New  sign-changing solutions ]New  sign-changing solutions
for the 2D
Lane-Emden problem with large exponents
A. Pistoia]Angela Pistoia
[A. Pistoia] Dipartimento SBAI, Sapienza Università di Roma,
Via Antonio Scarpa 16, 00161 Rome, Italy. Phone: +39 06 4976 6677
T. Ricciardi]Tonia Ricciardi
[T. Ricciardi] Dipartimento di Matematica e Applicazioni,
Università di Napoli Federico II, Via Cintia, Monte S. Angelo, 80126 Napoli, Italy. Phone: +39 081 628802
Work partially supported by the MUR-PRIN-20227HX33Z “Pattern formation in nonlinear phenomena” and the INdAM-GNAMPA group

We construct  a new family of sign-changing solutions  for a two-dimensional Lane-Emden problem
with large exponent whose shape resembles a tower  with alternating sign of  bubbles  solving different singular Liouville equations on the whole plane.

[2000]

§ INTRODUCTION AND STATEMENT OF THE MAIN RESULTS

We consider the classical Lane-Emden problem:

-Δu=   |u|^p-1u      in

u=   0      on ,
where $\Om\subset\rr^2$ is a smooth bounded domain and $p>1$.

Standard variational tools ensure the existence of     infinitely many (possibly sign-changing) solutions of problem (<ref>).

In the last decade, a fruitful line of research has been the  study the asymptotic behaviour  of solutions to (<ref>)
as $p$ approaches $+\infty$.
In the pioneering works [22, 23], Ren and Wei   study the profile
of the least energy solution $u_p$ (which is positive) and prove that it has a single point concentration and converges at zero locally uniformly outside the concentration point. Later, El Mehdi and Grossi [10] and Adimurthi and Grossi [1]   identify a limit problem by showing that
a suitable scaling   converges to a radial solution of

\begin{equation}\label{plim}
-\Delta U=e^U\ \hbox{in}\ \rr^2,\ \int\limits_{\rr^2}e^U=8\pi.\end{equation}

Concerning general positive solutions (i.e. not necessarily with least energy) the first asymptotic analysis was carried out by
De Marchis,  Ianni and Pacella [7] (see also [8, 6]). More recently, other contributions have been given by Kamburov and Sirakov [20] and
Thizy [24]. We can summarize the known results as follows: the set of positive solutions
is uniformly bounded in $p$, each positive solution
concentrates at a finite number $k$ of points $x_1,\dots,x_k$ in $\Omega$,  converges to zero locally uniformly outside the concentration set
and suitable scalings of it
about each peak converge to the   bubble $U$, i.e  solution of the limit problem (<ref>).
Moreover, the $k-$upla of peaks $ (x_1,\dots,x_k)$ is a critical point of the
Kirchhoff-Routh $\Psi_k:{\mathcal D}\to\mathbb R$, with ${\mathcal D} =   \{ (x_1, \dots , x_k ) \in \Omega^k \, : \,
x_i \not =x_j  \}$ defined by

\begin{equation}\label{kr}
\Psi_k(x_1,\dots,x_k):=\sum\limits_{i=1,\dots,k}\sigma_i^2H(x_i,x_i)+\sum\limits_{i,j=1,\dots,k\atop i\not=j}\sigma_i\sigma_jG(x_i,x_j). \end{equation}

with all the $\sigma_i$'s equal to $+1.$

Here $G(x,y)$ is  the Green's function
and   $H(x,y)$
its regular part. That is,
-Δ_xG(x,y)=_y   x,y,∈, x≠y

G(x,y)=0   x∈∂, y∈and
We denote by

h(x)=H(x,x),    x∈,
the Robin function.

