alpha_{i}}=d_{i,n}\rho_{n}^{\beta_{i}}$. Now, define
$\Phi_{i,n}(y):=\phi_{i,n}(\delta_{i}y)$ for
$y\in\Omega_{i,n}:=\delta_{i}^{-1}\Omega_{\epsilon_{n}}$. Thus, extending
$\phi_{n}=0$ in $\text{I\\!R}^{2}\setminus\Omega_{\epsilon_{n}}$ and arguing
in the same way as in [17, Claim 1, section 4] we can prove the following
fact. We provide a sketch of the proof.

######  Claim 5.1.

There holds that the sequence $\Phi_{i,n}$ converges (up to subsequence) to
$\Phi_{i}^{*}=a_{i}Y_{0i}$ for $i=1,\dots,m$, weakly in
$H_{\alpha_{i}}(\text{I\\!R}^{2})$ and strongly in
$L_{\alpha_{i}}(\text{I\\!R}^{2})$ as $n\to+\infty$ for some constant
$a_{i}\in\text{I\\!R}$, $i=1,\dots,m$.

###### Proof:.

First, notice that $\|\Phi_{i,n}\|_{H^{1}_{0}(\Omega_{i,n})}=1$, for
$i=1,\dots,m$. Then, we want to prove that there is a constant $M>0$ such for
all $n$ (up to a subsequence) $\|\Phi_{i,n}\|_{L_{\alpha_{i}}}^{2}\leq M$.
Notice that for any $i\in\\{1,\dots,m\\}$ we find that in $\Omega_{i,n}$

$\begin{split}\Delta\Phi_{i,n}&+\delta_{i}^{2}K(\delta_{i}y)\Phi_{i,n}=\delta_{i}^{2}h_{n}(\delta_{i}y).\end{split}$
(5.4)

Furthermore, it follows that $\Phi_{i,n}\to\Phi_{i}^{*}$ weakly in
$H^{1}_{0}(\Omega_{i,n})$ and strongly in $L^{p}(K)$ for any $K$ compact sets
in $\mathbb{R}^{2}$. Now, we multiply (5.4) by $\Phi_{i,n}$ for any
$i\in\\{1,\dots,m\\}$, integrate by parts and we obtain that

$\sum_{i=1}^{m}2\alpha_{i}^{2}\|\Phi_{i,n}\|_{L_{\alpha_{i}}}^{2}=\,1+o(1)$
(5.5)

since

$\delta_{i}^{2}K(\delta_{i}y)=\displaystyle{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+O\bigg{(}\sum_{j<i}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{j}}+\sum_{i<j}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{j}}\bigg{)}\quad\text{
for all }i=1,\dots,m$ (5.6)

uniformly for $y$ on compact subsets of $\text{I\\!R}^{2}\setminus\\{0\\}$.
Thus, we deduce that $\|\Phi_{i,n}\|_{L_{\alpha_{i}}}^{2}=O(1)$. Therefore,
the sequence $\\{\Phi_{i,n}\\}_{n}$ is bounded in
$H_{\alpha_{i}}(\text{I\\!R}^{2})$, so that there is a subsequence
$\\{\Phi_{i,n}\\}_{n}$ and functions $\Phi_{i}^{*}$, $i=1,2$ such that
$\\{\Phi_{i,n}\\}_{n}$ converges to $\Phi_{i}^{*}$ weakly in
$H_{\alpha_{i}}(\text{I\\!R}^{2})$ and strongly in
$L_{\alpha_{i}}(\text{I\\!R}^{2})$. Hence, taking into account (5.4) and (5.6)
we deduce that $\Phi^{*}_{i}$ a solution to

$\Delta\Phi+{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}\Phi=0,\qquad
i=1,\dots,m,\qquad\text{in $\text{I\\!R}^{2}\setminus\\{0\\}$}.$

It is standard that $\Phi_{i}^{*}$, $i=1,\dots,m$ extend to a solution in the
whole $\text{I\\!R}^{2}$. Hence, by using symmetry assumptions if necessary,
we get that $\Phi^{*}_{i}=a_{i}Y_{0i}$ for some constant
$a_{i}\in\text{I\\!R}$, $i=1,\dots,m$. ∎

For the next step consider for any $j\in\\{1,\dots,m\\}$ the functions
$Z_{0j}(x)=Y_{0j}(\delta_{j}^{-1}x)=\frac{\delta_{j}^{\alpha_{j}}-|x|^{\alpha_{j}}}{\delta_{j}^{\alpha_{j}}+|x|^{\alpha_{j}}}$
so that $-\Delta Z_{0j}=|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}.$ By similar
arguments as to obtain expansion in Lemma 2.1, it follows that
$P_{\epsilon}Z_{0j}=Z_{0j}+1-\gamma_{0}G(x,0)+O(\rho^{\tilde{\sigma}})$
uniformly in $\Omega_{\epsilon}$ for some $\tilde{\sigma}>0$, as a consequence
of [13, Lemma 4.1] with $m=1$ and $\xi_{1}=0$. Now, introduce the coefficient
$\gamma_{0}$ as

$\gamma_{0}\left[-{1\over 2\pi}\log\epsilon+H(0,0)\right]=2,$ (5.7)

so that from (2.21),
$\gamma_{0}=-\frac{4\pi(\alpha_{1}-2)}{\beta_{1}+1}\cdot\frac{1}{\log\rho}+O\big{(}{1\over|\log\rho|^{2}}\big{)}$.

For simplicity, we shall denote
$\Pi_{i}(y)=\frac{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}}{(1+|y|^{\alpha_{i}})^{2}}$,
for $i=1,\dots,m$.

######  Claim 5.2.

For all $j=1,\dots,m$, there exist the limit, up to subsequences,

$\mu_{j}=\lim_{n\to+\infty}\log\rho_{n}\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}(y)\,dy.$
(5.8)

Furthermore, it holds

$\begin{split}2\sum_{i=1}^{m}a_{i}=\bigg{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\sum_{i=1}^{m}(-1)^{i+1}\frac{\beta_{i}}{2\pi\alpha_{i}}\bigg{)}\mu_{1},\end{split}$
(5.9)

and for all $j=1,\dots,m-1$.

$\mu_{j}=(-1)^{j-1}\mu_{1}.$ (5.10)

###### Proof:.

Consider tests functions $\gamma_{0}^{-1}PZ_{0j}$’s and the assumption on
$h_{n}$, $|\log\rho_{n}|\ \|h_{n}\|_{*}=o(1)$. Hence, multiplying equation
(5.4) by $P_{\epsilon}Z_{0j}$ and integrating by parts we obtain that

$\begin{split}&\int_{\Omega_{\epsilon}}hPZ_{0j}=-\int_{\Omega_{\epsilon}}\phi|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}+\int_{\Omega_{\epsilon}}K\phi
P_{\epsilon}Z_{0j}=\int_{\Omega_{\epsilon}}\left(K-|x|^{\alpha_{j}-2}e^{w_{j}}\right)P_{\epsilon}Z_{0j}\phi\\\
&+\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\left(PZ_{0j}-Z_{0j}\right)\phi=\int_{\Omega_{\epsilon}}\phi\sum_{i=1\atop
i\neq
j}^{m}|x|^{\alpha_{i}-2}e^{w_{i}}PZ_{0j}+\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\left[1-\gamma_{0}G(x,0)+O(\rho^{\tilde{\sigma}})\right]\phi\end{split}$

in view of $P_{\epsilon}Z_{0j}=0$ on $\partial\Omega_{\epsilon_{n}}$ and
$\int_{\Omega_{\epsilon}}\Delta\phi
P_{\epsilon}Z_{0j}=\int_{\Omega_{\epsilon}}\phi\;\Delta
P_{\epsilon}Z_{0j}=\int_{\Omega_{\epsilon}}\phi\;\Delta Z_{0j}$. Now,
multiplying by $\gamma_{0}^{-1}$ and estimating every integral term we find
that
$\gamma_{0}^{-1}\int_{\Omega_{\epsilon}}hPZ_{0j}=O\left(\,|\log\rho|\,\|h\|_{p}\right)=o\left(1\right),$
in view of $PZ_{0j}=O(1)$ and
$\gamma_{0}^{-1}=-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}\log\rho+O(1)=O(|\log\rho|)$
in (5.7). Next, scaling we obtain that

$\gamma_{0}^{-1}\int_{\Omega_{\epsilon}}\phi|x|^{\alpha_{j}-2}e^{w_{j}}=-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}\log\rho\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Phi_{j,n}\,dy+o(1),$

in view of

$\lim\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Phi_{j,n}\,dy=a_{j}\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0,j}\,dy=0$

Also, by using (5.7) and expansion of $P_{\epsilon}Z_{0j}$, we get that for
$i<j$

$\begin{split}&\gamma_{0}^{-1}\int_{\Omega_{\epsilon}}|x|^{\alpha_{i}-2}e^{w_{i}}\phi
PZ_{0j}=\int_{\Omega_{i,n}}{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}\Phi_{i,n}\left[\frac{2\gamma_{0}^{-1}}{1+(\frac{\delta_{i}|y|}{\delta_{j}})^{\alpha_{j}}}+\frac{1}{2\pi}\log(\delta_{i}|y|)-H(\delta_{i}|y|,0)\right]\,dy\\\
&\,+O(\rho^{\tilde{\sigma}}|\log\rho|)=2\gamma_{0}^{-1}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy+\frac{1}{2\pi}\log\delta_{i}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy\\\
&\,+\frac{1}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\log|y|\,dy+o(1),\end{split}$

and for $i>j$ we have that

$\begin{split}&\gamma_{0}^{-1}\int_{\Omega_{\epsilon}}|x|^{\alpha_{i}-2}e^{w_{i}}\phi
PZ_{0j}=\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}\left[\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}\frac{2\gamma_{0}^{-1}}{(\frac{\delta_{j}}{\delta_{i}})^{\alpha_{j}}+|y|^{\alpha_{j}}}+\frac{1}{2\pi}\log(\delta_{i}|y|)-H(\delta_{i}|y|,0)\right]\,dy\\\
&\,+O(\rho^{\tilde{\sigma}}|\log\rho|)=\frac{1}{2\pi}\log\delta_{i}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy+\frac{1}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\log|y|\,dy+o(1),\end{split}$

in view of
$\big{(}\frac{\delta_{j}}{\delta_{i}}\big{)}^{\alpha_{j}}\gamma_{0}^{-1}=o(1)$.
Furthermore, it follows that

$\begin{split}\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}G(x,0)\phi=&\,-\frac{1}{2\pi}\log\delta_{j}\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}(y)\,dy-\frac{1}{2\pi}\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}(y)\log|y|\,dy+o(1).\end{split}$

From the choice of $\delta_{j}$’s in (2.2), it follows that
$\log\delta_{j}=\frac{1}{\alpha_{j}}\big{[}\log
d_{j}+\beta_{j}\log\rho\big{]}$ and from previous expansions we deduce that
for all $j=2,\dots,m-1$

$\begin{split}&\gamma_{0}^{-1}\int_{\Omega_{\epsilon}}hPZ_{0j}=\sum_{i<j}\bigg{[}\Big{(}-\frac{\beta_{1}+1}{2\pi(\alpha_{1}-2)}+\frac{\beta_{i}}{2\pi\alpha_{i}}\Big{)}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}\,dy+\frac{1}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\log|y|\,dy\bigg{]}\\\
&\,+\sum_{i>j}\bigg{[}\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy+\frac{1}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\log|y|\,dy\bigg{]}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}\log\rho\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j,n}\,dy\\\
&\,+\frac{\beta_{j}}{2\pi\alpha_{j}}\log\rho\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}\,dy+\frac{1}{2\pi}\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}(y)\log|y|\,dy+o(1),\end{split}$

Since for all $i=1,\dots,m$

$\lim\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\log|y|\,dy=a_{i}\int_{\text{I\\!R}^{2}}\Pi_{i}Y_{0,i}(y)\log|y|\,dy=-4\pi
a_{i}$ (5.11)

we obtain that for all $i=2,\dots,m-1$

$\begin{split}2\sum_{i=1}^{m}a_{i}=&\,\sum_{i<j}\bigg{[}\Big{(}-\frac{\beta_{1}+1}{2\pi(\alpha_{1}-2)}+\frac{\beta_{i}}{2\pi\alpha_{i}}\Big{)}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}\,dy\bigg{]}+\sum_{i>j}\bigg{[}\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy\bigg{]}\\\
&\,+\Big{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\frac{\beta_{j}}{2\pi\alpha_{j}}\Big{)}\log\rho\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}\,dy+o(1).\end{split}$
(5.12)

Choosing $j\in\\{2,\dots,m-2\\}$ we rewrite previous expression for $j+1$ as

$\begin{split}&2\sum_{i=1}^{m}a_{i}=\sum_{i<j}\bigg{[}\Big{(}-\frac{\beta_{1}+1}{2\pi(\alpha_{1}-2)}+\frac{\beta_{i}}{2\pi\alpha_{i}}\Big{)}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}\bigg{]}+\Big{(}-\frac{\beta_{1}+1}{2\pi(\alpha_{1}-2)}+\frac{\beta_{j}}{2\pi\alpha_{j}}\Big{)}\log\rho\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j,n}\\\
&\,+\sum_{i>j+1}\bigg{[}\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}\bigg{]}+\Big{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\frac{\beta_{j+1}}{2\pi\alpha_{j+1}}\Big{)}\log\rho\int_{\Omega_{j+1,n}}\Pi_{j+1}\Phi_{j+1,n}+o(1).\end{split}$
(5.13)

Hence, by subtracting (5.12) from (5.13) we obtain that

$0=-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}\log\rho\left[\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}\,dy+\int_{\Omega_{j+1,n}}\Pi_{j+1}(y)\Phi_{j+1,n}\,dy\right]+o(1),$

so that, for all $j=2,\dots,m-2$

$\lim\log\rho_{n}\left[\int_{\Omega_{j,n}}\Pi_{j}(y)\Phi_{j,n}\,dy+\int_{\Omega_{j+1,n}}\Pi_{j+1}(y)\Phi_{j+1,n}\,dy\right]=0.$
(5.14)

Similarly as above it extends to $j=1$

$\begin{split}2\sum_{i=1}^{m}a_{i}=&\,\sum_{i>1}\bigg{[}\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}dy\bigg{]}+\Big{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\frac{\beta_{1}}{2\pi\alpha_{1}}\Big{)}\log\rho\int_{\Omega_{1,n}}\Pi_{1}\Phi_{1,n}dy+o(1)\end{split}$

and

$\lim\log\rho_{n}\left[\int_{\Omega_{1,n}}\Pi_{1}(y)\Phi_{1,n}\,dy+\int_{\Omega_{2,n}}\Pi_{2}(y)\Phi_{2,n}\,dy\right]=0;$

and to $j=m$

$\begin{split}2\sum_{i=1}^{m}a_{i}=&\,\sum_{i<m}\bigg{[}\Big{(}-\frac{\beta_{1}+1}{2\pi(\alpha_{1}-2)}+\frac{\beta_{i}}{2\pi\alpha_{i}}\Big{)}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}\,dy\bigg{]}\\\
&\,+\Big{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\frac{\beta_{m}}{2\pi\alpha_{m}}\Big{)}\log\rho\int_{\Omega_{m,n}}\Pi_{m}(y)\Phi_{m,n}\,dy+o(1)\end{split}$

and

$\lim\log\rho_{n}\left[\int_{\Omega_{m-1,n}}\Pi_{m-1}(y)\Phi_{m-1,n}\,dy+\int_{\Omega_{m,n}}\Pi_{m}(y)\Phi_{m,n}\,dy\right]=0.$

Now, note that for $j=1$ we deduce that

$\begin{split}&2\sum_{i=1}^{m}a_{i}=\lim\bigg{[}\sum_{i=2}^{m}\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}dy+\Big{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\frac{\beta_{1}}{2\pi\alpha_{1}}\Big{)}\log\rho\int_{\Omega_{1,n}}\Pi_{1}\Phi_{1,n}dy\bigg{]}\\\
&=\lim\bigg{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\sum_{i=1}^{m}(-1)^{i+1}\frac{\beta_{i}}{2\pi\alpha_{i}}\bigg{)}\log\rho\int_{\Omega_{1,n}}\Pi_{1}\Phi_{1,n}\,dy=\bigg{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\sum_{i=1}^{m}(-1)^{i+1}\frac{\beta_{i}}{2\pi\alpha_{i}}\bigg{)}\mu_{1},\end{split}$

and there exists
$\mu_{1}=\lim\log\rho_{n}\int_{\Omega_{1,n}}\Pi_{1}(y)\Phi_{1,n}\,dy$. In
fact, by using (5.14) we have that