The first contribution to the analysis of the concentration phenomenon
for sign-changing solutions of problem (<ref>) as  $p\to+\infty$ is due to Grossi, Grumiau and Pacella in [13] who study  the profile and qualitative properties of the least energy nodal solution. In particular, the authors show that it  concentrates at two different points  $x_1$ and $x_2$ in $\Omega$
and up to  suitable scalings around
each peak it converges to a positive and a negative   bubble $U$  (see (<ref>)).
Moreover, the pair of peaks $ (x_1,x_2)$ is a critical point of the
Kirchhoff-Routh  function (<ref>) with $k=2$ and $\sigma_1=+1$, $\sigma_2=-1.$

Successively, the same  authors Grossi, Grumiau and Pacella in [14]
analyze the
asymptotic behavior of the least energy radial nodal solution  in the ball and prove that
its positive and negative parts  concentrate at the origin    and the limit profile looks like a tower of two bubbles given by a superposition of the   radial solution to (<ref>) and  the radial solution of the  singular Liouville problem in $\mathbb R^2$

\begin{equation}\label{plims}
-\Delta U=|x|^\beta e^U\ \hbox{in}\ \rr^2,\ \int\limits_{\rr^2}e^U=4\pi(\beta+2) \end{equation}

for a suitable non-integer and positive $\beta$.
Later, such a result  has been generalized   to other symmetric domains by De Marchis, Ianni and Pacella [7].
Recently, Ianni and Saldana [18] provide a complete and extremely accurate asymptotic analysis of the radial solutions of problem (<ref>) when the domain is the unit ball.
In particular, they prove that the radial solution with $m$ nodal lines looks like a superposition  (with alternating sign) of  the  radial  solution to (<ref>) and the radial solutions of $m$ different  Liouville problem (<ref>) with different non-integers and positive $\beta_1,\dots,\beta_m$.

As far as it concerns the existence of positive and sign-changing solutions with $k$ concentration points, Esposito, Musso and Pistoia in [11, 12]
proved that any non-degenerate critical point $ (x^*_1,\dots,x^*_k)$ of the Kirchhoff-Routh function (<ref>) with $\sigma_i\in\{-1,+1\}$,
generates a solutions with $k-$peaks
approaching the points $  x^*_1,\dots,x^*_k$  as $p$ is large enough being the peak $x^*_i$ positive or negative if $\sigma_i=+1$ or $\sigma_i=-1$, respectively.
In this regard, it is useful to point out that Micheletti and Pistoia [21] and Bartsch, Micheletti and Pistoia [2] proved that for   generic domains $\Omega$ all the critical points of the  Kirchhoff-Routh function  are non-degenerate. The limit profile of the solutions built in [11, 12] resembles the sum of $k$  bubbles solutions to (<ref>) concentrated in  the different points $  x^*_i$.
In particular,  in [11] it is also shown that if the domain $\Omega$ is has a rich geometry (namely it is not contractible or it has a dumbell shape)  positive multipeak   solutions can be found. On the contrary if the domain $\Omega$ is convex, positive solutions with more than one peak do not exist as proved by Grossi and Takahashi in [17]. The scenery of sign-changing solutions is completely different. Indeed,
in any domain $\Omega$ there exist  at least two pairs of solutions with one positive peak and one negative peak as proved  in [12]. In particular, Bartsch, Pistoia and Weth [3, 4] prove that  if the domain is symmetric with respect to the  line $\mathbb R\times \{0\}$
there exist  infinitely many sign-changing solutions whose peaks are  aligned on the symmetry line $\Omega\cap \mathbb R\times \{0\}$. We point out that in the case of the ball these solutions are not radially symmetric.

In the present paper, we will build a new type of sign-changing solutions in a general domain whose profile resembles the tower of bubbles as predicted in the radial case by [18].

For the sake of simplicity, we will assume that the domain $\Omega$ is symmetric with respect to the origin $0\in\Omega$, i.e. $x\in\Omega$ iff $-x\in\Omega.$
We aim to construct a sign-changing tower of peaks for (<ref>) concentrating at $0\in\Om$, in the spirit of [11, 16].

In order to state our main result, we briefly recall some known definitions and facts.

In view of the standard expansion:

(1+v/p)^p=e^v{1+O(1/p)}        as p→+∞,
for any fixed $v\in\rr$, the equation
$-\Delta v=v^p$ (corresponding to (<ref>) for $u\ge0$) may be viewed as a perturbation of the Liouville equation $-\Delta v=e^v$.
Therefore, a family of solutions $u_p$ to (<ref>), with a peak at $0\in\Om$ as $p\to+\infty$,  is expected to exist in the form

where $\tau,\de>0$ are parameters depending on $p$ with $\tau,\delta\to0$, $P$ denotes the standard projection on $H_0^1(\Om)$,

\[
\]

denotes the concentrating family of radial solutions to the Liouville equation (<ref>)
and $\omp$ is an error, see (<ref>) in the Appendix for details.
However, as observed in [11, 12], where multiple isolated peak solutions
to (<ref>) are constructed,
the first order expansion (<ref>)
is not sufficiently accurate,
so that two higher order correction terms $w^0$ and $w^1$ must be included in ansatz (<ref>).
More precisely, the third order expansion of the nonlinearity in (<ref>) is needed. For fixed $v,w^0,w^1\in\rr$ such
an expansion takes the form

where $\vphio,\vphii$ are the functions defined by

see Lemma <ref> in the Appendix for more details on this expansion.
Thus, ansatz (<ref>) is replaced by the following ansatz employed in [11, 12]:
where $w^0$, $w^1$ are the smooth radial solutions, whose existence is established in [5] (see also Lemma <ref> in the Appendix), to the linearized equations
and    Δw^1+e^vw^1=e^v(v,w^0)    on ^2,
w^ℓ(y)=C^ℓln|y|+O(1/|y|)    as |y|→+∞, ℓ=0,1.
The first order approximation for the tower of peaks
is given by a superposition of a finite number of   bubbles $\Udai$,
solutions to the singular Liouville problems

-ΔU=|y|^-2e^U      on ^2

∫_^2|y|^-2e^U dx=4π,
given by
where $\ai\ge2$, $\dei>0$ are suitably chosen and

\[
\vi(y)=\ln\frac{2\ai^2}{(1+|y|^{\ai})^2},\quad i=1,2,\ldots,k.
\]

The corresponding lower order corrections $\woai$, $\wiai$, are given by the radial solutions
to the linearized equations

Δ+|y|^-2e^=|y|^-2e^ (+ln|y|^-2),    on ^2,

Δ+|y|^-2e^=|y|^-2e^ (+ln|y|^-2,)
on ^2,

(y)=ln|y|+O(|y|^-1)    as  |y|→+∞,     ℓ=0,1,
whose existence is established in [5].
With this notation our main result may be stated as follows.

For any $k\in\mathbb N$ there exists a sufficiently large $p_0>0$ such that
for all $p\ge p_0$ there exists a sign-changing tower of peaks solution to
(<ref>) of the form $\up=\Up+\om_p$,
where, for $i=1,2,\ldots,k$, there holds $\ti=O(p^{-1})$, $2=\al_1<\al_2<\ldots<\ai<\al_{i+1}\ldots<\ak$,
$\dei=C_ie^{-b_ip}$ with $b_1>b_2>\ldots>b_k>0$ and $C_i>0$ and $\woai$, $\wiai$ are defined by

We point out that by the building process we easily deduce that the solution $\up=\Up+\om_p$ has $k$ nodal regions  which shrink to the origin as $p\to\infty.$
It is also worthwhile noticing that the main order term of our solution, i.e. the sum of   bubbles $\sum _{i}(-1)^i\tau_iP\Udai$ with alternating sign,  coincide  in the radial case with the profile   described by
Ianni and Saldana in <cit.>.