$\begin{split}\sum_{i=1}^{m}&\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy\\\
=&\,\sum_{i=1}^{m-2}\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy+\Big{(}\frac{\beta_{m-1}}{2\pi\alpha_{m-1}}-\frac{\beta_{m}}{2\pi\alpha_{m}}\Big{)}\log\rho\int_{\Omega_{m-1,n}}\Pi_{m-1}(y)\Phi_{m-1,n}\,dy\\\
&\,+\frac{\beta_{m}}{2\pi\alpha_{m}}\underbrace{\log\rho\bigg{(}\int_{\Omega_{m-1,n}}\Pi_{m-1}(y)\Phi_{m-1,n}\,dy+\int_{\Omega_{m,n}}\Pi_{m}(y)\Phi_{m,n}\,dy\bigg{)}}_{=o(1)}\\\
=&\,\sum_{i=1}^{m-3}\frac{\beta_{i}}{2\pi\alpha_{i}}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i,n}(y)\,dy+\frac{1}{2\pi}\Big{(}\frac{\beta_{m-2}}{\alpha_{m-2}}-\frac{\beta_{m-1}}{\alpha_{m-1}}+\frac{\beta_{m}}{\alpha_{m}}\Big{)}\log\rho\int_{\Omega_{m-2,n}}\Pi_{m-2}(y)\Phi_{m-2}(y)\,dy\\\
&+\Big{(}\frac{\beta_{m-1}}{2\pi\alpha_{m-1}}-\frac{\beta_{m}}{2\pi\alpha_{m}}\Big{)}\underbrace{\log\rho\bigg{(}\int_{\Omega_{m-2,n}}\Pi_{m-2}(y)\Phi_{m-2}(y)\,dy+\int_{\Omega_{m-1,n}}\Pi_{m-1}(y)\Phi_{m-1,n}\,dy\bigg{)}}_{=o(1)}+o(1)\\\
&\vdots\\\
=&\,\frac{1}{2\pi}\Big{(}\frac{\beta_{1}}{\alpha_{1}}-\frac{\beta_{2}}{\alpha_{2}}+\frac{\beta_{3}}{\alpha_{3}}+\cdots+(-1)^{m+1}\frac{\beta_{m}}{\alpha_{m}}\Big{)}\log\rho\int_{\Omega_{1,n}}\Pi_{1}(y)\Phi_{1,n}(y)\,dy+o(1).\end{split}$

Thus, we get (5.9). From the existence of $\mu_{1}$ and (5.14), it follows the
existence of $\mu_{2},\dots,\mu_{m}$ satisfying $\mu_{j}+\mu_{j+1}=0$ for all
$j=1,\dots,m-1$. Therefore, it is readily checked that (5.10) it holds. ∎

######  Claim 5.3.

There hold that $a_{j}=0$ for all $j=1,\dots,m$.

###### Proof:.

To this aim we will use as tests functions $P_{\epsilon}w_{j}$ and the
assumption on $h_{n}$, $|\log\rho_{n}|\ \|h_{n}\|_{*}=o(1)$. Hence,
multiplying equation (5.4) by $P_{\epsilon}w_{j}$ and integrating by parts we
obtain that

$\begin{split}\int_{\Omega_{\epsilon}}&hPw_{j}=\int_{\Omega_{\epsilon}}\Delta
w_{j}\phi+\int_{\Omega_{\epsilon}}K\phi
P_{\epsilon}w_{j}=-\int_{\Omega_{\epsilon}}\phi|x|^{\alpha_{j}-2}e^{w_{j}}+\int_{\Omega_{\epsilon}}K\phi
P_{\epsilon}w_{j}=-\int_{\Omega_{\epsilon}}\phi|x|^{\alpha_{j}-2}e^{w_{j}}\\\
&\,+\sum_{i=1}^{m}\int_{\Omega_{\epsilon}}|x|^{\alpha_{i}-2}e^{w_{i}}\phi\left[-2\log\big{(}\delta_{j}^{\alpha_{j}}+|x|^{\alpha_{j}}\big{)}+(4\pi\alpha_{j}-\gamma_{j})H(x,0)+\frac{\gamma_{j}}{2\pi}\log|x|+O(\rho^{\eta})\right]\end{split}$

in view of $P_{\epsilon}w_{j}=0$ on $\partial\Omega_{\epsilon_{n}}$ and
$\int_{\Omega_{\epsilon}}\Delta\phi
P_{\epsilon}w_{j}=\int_{\Omega_{\epsilon}}\phi\;\Delta P_{\epsilon}w_{j}$.
Now, estimating every integral term we find that

$\int_{\Omega_{\epsilon}}hPw_{j}=O\left(\,|\log\rho|\,\|h\|_{p}\right)=o\left(1\right),$

in view of $Pw_{j}=O(|\log\rho|)$ and
$G(x,0)=O(|\log\epsilon|)=O(|\log\rho|)$. Next, scaling we obtain that

$\int_{\Omega_{\epsilon}}\phi|x|^{\alpha_{j}-2}e^{w_{j}}=\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Phi_{j,n}=a_{j}\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}\,dy+o(1)=o(1).$

For $i=j$ we have that

$\begin{split}&\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\phi
P_{\epsilon}w_{j}=\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}\Big{[}-2\alpha_{j}\log\delta_{j}-2\log(1+|y|^{\alpha_{j}})+(4\pi\alpha_{j}-\gamma_{j})H(\delta_{j}y,0)+\frac{\gamma_{j}}{2\pi}\log|\delta_{j}y|\\\
&\,+O(\rho^{\eta})\Big{]}=-2\alpha_{j}\log\delta_{j}\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}-2\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}\log(1+|y|^{\alpha_{j}})+(4\pi\alpha_{j}-\gamma_{j})\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}H(\delta_{j}y,0)\\\
&\,+\frac{\gamma_{j}\log\delta_{j}}{2\pi}\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}+\frac{\gamma_{j}}{2\pi}\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}\log|y|+o(1),\end{split}$

similarly, for $i>j$ we obtain that

$\begin{split}&\int_{\Omega_{\epsilon}}|x|^{\alpha_{i}-2}e^{w_{i}}\phi
P_{\epsilon}w_{j}=\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\Big{[}-2\alpha_{j}\log\delta_{i}-2\log\Big{(}\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}\Big{)}+(4\pi\alpha_{j}-\gamma_{j})H(\delta_{i}y,0)\\\
&\,+\frac{\gamma_{j}}{2\pi}\log|\delta_{i}y|+O(\rho^{\eta})\Big{]}=-2\alpha_{j}\log\delta_{i}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}-2\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log\Big{(}\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}\Big{)}\\\
&\,+(4\pi\alpha_{j}-\gamma_{j})\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}H(\delta_{i}y,0)+\frac{\gamma_{j}\log\delta_{i}}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}+\frac{\gamma_{j}}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log|y|+o(1),\end{split}$

and for $i<j$

$\begin{split}&\int_{\Omega_{\epsilon}}|x|^{\alpha_{i}-2}e^{w_{i}}\phi
P_{\epsilon}w_{j}=\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\Big{[}-2\alpha_{j}\log\delta_{j}-2\log\Big{(}1+\Big{(}\frac{\delta_{i}|y|}{\delta_{j}}\Big{)}^{\alpha_{j}}\Big{)}+(4\pi\alpha_{j}-\gamma_{j})H(\delta_{i}y,0)+\frac{\gamma_{j}}{2\pi}\log|\delta_{i}y|\\\
&\,+O(\rho^{\eta})\Big{]}=-2\alpha_{j}\log\delta_{j}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}-2\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log\Big{(}1+\Big{(}\frac{\delta_{i}|y|}{\delta_{j}}\Big{)}^{\alpha_{j}}\Big{)}+(4\pi\alpha_{j}-\gamma_{j})\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}H(\delta_{i}y,0)\\\
&\,+\frac{\gamma_{j}\log\delta_{i}}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}+\frac{\gamma_{j}}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log|y|+o(1),\end{split}$

Now, by using the definition of $\gamma_{j}$’s in (2.20) and Lemma 2.1 we find
that
$\frac{\gamma_{j}}{2\pi}=2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}+O\big{(}\frac{1}{|\log\rho|}\big{)}$
so that from the choice of $\delta_{j}$’s we obtain that
$\frac{\gamma_{j}}{2\pi}\log\delta_{i}=2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\log\rho+O(1).$
Hence, from previous expansions we deduce that for all $j=2,\dots,m-1$

$\begin{split}o(1)&=\sum_{i<j}\Bigg{[}\Big{(}-2\alpha_{j}\log\delta_{j}+\frac{\gamma_{j}\log\delta_{i}}{2\pi}\Big{)}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}+\frac{\gamma_{j}}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log|y|\bigg{]}\\\
&\,+\Big{(}-2\alpha_{j}\log\delta_{j}+\frac{\gamma_{j}\log\delta_{j}}{2\pi}\Big{)}\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}+\frac{\gamma_{j}}{2\pi}\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}\log|y|-2\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}\log(1+|y|^{\alpha_{j}})\\\
&\,+\sum_{i>j}\Bigg{[}\Big{(}-2\alpha_{j}\log\delta_{i}+\frac{\gamma_{j}\log\delta_{i}}{2\pi}\Big{)}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}+\frac{\gamma_{j}}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log|y|\\\
&\,-2\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log\Big{(}\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}\Big{)}\bigg{]}=-2\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}\log(1+|y|^{\alpha_{j}})\\\
&\,+\sum_{i\leq
j}\Big{(}-2\beta_{j}+2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}+\sum_{i=1}^{m}2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log|y|+o(1)\\\
&\,+\sum_{i>j}\bigg{[}\Big{(}-2\alpha_{j}\frac{\beta_{i}}{\alpha_{i}}+2\beta_{j}
\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\log\rho\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}-2\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log\Big{(}\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}\Big{)}\bigg{]}.\end{split}$
(5.15)