An interesting problem would be to show that all the  solutions to  (<ref>) having $k$ nodal regions shrinking to the origin as $p\to\infty$ look like the first order approximation term  $\Up$. It should be the first step to prove the local uniqueness of the nodal solution   in  the same spirit of  Grossi, Ianni, Luo and Yan [15]. We observe that the  unicity of the radial solution with $k$ nodal regions has been proved by Kajikiya in [19] using ODE techniques.

Our   result claims the existence of symmetric  sign-changing solutions which look like a tower of bubbles centered at the origin  being the domain $\Omega$ symmetric with respect to it.
The symmetry assumptions   simplify  the computations a lot because the linearized operator is invertible in the space of symmetric functions  and the solution can be found merely using a  contraction mapping argument. In general
we strongly believe that it is possible to carry out the construction around at any non-degenerate critical point $x_0$ of the Robin's function (<ref>).   This   could be managed  introducing  new parameters which are
the centers of the  bubbles  close to $x_0$,  which at the prices of heavy technicality should allow to perform a classical Ljapunov-Schmidt reduction.

This article is organized as follows.
Section <ref> is dedicated to the choice of the parameters $\aj,\dej,\tj$ appearing in Theorem <ref>. More precisely,
we first derive conditions for the parameters in the form of a nonlinear a system (see (<ref>)) which ensure smallness of the error" $\mathcal R_p:=\Delta\Up+\gp(\Up)$ with respect to
a suitable $L^\infty$-weighted norm $\rp$. By rather involved ad hoc arguments we prove solvability of the system and check that $\aj\not\in\mathbb N$ for $j\ge2$,
which is essential for the invertibility of the linearized operator.
In Section <ref> we carry out the details of the estimates of the error $\Rp$. To this end, we partition $\Om$ into shrinking annuli" $\Aj$ in the spirit of [16],
although our choice of the $\Aj$'s is more delicate.
Sections 4 to 7 are dedicated to the analysis of the linearized operator $\Lp=\Delta+\gp'(\Up)$.
In particular, Sections 4–5 are concerned with estimations in shrinking rings, and include delicate ad hoc arguments to compute the integrals involving radial eigenfunctions.
Section 7 extends the estimates by a barrier function in the spirit of [9], with new ingredients.
Finally, in Section 8 we conclude the proof of Theorem <ref>
by a fixed point argument. For the reader's convenience, we collect in the Appendix some technical estimates as well as some known results.

§.§ Acknowledgement
We would like to thank Luca Battaglia for his   contribution in the proof of Proposition <ref>.

§ ANSATZ FOR THE $K$-PEAK TOWER AND CHOICE OF PARAMETERS

In what follows, we denote by $\gp=\gp(t)$ the nonlinearity appearing in (<ref>), namely we set:

(t):=|t|^p-1t,     t∈.

We recall from Section <ref> that we seek solutions to (<ref>) of the
sign-changing tower of peaks" form:

where for $\ai\ge2$, $\dei>0$, we have that

is the family of radial solutions to (<ref>),
(x)=(x/),    ℓ=0,1,  x∈,
where $\woai$, $\wiai$ are the correction
terms defined in (<ref>), see Lemma <ref> for details.

§.§ Notation

Henceforth, we denote by $C>0$ a general constant independent of $p$.
For any measurable set $E\subset\Om$ we denote by $\chi_E$ the characteristic function of $E$.
By uniform convergence in $E$ we understand uniform convergence in $\overline E$.
For any two families of sets $A_p,B_p\subset\rr^2$ depending on $p$, by $A_p\cong B_p$
we understand that there exists a constant $K>0$ independent of $p$ such that