Since for all $j=1,\dots,m$

$\lim\int_{\Omega_{j,n}}\Pi_{j}\Phi_{j}\log(1+|y|^{\alpha_{j}})=a_{j}\int_{\text{I\\!R}^{2}}\Pi_{j}Y_{0j}\log(1+|y|^{\alpha_{j}})=-2\pi\alpha_{j}a_{j},$
(5.16)

by using (5.8)

$\lim\int_{\Omega_{i,n}}\Pi_{i}\Phi_{i}\log\Big{(}\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}\Big{)}=\alpha_{j}a_{i}\int_{\text{I\\!R}^{2}}\Pi_{i}Y_{0i}\log|y|=-4\pi\alpha_{j}a_{i},$

in view of $\frac{\delta_{j}}{\delta_{i}}=o(1)$, and (5.11), as $n\to+\infty$
we get that

$\begin{split}0=&4\pi\alpha_{j}a_{j}+\sum_{i\leq
j}\Big{(}-2\beta_{j}+2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\mu_{i}+\sum_{i=1}^{m}2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}(-4\pi
a_{i})\\\
&\,+\sum_{i>j}\bigg{[}\Big{(}-2\alpha_{j}\frac{\beta_{i}}{\alpha_{i}}+2\beta_{j}
\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\mu_{i}-2(-4\pi\alpha_{j}a_{i})\bigg{]}\end{split}$

so that

$\begin{split}0=&\,4\pi\alpha_{j}a_{j}+\sum_{i\leq
j}\Big{(}-2\beta_{j}+2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\mu_{i}-8\pi\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\sum_{i=1}^{m}a_{i}\\\
&\,+\sum_{i>j}\Big{(}-2\alpha_{j}\frac{\beta_{i}}{\alpha_{i}}+2\beta_{j}
\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\mu_{i}+8\pi\alpha_{j}\sum_{i>j}a_{i}.\end{split}$

It is readily checked that
$8\pi\alpha_{j}\sum_{i>j}a_{i}=4\pi\alpha_{j}\sum_{i>j}a_{i}+4\pi\alpha_{j}\sum_{i\geq
j+1}a_{i},$ so we rewrite

$\begin{split}&0=4\pi\alpha_{j}\sum_{i\geq j}a_{i}+4\pi\alpha_{j}\sum_{i\geq
j+1}a_{i}-8\pi\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\sum_{i=1}^{m}a_{i}-2\beta_{j}\mu_{1}\sum_{i\leq
j}(-1)^{i+1}-2\alpha_{j}\mu_{1}\sum_{i>j}(-1)^{i+1}\frac{\beta_{i}}{\alpha_{i}}\\\
&\,+2\beta_{j}
\frac{\alpha_{1}-2}{\beta_{1}+1}\mu_{1}\sum_{i=1}^{m}(-1)^{i+1}\frac{\beta_{i}}{\alpha_{i}}=4\pi\alpha_{j}\sum_{i\geq
j}a_{i}+4\pi\alpha_{j}\sum_{i\geq
j+1}a_{i}-8\pi\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\sum_{i=1}^{m}a_{i}\\\
&\,+\mu_{1}\bigg{[}-2\beta_{j}\frac{1+(-1)^{j+1}}{2}-2\alpha_{j}\sum_{i>j}(-1)^{i+1}\frac{\beta_{i}}{\alpha_{i}}+2\beta_{j}
\frac{\alpha_{1}-2}{\beta_{1}+1}\sum_{i=1}^{m}(-1)^{i+1}\frac{\beta_{i}}{\alpha_{i}}\bigg{]},\end{split}$

in view of (5.10). For simplicity we shall denote for $j=1,\dots,m$

$x_{j}=\sum_{i=j}^{m}a_{i},\qquad
A_{j}=-\frac{\beta_{j}}{2\pi}\frac{1+(-1)^{j+1}}{2}-\frac{\alpha_{j}}{2\pi}\sum_{i>j}(-1)^{i+1}\frac{\beta_{i}}{\alpha_{i}}+\frac{\beta_{j}}{2\pi}
\frac{\alpha_{1}-2}{\beta_{1}+1}B,$ $\text{and}\quad
A_{m+1}=-\frac{1}{2\pi}\Big{(}-\frac{\beta_{1}+1}{\alpha_{1}-2}+B\Big{)},\quad\text{with}\quad
B=\sum_{i=1}^{m}(-1)^{i+1}\frac{\beta_{i}}{\alpha_{i}}.$

Hence, taking into account (5.9) and dividing by $4\pi$, we have the following
linear system in $m+1$ variables $X=[x_{2}\ x_{3}\ \dots\ x_{m}\ x_{1}\
\mu_{1}]^{T}$

$\begin{split}0=&\,\alpha_{j}x_{j}+\alpha_{j}x_{j+1}-2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}x_{1}+A_{j}\mu_{1},\quad
j=2,\dots,m-1\\\
0=&\,\alpha_{m}x_{m}-2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}x_{1}+A_{m}\mu_{m}\\\
0=&\,\alpha_{1}x_{2}+\Big{(}\alpha_{1}-2\beta_{1}\frac{\alpha_{1}-2}{\beta_{1}+1}\Big{)}x_{1}+A_{1}\mu_{1}\\\
0=&\,2x_{1}+A_{m+1}\mu_{1}\end{split}$

which in matrix form becomes $AX=\vec{0}$ with

$A=\left[\begin{array}[]{ccccccc}\alpha_{2}&\alpha_{2}&0&\cdots&0&-2\beta_{2}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{2}\\\
0&\alpha_{3}&\alpha_{3}&\cdots&0&-2\beta_{3}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{3}\\\
0&0&\alpha_{4}&\cdots&0&-2\beta_{4}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{4}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&\cdots&\alpha_{m}&-2\beta_{m}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{m}\\\
\alpha_{1}&0&0&\cdots&0&\alpha_{1}-2\beta_{1}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{1}\\\
0&0&0&\cdots&0&2&A_{m+1}\end{array}\right]=\left[\begin{array}[]{c}f_{1}\\\
f_{2}\\\ \vdots\\\ f_{m}\\\ f_{m+1}\end{array}\right]$

Operating with rows $f_{j}$’s of $A$ in the following form

$f_{m}\to f_{m}+(-1)^{j}\frac{\alpha_{1}}{\alpha_{j+1}}f_{j},\qquad
j=1,\dots,m-1$

we are lead to consider the matrix

$\left[\begin{array}[]{ccccccc}\alpha_{2}&\alpha_{2}&0&\cdots&0&-2\beta_{2}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{2}\\\
0&\alpha_{3}&\alpha_{3}&\cdots&0&-2\beta_{3}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{3}\\\
0&0&\alpha_{4}&\cdots&0&-2\beta_{4}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{4}\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\\
0&0&0&\cdots&\alpha_{m}&-2\beta_{m}\frac{\alpha_{1}-2}{\beta_{1}+1}&A_{m}\\\
0&0&0&\cdots&0&\alpha_{1}-2\alpha_{1}B\frac{\alpha_{1}-2}{\beta_{1}+1}&\frac{\alpha_{1}}{2\pi}\Big{(}-1+\frac{\alpha_{1}-2}{\beta_{1}+1}B\Big{)}\\\
0&0&0&\cdots&0&2&A_{m+1}\end{array}\right]$

since

$\alpha_{1}-2\beta_{1}\frac{\alpha_{1}-2}{\beta_{1}+1}+\sum_{j=1}^{m-1}(-1)^{j}\frac{\alpha_{1}}{\alpha_{j+1}}(-2\beta_{j+1})\frac{\alpha_{1}-2}{\beta_{1}+1}=\alpha_{1}-2\alpha_{1}B\frac{\alpha_{1}-2}{\beta_{1}+1}$