K^-1B_p⊂A_p⊂KA_p,    p→∞where $KB=\{Kx:x\in B\}$.
For $i=1,2,\ldots,k$ it will be convenient to set

\[
\vi(y):=U_{\ai,1}(y)=\ln\frac{2\ai^2}{(1+|y|^\ai)^2},\qquad y\in\rr^2,
\]

so that the scaling property may be written in the form

It is convenient to set

so that we may write (<ref>) in the form:
Δ+e^(y)=e^(y)φ^0()    on ^2

Δ+e^(y)=e^(y)φ^1(,)    on ^2.
The mass" takes the form:
We set

(x)=(x/),    (x)=(x/),    i=1,2,…,k.
It follows that $\wodai$, $\widai$, satisfy

Δ+|x|^-2e^=|x|^-2e^((x/)),    x∈

Δ+|x|^-2e^=|x|^-2e^((x/),(x)),    x∈.
For later estimates it is essential to observe that $\Vai$ satisfies

for some $C>0$ independent of $y\in\rr^2$.
We observe that

-Δ   (+/p+/p^2)

Occasionally within some proofs it will be convenient to denote by $\tUdai$, $\tvi$, $\tai$,
the corrections" by the lower order terms $\wodai$, $\widai$, or by the related constants $\Coai$, $\Ciai$:

§.§ The choice of parameters

Our aim in this section is to
choose $\al_i,\de_i,\ti$, so that the error" $\Rp$ defined by

vanishes with order $p^{-4}$ in a suitable weighted norm.
More precisely, in Proposition <ref> we derive conditions for the parameters $\ai$, $\dei$, $\ti$,
in the form of a nonlinear system, which imply smallness" of $\Rp$.
In Proposition <ref> we prove the solvability of the system.
Finally, in Proposition <ref> we show that $\ai\not\in\mathbb N$ for $i\ge2$,
which is essential to prove invertibility of the linearized problem.
We make the following assumptions on the parameters $\al_i=\al_i(p)$, $\de_i=\de_i(p)$, $\ti=\ti(p)>0$, $i=1,2,\ldots,k$:

The singularity coefficients $\al_i$ increase as the index $i$ increases:

the concentration parameters $\de_i$ vanish exponentially with respect to $p$,
with decreasing speed with respect to $i$:

_i=C_ie^-b_ip,  i=1,2,…,k;   C^-1≤b_k<b_k-1<…<b_2<b_1≤C,
where $C^{-1}\le C_i=C_i(p)\le C$, $b_i=b_i(p)$, $i=1,2,\ldots,k$.
In particular,

_i=o(_i+1),  i=1,2,…,k-1;

the coefficients $\ti$ satisfy:

=O(1/p),     C^-1≤/≤C,    i,j=1,2,…,k.

Ansatz (A3) implies that $\ti P\Udai$ is $L^\infty$-bounded, consistently
with the estimates in [22, 23].

§.§ A partition of $\Om$: the shrinking annuli" $\Aj$

In order to estimate the error $\Rp$ defined in (<ref>), extending ideas in [16], we partition $\Om$ into suitable
sets $\Aj$, $j=1,2,\ldots,k$, such that the $j$-th bubble
$P\tUdaj$ is the dominant bubble in $\Aj$. Our choice of $\Aj$ is more delicate than in [16],
and is optimal with respect to the property:

(y)≥-p-C uniformly for y∈/,
for some $C>0$ independent of $p\to+\infty$, see
Lemma <ref>.
For $j=1,\ldots,k-1$, let $0<\ej<1$ be defined by

where $\tj$ is the constant appearing in (<ref>).
We define

A_1:=   {x∈: 0≤|x|<_1^_1_2^1-_1}

:=   {x∈: _j-1^_j-1_j^1-_j-1≤|x|<_j^_j+1^1-},

:=   {x∈: |x|≥_k-1^_k-1_k^1-_k-1}.
We note that the set $A_1$ is a shrinking ball, the sets $\Aj$, $j=2,\ldots,k-1$ are shrinking annuli
and the set $A_k$ is an annulus invading $\Om$.
For later use we observe that