and

$\begin{split}&A_{1}+\sum_{j=1}^{m-1}(-1)^{j}\frac{\alpha_{1}}{\alpha_{j+1}}A_{j+1}=-\frac{\alpha_{1}}{2\pi}\bigg{(}\underbrace{\frac{\beta_{1}}{\alpha_{1}}+\sum_{i=2}^{m}(-1)^{i+1}\frac{\beta_{i}}{\alpha_{i}}}_{=B}\bigg{)}+\underbrace{\frac{\beta_{1}}{2\pi}
\frac{\alpha_{1}-2}{\beta_{1}+1}B+\frac{\alpha_{1}}{2\pi}\frac{\alpha_{1}-2}{\beta_{1}+1}B\sum_{j=1}^{m-1}(-1)^{j}\frac{\beta_{j+1}}{\alpha_{j+1}}}_{=\frac{\alpha_{1}}{2\pi}\frac{\alpha_{1}-2}{\beta_{1}+1}B^{2}}\\\
&\,+\frac{\alpha_{1}}{2\pi}\underbrace{\sum_{j=1}^{m-1}(-1)^{j}\bigg{[}-\frac{\beta_{j+1}}{\alpha_{j+1}}\frac{1+(-1)^{j}}{2}+\sum_{i>j+1}(-1)^{i}\frac{\beta_{i}}{\alpha_{i}}\bigg{]}}_{=0}=\frac{\alpha_{1}}{2\pi}B\Big{(}-1+\frac{\alpha_{1}-2}{\beta_{1}+1}B\Big{)}.\end{split}$

Notice that $A$ is invertible, since

$\text{det}\left[\begin{array}[]{cc}\alpha_{1}-2\alpha_{1}B\frac{\alpha_{1}-2}{\beta_{1}+1}&\frac{\alpha_{1}}{2\pi}\Big{(}-1+\frac{\alpha_{1}-2}{\beta_{1}+1}B\Big{)}\\\
2&A_{m+1}\end{array}\right]=\frac{\alpha_{1}}{4\pi}\cdot\frac{\beta_{1}+1}{\alpha_{1}-2}>0.$

Therefore, $x_{2}=x_{3}=\dots=x_{m}=x_{1}=\mu_{1}=0$, which implies our claim.
∎

Now, by using Claims 5.1-5.3 we deduce that $\Phi_{j,n}$ converges to zero
weakly in $H_{\alpha_{j}}(\text{I\\!R}^{2})$ and strongly in
$L_{\alpha_{j}}(\text{I\\!R}^{2})$ as $n\to+\infty$. Thus, we arrives at a
contradiction with (5.5) and it follows the priori estimate $\|\phi\|\leq
C|\log\rho|\,\|h\|_{p}$. It only remains to prove the solvability assertion.
As usual, expressing (3.17) in weak form, with the aid of Riesz’s
representation theorem and Fredholm’s alternative, the existence of a unique
solution follows from the a priori estimate (3.18). This finishes the proof of
Proposition 3.1. ∎

###### Proof of the Proposition 4.1.

Arguing as above, it is enough to prove the a-priori estimate (4.18). Let us
assume by contradiction the existence of $p>1$, sequences $\rho_{n}\to 0$,
functions $h_{n}\in L^{p}(\Omega_{\epsilon_{n}})$, $\phi_{n}\in
W^{2,2}(\Omega_{\epsilon_{n}})$ such that

$\mathcal{L}(\phi_{n})=h_{n}\ \ \text{in}\ \ \Omega_{\epsilon_{n}},\ \
\phi_{n}=0\ \ \text{on}\ \ \partial\Omega_{\epsilon_{n}},$ (5.17)

with $\|\phi_{n}\|=1$ and $|\log\rho_{n}|\ \|h_{n}\|_{p}=o(1)$ as
$n\to+\infty$. We will omit the subscript $n$ in $\delta_{i,n}=\delta_{i}$.
Recall that $\delta_{i}^{\alpha_{i}}=d_{i,n}\rho_{n}^{\beta_{i}}$. Now, define
$\Phi_{i,n}(y):=\phi_{n}(\delta_{i}y)$ for
$y\in\Omega_{i,n}:=\delta_{i}^{-1}\Omega_{\epsilon_{n}}$, $i=1,\dots,m$ and
extend $\phi_{n}=0$ in $\text{I\\!R}^{2}\setminus\Omega_{\epsilon_{n}}$.
Arguing in the same way as in [13, Claim 1, section 4], we can prove the
following fact. We provide a sketch of the proof.

######  Claim 5.4.

The sequence $\\{\Phi_{i,n}\\}_{n}$ converges (up to a subsequence) to
$\Phi_{i}^{*}$ weakly in $H_{\alpha_{i}}(\text{I\\!R}^{2})$ and strongly in
$L_{\alpha_{i}}(\text{I\\!R}^{2})$.

###### Proof:.

First, we shall show that the sequence $\\{\Phi_{i,n}\\}_{n}$ is bounded in
$H_{\alpha_{i}}(\text{I\\!R}^{2})$. Notice that
$\|\Phi_{i,n}\|_{H^{1}_{0}(\Omega_{i,n})}=1$ for $i=1,\dots,m$. Thus, we want
to prove that there is a constant $M>0$ such for all $n$ (up to a subsequence)
$\|\Phi_{i,n}\|_{L_{\alpha_{i}}}^{2}\leq M.$ Notice that for any
$i\in\\{1,\dots,m\\}$ we find that in $\Omega_{i,n}$

$\begin{split}\Delta\Phi_{i,n}&+\delta_{i}^{2}K_{0}(\delta_{i}y)\left(\Phi_{i,n}+\tilde{c}_{0,n}\right)+\delta_{i}^{2}K_{1}(\delta_{i}y)\left(\Phi_{i,n}+\tilde{c}_{1,n}\right)=\delta_{i}^{2}h_{n}(\delta_{i}y),\end{split}$
(5.18)

where for simplicity we denote $\tilde{c}_{j,n}=\tilde{c}_{j}(\phi_{n})$, with
$\tilde{c}_{j}$ given by (5.1). Furthermore, it follows that
$\Phi_{i,n}\to\Phi_{i}^{*}$ weakly in $H^{1}_{0}(\Omega_{i,n})$ and strongly
in $L^{p}(K)$ for any $K$ compact sets in $\mathbb{R}^{2}$. Now, let $\chi$ a
smooth function with compact support in $\mathbb{R}^{2}.$ We multiply (5.18)
by $\chi$, integrate by parts and we obtain that

$\begin{split}-\int_{\Omega_{i,n}}&\nabla\Phi_{i,n}\nabla\chi+\int_{\Omega_{i,n}}\bigg{[}\displaystyle{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+o(1)\bigg{]}\Phi_{i,n}\chi+\tilde{c}_{0,n}\int_{\Omega_{i,n}}\bigg{[}\displaystyle{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+o(1)\bigg{]}\chi\\\
&+\int_{\Omega_{i,n}}o(1)\Phi_{i,n}\chi+\tilde{c}_{1,n}\int_{\Omega_{i,n}}o(1)\chi=\int_{\Omega_{i,n}}\delta_{i}^{2}h_{n}(\delta_{i}y)\chi\end{split}$

for $i$ odd and

$\begin{split}-\int_{\Omega_{i,n}}&\nabla\Phi_{i,n}\nabla\chi+\int_{\Omega_{i,n}}o(1)\Phi_{i,n}\chi+\tilde{c}_{0,n}\int_{\Omega_{i,n}}o(1)\chi+\int_{\Omega_{i,n}}\bigg{[}\displaystyle{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+o(1)\bigg{]}\Phi_{i,n}\chi\\\
&+\tilde{c}_{1,n}\int_{\Omega_{i,n}}\bigg{[}\displaystyle{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+o(1)\bigg{]}\chi=\int_{\Omega_{i,n}}\delta_{i}^{2}h_{n}(\delta_{i}y)\chi\end{split}$

for $i$ even, in view of

$\delta_{i}^{2}K_{0}(\delta_{i}y)=\begin{cases}\displaystyle{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+O\bigg{(}\sum_{j<i\atop
j\text{
odd}}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{j}}+\sum_{i<j\atop
j\text{
odd}}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{j}}\bigg{)}&\text{if
$i$ is odd}\\\\[17.07182pt] \displaystyle O\bigg{(}\sum_{j<i\atop j\text{
odd}}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{j}}+\sum_{i<j\atop
j\text{
odd}}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{j}}\bigg{)}&\text{ if
$i$ is even}\end{cases}$ (5.19)

and

$\delta_{i}^{2}K_{1}(\delta_{i}y)=\begin{cases}\displaystyle
O\bigg{(}\sum_{j<i\atop j\text{
even}}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{j}}+\sum_{i<j\atop
j\text{
even}}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{j}}\bigg{)}&\text{ if
$i$ is odd}\\\\[17.07182pt]
\displaystyle{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+O\bigg{(}\sum_{j<i\atop
j\text{
even}}\Big{(}{\delta_{j}\over\delta_{i}}\Big{)}^{\alpha_{j}}+\sum_{i<j\atop
j\text{
even}}\Big{(}{\delta_{i}\over\delta_{j}}\Big{)}^{\alpha_{j}}\bigg{)}&\text{if
$i$ is even}\end{cases}$ (5.20)

uniformly on compact subsets of $\text{I\\!R}^{2}\setminus\\{0\\}$. We re-
write the system for $\tilde{c}_{0,n}$ and $\tilde{c}_{1,n}$ as a diagonal
dominant one as $n\to+\infty$

$\begin{split}\tilde{c}_{0,n}\int_{\Omega_{i,n}}\bigg{[}{2\alpha_{i}^{2}|y|^{\alpha_{i}-2}\over(1+|y|^{\alpha_{i}})^{2}}+o(1)\bigg{]}\chi+o(1)\tilde{c}_{1,n}=&\,O(1)\\\\[11.38092pt]
o(1)\
\tilde{c}_{0,n}+\tilde{c}_{1,n}\int_{\Omega_{j,n}}\bigg{[}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}+o(1)\bigg{]}\chi=&\,O(1),\end{split}$

choosing $i$ odd and $j$ even. Thus, if we choose $\chi$ so that
$\int_{\mathbb{R}^{2}}{2\alpha_{k}^{2}|y|^{\alpha_{k}-2}\over(1+|y|^{\alpha_{k}})^{2}}\chi\
dy\not=0$ for $k=i,j$ then we obtain that $\tilde{c}_{i,n}=O(1)$, for $i=0,1$.
Now, we multiply (5.18) by $\Phi_{i,n}$ for any $i\in\\{1,\dots,m\\}$,
integrate by parts and we get

$\sum_{i=1}^{m}2\alpha_{i}^{2}\|\Phi_{i,n}\|_{L_{\alpha_{i}}}^{2}=\,1+\lambda_{0}(\tilde{c}_{0,n})^{2}+\lambda_{1}\tau^{2}(\tilde{c}_{1,n})^{2}+o(1).$
(5.21)

Therefore, the sequence $\\{\Phi_{i,n}\\}_{n}$ is bounded in
$H_{\alpha_{i}}(\text{I\\!R}^{2})$, so that there is a subsequence
$\\{\Phi_{i,n}\\}_{n}$ and functions $\Phi_{i}^{*}$, $i=1,\dots,m$ such that
$\\{\Phi_{i,n}\\}_{n}$ converges to $\Phi_{i}^{*}$ weakly in
$H_{\alpha_{i}}(\text{I\\!R}^{2})$ and strongly in
$L_{\alpha_{i}}(\text{I\\!R}^{2})$. That proves our claim. ∎

Define the sequences $\psi_{i,n}=\phi_{n}+\tilde{c}_{i,n}$, $i=0,1$. Notice
that clearly

$\Delta\psi_{i,n}+K_{0}\psi_{1,n}+K_{1}\psi_{2,n}=h_{n}\quad\text{in}\ \
\Omega_{\epsilon_{n}},\qquad i=0,1.$ (5.22)

Now, define $\Psi_{i,j,n}(y):=\psi_{i,n}(\delta_{j}y)$ for $y\in\Omega_{j,n}$,
$i=0,1$ and $j=1,\dots,m$. Note that
$\Psi_{i,j,n}=\Phi_{j,n}+\tilde{c}_{i,n}$. Thus, we can prove the following
fact.

######  Claim 5.5.

$\Psi_{0,j,n}\to a_{j}Y_{0j}$ for $j$ odd and $\Psi_{1,j,n}\to a_{j}Y_{0j}$
for $j$ even, weakly in $H_{\alpha_{j}}(\text{I\\!R}^{2})$ and strongly in
$L_{\alpha_{j}}(\text{I\\!R}^{2})$ as $n\to+\infty$ for some constant
$a_{j}\in\text{I\\!R}$, $j=1,\dots,m$.

###### Proof:.

From the previous computations, it is clear that in $\Omega_{j,n}$

$\Delta\Psi_{0,j,n}+\delta_{j}^{2}K_{0}(\delta_{j}y)\Psi_{0,j,n}+\delta_{j}^{2}K_{1}(\delta_{j}y)\left(\Psi_{0,j,n}-\tilde{c}_{0,n}+\tilde{c}_{1,n}\right)=\delta_{j}^{2}h_{n}(\delta_{j}y)$

and

$\Delta\Psi_{1,j,n}+\delta_{j}^{2}K_{0}(\delta_{j}y)\left(\Psi_{1,n}-\tilde{c}_{1,n}+\tilde{c}_{0,n}\right)+\delta_{j}^{2}K_{1}(\delta_{j}y)\Psi_{1,j,n}=\delta_{j}^{2}h_{n}(\delta_{j}y).$

Furthermore, $\\{\tilde{c}_{i,n}\\}$ is a bounded sequence in IR, so it
follows that $\\{\Psi_{i,j,n}\\}_{n}$ is bounded in
$H_{\alpha_{j}}(\text{I\\!R}^{2})$ for $i=1,2$ and $j=1,\dots,m$. Also, we
have that

$\int_{\Omega_{j,n}}(\delta_{j}^{2}|h_{n}(\delta_{j}y)|)^{p}\,dy=\delta_{j}^{2p-2}\int_{\Omega_{\epsilon_{n}}}|h_{n}(x)|^{p}\,dx=\delta_{i}^{2p-2}\|h_{n}\|_{p}^{p}=o(1).$

Therefore, taking into account (5.19) and (5.20) we deduce that
$\Psi_{i,j,n}\to\Psi_{j}^{*}$ as $n\to+\infty$ with $i=1$ if $j=1,\dots,m_{1}$
and $i=2$ if $j=m_{1}+1,\dots,m$, where $\Psi^{*}_{j}$ is a solution to

$\Delta\Psi+{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi=0,\qquad
j=1,\dots,m,\qquad\text{in $\text{I\\!R}^{2}\setminus\\{0\\}$}.$

It is standard that $\Psi_{j}^{*}$, $j=1,\dots,m$, extends to a solution in
the whole $\text{I\\!R}^{2}$. Hence, by using symmetry assumptions if
necessary, we get that $\Psi^{*}_{j}=a_{j}Y_{0j}$ for some constant
$a_{j}\in\text{I\\!R}$, $j=1,\dots,m$. ∎

Denote $\tilde{c}_{i}=\displaystyle\lim_{n\to+\infty}\tilde{c}_{i,n}$ for
$i=0,1$, up to a subsequence if necessary. Hence, we get that

$\Phi_{j,n}\to a_{j}Y_{0j}-\tilde{c}_{0},\ \text{ for $j$
odd}\quad\text{and}\quad\Phi_{j,n}\to a_{j}Y_{0j}-\tilde{c}_{1},\ \text{ for
$j$ even,}$ (5.23)

weakly in $H_{\alpha_{j}}(\text{I\\!R}^{2})$ and strongly in
$L_{\alpha_{j}}(\text{I\\!R}^{2})$, since
$\Phi_{j,n}=\Psi_{i,j,n}-\tilde{c}_{i,n}$.

######  Claim 5.6.

For all $j=1,\dots,m$, there exist the limit, up to subsequences,

$\mu_{j}=\lim_{n\to+\infty}\log\rho_{n}\int_{\Omega_{j,n}}\Pi_{j}(y)\Psi_{l,j,n}(y)\,dy=0,$
(5.24)

where $l=0$ for $j$ odd and $l=1$ for $j$ even. Furthermore, it holds

$\begin{split}\sum_{i=1}^{m}a_{i}=0.\end{split}$ (5.25)

###### Proof:.

To this aim we use the test function $P_{\epsilon}Z_{0j}$, where $Z_{0j}$ as
in Claim 5.3. Thus, from the assumption on $h_{n}$, $|\log\rho_{n}|\
\|h_{n}\|_{*}=o(1)$, we get (5.24)-(5.25).

Assume that either $l=0$ for all $j$ odd or $l=1$ for all $j$ even.
Multiplying equation (5.17) by $P_{\epsilon}Z_{0j}$ and integrating by parts
we obtain that

$\begin{split}\int_{\Omega_{\epsilon}}hP_{\epsilon}Z_{0j}=&\
\int_{\Omega_{\epsilon}}\Delta
Z_{0j}\left[\psi_{l}-\tilde{c}_{l,n}\right]+\int_{\Omega_{\epsilon}}\left[K_{0}\psi_{0}+K_{1}\psi_{1}\right]P_{\epsilon}Z_{0j},\end{split}$

in view of $P_{\epsilon}Z_{0j}=0$ and $\psi_{l}=\tilde{c}_{l,n}$ on
$\partial\Omega_{\epsilon}$ and

$\begin{split}\int_{\Omega_{\epsilon}}\Delta\psi_{l}P_{\epsilon}Z_{0j}&=\int_{\Omega_{\epsilon}}\psi_{l}\Delta
P_{\epsilon}Z_{0j}-\psi_{l}\bigg{|}_{\partial\Omega_{\epsilon}}\int_{\Omega_{e}}\Delta
PZ_{0j}=\int_{\Omega_{\epsilon}}\Delta
Z_{0j}\left[\psi_{l}-\tilde{c}_{l,n}\right].\end{split}$

Furthermore, we have that

$\begin{split}&\int_{\Omega_{\epsilon}}hPZ_{0j}=-\int_{\Omega_{\epsilon}}\left[\psi_{l}-\tilde{c}_{l,n}\right]|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}+\int_{\Omega_{\epsilon}}\left[K_{0}\psi_{0}+K_{1}\psi_{1}\right]P_{\epsilon}Z_{0j}\\\
&=\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\psi_{l}\left(PZ_{0j}-Z_{0j}\right),+\tilde{c}_{l,n}\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}+\int_{\Omega_{\epsilon}}\left(K_{0}\psi_{0}+K_{1}\psi_{1}-|x|^{\alpha_{j}-2}e^{w_{j}}\psi_{l}\right)P_{\epsilon}Z_{0j}.\end{split}$

Multiplying by $\gamma_{0}^{-1}$ and arguing in the same way as in Claim 5.2
(replacing $\Psi_{l,i,n}$ by $\Phi_{i,n}$ with $l=0$ for $j$ odd and $l=1$ for
$j$ even in (5.12) and (5.13)) we deduce

$\begin{split}2\sum_{i=1}^{m}a_{i}=\bigg{(}-\frac{\beta_{1}+1}{4\pi(\alpha_{1}-2)}+\sum_{i=1}^{m}(-1)^{i+1}\frac{\beta_{i}}{2\pi\alpha_{i}}\bigg{)}\mu_{1},\end{split}$

and $\mu_{j}=(-1)^{j-1}\mu_{1}$ for all $j=1,\dots,m-1$, in view of

$\gamma_{0}^{-1}\int_{\Omega_{\epsilon}}hPZ_{0j}=O\left(\,|\log\rho|\,\|h\|_{p}\right)=o\left(1\right)\quad\text{and}\quad\gamma_{0}^{-1}\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}=O(\rho^{\tilde{\sigma}}|\log\rho|).$

On the other hand, it is readily checked that

$\begin{split}\int_{\Omega_{\epsilon}}K_{0}&=\sum_{k=1\atop k\text{
odd}}^{m}\int_{\Omega_{\epsilon}}|x|^{\alpha_{k}-2}e^{w_{k}}\
dx=\sum_{k=1\atop k\text{
odd}}^{m}\left[4\pi\alpha_{k}+O(\delta_{k}^{\alpha_{k}})+O\Big{(}{\epsilon^{2}\over\delta_{k}^{2}}\Big{)}\right]=\lambda_{0}+O(\rho^{\tilde{\sigma}})\end{split}$

and similarly
$\int_{\Omega_{\epsilon}}K_{1}=\lambda_{2}\tau^{2}+O(\rho^{\tilde{\sigma}})$
for some $\tilde{\sigma}>0$, so that for $\psi_{0}$ and $\psi_{1}$ we have
that

$\int_{\Omega_{\epsilon}}K_{i}\psi_{i}=\left(1-{1\over\lambda_{i}\tau^{2i}}\int_{\Omega_{\epsilon}}K_{i}\right)\int_{\Omega_{\epsilon}}K_{i}\phi=O(\rho^{\tilde{\sigma}})\int_{\Omega_{\epsilon}}K_{i}\phi=O(\rho^{\tilde{\sigma}}).$

Also, we get that

$\int_{\Omega_{\epsilon}}K_{0}\psi_{0}=\sum_{j=1\atop j\text{
odd}}^{m}\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\psi_{i}=\sum_{j=1\atop
j\text{
odd}}^{m_{1}}\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{0,j,n}(y)\,dy$

and

$\int_{\Omega_{\epsilon}}K_{1}\psi_{1}=\sum_{j=1\atop j\text{
even}}^{m}\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{1,j,n}(y)\,dy.$

Hence, it follows that

$\lim_{n\to+\infty}\log\rho_{n}\int_{\Omega_{\epsilon_{n}}}K_{0}\psi_{0}=0=\sum_{j=1\atop
j\text{ odd}}^{m}\mu_{j}=\mu_{1}\sum_{j=1\atop j\text{ odd}}^{m}1,$

so that, $\mu_{j}=0$ for all $j=1,\dots,m$. ∎

######  Claim 5.7.

There hold that

$0=\tilde{c}_{l}+a_{j}+2\sum_{i>j}a_{i},$ (5.26)

where $l=0$ for $j$ odd and $l=1$ for $j$ even.

###### Proof:.

To this aim we will use as tests functions $P_{\epsilon}w_{j}$. Hence,
multiplying equation (5.4) by $P_{\epsilon}w_{j}$ and integrating by parts as
in the previous Claim we obtain that

$\begin{split}\int_{\Omega_{\epsilon}}hPw_{j}=&\,-\int_{\Omega_{\epsilon}}[\psi_{l}-\tilde{c}_{l,n}]|x|^{\alpha_{j}-2}e^{w_{j}}+\int_{\Omega_{\epsilon}}[K_{0}\psi_{0}+K_{1}\psi_{j}]P_{\epsilon}w_{j}\\\
=&\,-\int_{\Omega_{\epsilon}}\psi_{l}|x|^{\alpha_{j}-2}e^{w_{j}}+\tilde{c}_{l,n}\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}+\int_{\Omega_{\epsilon}}[K_{0}\psi_{0}+K_{1}\psi_{1}]P_{\epsilon}w_{j}\end{split}$

in view of $P_{\epsilon}w_{j}=0$ on $\partial\Omega_{\epsilon_{n}}$ and
$\int_{\Omega_{\epsilon}}\Delta\psi_{l}P_{\epsilon}w_{j}=\int_{\Omega_{\epsilon}}\psi_{l}\;\Delta
P_{\epsilon}w_{j}$. Arguing in the same way as in Claim 5.3 (replacing
$\Psi_{l,i,n}$ by $\Phi_{i,n}$ with $l=0$ for $j$ odd and $l=1$ for $j$ even
in (5.15) and (5.16)) we find that

$\begin{split}0=&\,4\pi\alpha_{j}\tilde{c}_{l}+4\pi\alpha_{j}a_{j}+\sum_{i\leq
j}\Big{(}-2\beta_{j}+2\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\mu_{i}-8\pi\beta_{j}\frac{\alpha_{1}-2}{\beta_{1}+1}\sum_{i=1}^{m}a_{i}\\\
&\,+\sum_{i>j}\Big{(}-2\alpha_{j}\frac{\beta_{i}}{\alpha_{i}}+2\beta_{j}
\frac{\alpha_{1}-2}{\beta_{1}+1}\frac{\beta_{i}}{\alpha_{i}}\Big{)}\mu_{i}+8\pi\alpha_{j}\sum_{i>j}a_{i},\end{split}$

in view of

$\int_{\Omega_{\epsilon}}hPw_{j}=O\left(\,|\log\rho|\,\|h\|_{p}\right)=o\left(1\right),$
$\int_{\Omega_{\epsilon}}\psi_{l}|x|^{\alpha_{j}-2}e^{w_{j}}=\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{l,j,n}=a_{j}\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}\,dy+o(1)=o(1)$

and

$\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}=\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\,dy=\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\,dy+o(1)=4\pi\alpha_{j}+o(1).$

By using (5.24) and (5.25) we obtain that (5.26) for $l=0$ with $j$ odd and
$l=1$ with $j$ even. ∎

For the next step, consider the function $\eta_{j}(x)={4\over
3}\log(\delta_{j}^{\alpha_{j}}+|x|^{\alpha_{j}}){\delta_{j}^{\alpha_{j}}-|x|^{\alpha_{j}}\over\delta_{j}^{\alpha_{j}}+|x|^{\alpha_{j}}}+{8\over
3}{\delta_{j}^{\alpha_{j}}\over\delta_{j}^{\alpha_{j}}+|x|^{\alpha_{j}}},$ for
any $j\in\\{1,\dots,m\\}$ so that
$\Delta\eta_{j}+|x|^{\alpha_{j}-2}e^{w_{j}}\eta_{j}=|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}.$
Notice that

$P_{\epsilon}\eta_{j}=\eta_{j}+{8\pi\over
3}\alpha_{j}H(x,0)-\tilde{\gamma}_{j}G(x,0)+O(\rho^{\tilde{\sigma}})\quad\text{uniformly
in $\Omega_{\epsilon}$ for some $\tilde{\sigma}>0$},$

by using similar arguments as to obtain expansion in Lemma 2.1, as shown in
[13, Lemma 4.1] with $m=1$ and $\xi=0$, where the coefficients
$\tilde{\gamma}_{i}$’s, $i=1,\dots,m$, are given by

$\tilde{\gamma}_{i}\left[-{1\over 2\pi}\log\epsilon+H(0,0)\right]={4\over
3}\alpha_{i}\log\delta_{i}+{8\over 3}+{8\pi\over 3}\alpha_{i}H(0,0)$ (5.27)

From (2.2) it follows that
$\tilde{\gamma}_{i}=-\frac{8\pi(\alpha_{1}-2)\beta_{i}}{3(\beta_{1}+1)}+O\big{(}{1\over|\log\rho|}\big{)},$.

######  Claim 5.8.

There hold that $a_{j}+4\sum_{i>j}a_{i}+2\tilde{c}_{l}=0$ with $l=0$ for all
$j$ odd and $l=1$ for all $j$ even. Consequently, combining with (5.26) it
follows that $a_{i}=0$ for all $i=1,\dots,m$.

###### Proof:.

We use the following test function $P_{\epsilon}\eta_{j}$. Thus, from the
assumption on $h_{n}$, $|\log\rho_{n}|\ \|h_{n}\|_{*}=o(1)$, we get the above
relation between $a_{j}$ and $\tilde{c}_{i}$ either for $l=0$ and all $j$ odd
or for $l=1$ and all $j$ even. Assume that $l=0$ for all $j$ odd or $l=1$ for
all $j$ even. Multiplying equation (5.17) by $P_{\epsilon}\eta_{j}$ and again
integrating by parts we obtain that

$\begin{split}&\int_{\Omega_{\epsilon}}hP\eta_{j}\\!=\\!\int_{\Omega_{\epsilon}}\\!\left[\psi_{l}-\tilde{c}_{l,n}\right]\Big{[}|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}-|x|^{\alpha_{j}-2}e^{w_{j}}\eta_{j}\Big{]}\\!+\\!\int_{\Omega_{\epsilon}}\\!\left[K_{0}\psi_{0}+K_{1}\psi_{1}\right]P_{\epsilon}\eta_{j}\\!=\\!\int_{\Omega_{\epsilon}}\\!\psi_{i}|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}\\\
&+\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\psi_{l}\left(P\eta_{j}-\eta_{j}\right)+\int_{\Omega_{\epsilon}}\left(K_{0}\psi_{0}+K_{1}\psi_{1}-|x|^{\alpha_{j}-2}e^{w_{j}}\psi_{l}\right)P_{\epsilon}\eta_{j}-\tilde{c}_{l,n}\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\left(Z_{0j}-\eta_{j}\right),\end{split}$

Now, estimating every integral term we find that
$\int_{\Omega_{\epsilon}}hP\eta_{j}=O\left(\,|\log\rho|\,\|h\|_{p}\right)=o\left(1\right)$
for all $j=1,\dots,m$, in view of $P\eta_{j}=O(|\log\rho|)$ and
$G(x,0)=O(|\log\rho|)$. Next, by scaling we obtain that either for $l=0$ and
all $j$ odd or $l=1$ and all $j$ even, it holds

$\int_{\Omega_{\epsilon}}\psi_{l}|x|^{\alpha_{j}-2}e^{w_{j}}Z_{0j}=\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{l,j,n}Y_{0j}\,dy=a_{j}\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}^{2}\,dy+o(1).$

Note that

$\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}^{2}=\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\left({1-|y|^{\alpha_{j}}\over
1+|y|^{\alpha_{j}}}\right)^{2}dy={4\pi\over 3}\alpha_{j}$

and

$\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}\log|y|=\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\
{1-|y|^{\alpha_{j}}\over 1+|y|^{\alpha_{j}}}\ \log|y|\,dy=-4\pi.$

Also, by using (5.27) we get that

$\begin{split}&\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\psi_{l}\left(P_{\epsilon}\eta_{j}-\eta_{j}\right)=\int_{\Omega_{\epsilon}}|x|^{\alpha_{j}-2}e^{w_{j}}\psi_{l}\bigg{[}\left({8\pi\over
3}\alpha_{j}-\tilde{\gamma}_{j}\right)H(x,0)+{1\over
2\pi}\tilde{\gamma}_{j}\log|x|+O(\rho^{\tilde{\sigma}})\bigg{]}\\\
&=\left({8\pi\over
3}\alpha_{j}-\tilde{\gamma}_{j}\right)\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{l,j,n}H(\delta_{j}y,0)\,dy+{\tilde{\gamma}_{j}\over
2\pi}\log\delta_{j}\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{l,j,n}\,dy\\\
&\,+{\tilde{\gamma}_{j}\over
2\pi}\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{l,j,n}\log|y|dy+O(\rho^{\tilde{\sigma}})=-\frac{4(\alpha_{1}-2)\beta_{j}}{3(\beta_{!}+1)}a_{j}\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}\log|y|dy\\!+\\!o(1),\end{split}$

in view of (5.24). Furthermore, using (4.15) we have that

Notice that

$\int_{\Omega_{j,n}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\Psi_{i,j,n}(y)\,dy=a_{j}\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}(y)\,dy+o(1)=o(1),$

since

$\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}Y_{0j}(y)\,dy=\int_{\text{I\\!R}^{2}}{2\alpha_{j}^{2}|y|^{\alpha_{j}-2}\over(1+|y|^{\alpha_{j}})^{2}}\cdot{1-|y|^{\alpha_{j}}\over
1+|y|^{\alpha_{j}}}\,dy=0.$

If $i<j$ then

$\begin{split}P_{\epsilon}\eta_{j}(\delta_{i}y)=&\,\bigg{[}\frac{4}{3}\alpha_{j}\log\delta_{j}+\frac{4}{3}\log\Big{(}1+\Big{(}\frac{\delta_{i}|y|}{\delta_{j}}\Big{)}^{\alpha_{j}}\Big{)}\bigg{]}\frac{1-\big{(}\frac{\delta_{i}|y|}{\delta_{j}}\big{)}^{\alpha_{j}}}{1+\big{(}\frac{\delta_{i}|y|}{\delta_{j}}\big{)}^{\alpha_{j}}}+\frac{8}{3}\cdot\frac{1}{1+\big{(}\frac{\delta_{i}|y|}{\delta_{j}}\big{)}^{\alpha_{j}}}\\\
&\,+\frac{\tilde{\gamma}_{j}}{2\pi}\log(\delta_{i}|y|)+\Big{(}\frac{8\pi}{3}\alpha_{j}-\tilde{\gamma}_{j}\Big{)}H(\delta_{i}y,0)+O(\rho^{\tilde{\sigma}})\end{split}$

and

$\begin{split}\int_{\Omega_{\epsilon}}|x|^{\alpha_{i}-2}e^{w_{i}}\psi_{l}P_{\epsilon}\eta_{j}=&\,\int_{\Omega_{i,n}}\Pi_{i}\Psi_{l,i,n}P_{\epsilon}\eta_{j}(\delta_{i}y)=\frac{\tilde{\gamma}_{j}}{2\pi}\int_{\Omega_{i,n}}\Pi_{i}\Psi_{l,i,n}(y)\log|y|+o(1)\\\
=&\,\frac{16\pi(\alpha_{1}-2)\beta_{j}}{3(\beta_{1}+1)}a_{i}+o(1)\end{split}$

in view of $\frac{\delta_{i}|y|}{\delta_{j}}=o(1)$ uniformly for $y$ on
compact subsets, (5.24) and dominated convergence. Similarly, if $i>j$ then

$\begin{split}P_{\epsilon}\eta_{j}(\delta_{i}y)=&\,\bigg{[}\frac{4}{3}\alpha_{j}\log\delta_{j}+\frac{4}{3}\log\Big{(}\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}\Big{)}\bigg{]}\frac{\big{(}\frac{\delta_{j}}{\delta_{i}}\big{)}^{\alpha_{j}}-|y|^{\alpha_{j}}}{\big{(}\frac{\delta_{j}}{\delta_{i}}\big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}}+\frac{8}{3}\Big{(}\frac{\delta_{j}}{\delta_{i}}\Big{)}^{\alpha_{j}}\cdot\frac{1}{\big{(}\frac{\delta_{j}}{\delta_{i}}\big{)}^{\alpha_{j}}+|y|^{\alpha_{j}}}\\\