A_1/_1=   {y∈^2: _1 y∈ and

/=   {y∈^2: y∈ and  (_j-1/)^_j-1

/=   {y∈^2: _k y∈ and  |y|≥(_k-1/)^_k-1
In particular, the rescaled sets $\Aj/\dej$, $j=1,2,\ldots,k$, invade the whole space $\rr^2$ as $p\to+\infty$.
We also check that
by choice of $\aj$, $\dej$, there holds

A_1/_1≅{y∈^2: _1 y∈ and 0≤|y|< e^p/4},

/≅{y∈^2: y∈ and e^-p/-2≤|y|<e^p/+2},

/≅{y∈^2: y∈ and |y|≥e^-p/-2},
where the relation $\cong$ is defined in Section <ref>.
We note that the expansion rate of $A_1/\de_1$ in (<ref>) is consistent with [11].

§.§ A family of level sets for $\Vaj$: the sets $\Bj$

We shall  need the following subsets $\Bj\subset\Om$,
where the Taylor expansion as stated in Lemma <ref> holds uniformly:

:=   {x∈: 1+(x)+ln|x|^-2+ln^2/p>1/2}
={x∈: (x/)>-p/2}.
In view of the form of $\Vaj$ as in (<ref>), it is clear that the shrinking rate of $\Bj$ is given by
∼={x∈: 0≤|x|<C_1e^p/8},

∼={x∈: C^-1e^-p/2(-2)<|x|<Ce^p/2(+2)},
see Lemma <ref> below for the precise statement.
In view of (<ref>) we have
⊂,    j=1,2,…,k.

§.§ The $j$-th error $\Rj$

We recall from (<ref>) that
For every $j=1,2,\ldots,k$ we define the $j$-th error”

x∈so that we may write

For $j=1,2,\ldots,k$, let

where $\tai$, $h$ are defined in (<ref>), (<ref>), respectively.
Our first aim in this section is to establish the following.

Let $\Up$ be defined in (<ref>), where the parameters
$\al_i=\al_i(p)$, $\de_i=\de_i(p)$, $\ti=\ti(p)$, $i=1,2,\ldots,k$, are solutions to the system:

and satisfy assumptions (A1)–(A2)–(A3),
and where the correction profiles $w_{\al_i}^0$, $w_{\al_i}^1$, $i=1,2,\ldots,k$
are defined by (<ref>).
Then, there holds the estimate

(x)=   |x|^-2e^U_,(x)O(|(-2)ln|x|/|^6+ln^6(2+|x|/)/p^4)

=   |y|^-2e^(y)/^2O(|(-2)ln|y||^6+ln^6(2+|y|)/p^4),
uniformly for $x=\dej y\in\Bj$.

The form of $\Rj$ determines the form of the suitable weighted norm $\rp$ to be introduced later.
In particular, we note that since $|x|^{\aj-2}e^{U_{\aj,\dej}(x)}=\dej^{-2}|y|^{\aj-2}e^{\vj(y)}$,
$x=\dej y$, we expect a $\dej^2$ term in $\rp$.

We observe that in the one bubble case" $k=1$, $\al_1=2$, the error estimate in (<ref>) takes the form
ℛ_1(x)=8/_1^2(1+|y|^2)^2O(ln^6(2+|y|)/p^4),    y=_1^-1x,
in agreement with [11].

Our next aim is to prove the solvability of system (<ref>).
To this end, it is convenient to define the constants $s_j=s_j(p)>0$, $j=1,2,\ldots,k-1$,
by setting:

where the last equality holds true in view of the fifth equation in system (<ref>).
With this notation, we may write

For every $k\in\mathbb N$ there exists $p_0>0$ such that:

(Existence): For all $p>p_0$ there exists a solution
$(\aj,\dej,\tj)=(\aj(p),\dej(p),\tj(p))$, $j=1,2,\ldots,k$, to system (<ref>);

(Basic properties): The solution $(\aj,\dej,\tj)$ obtained in (i) satisfies
assumptions (A1)–(A2)–(A3);

(iii)(Properties of the $\sj$'s)
The $\sj$'s form a bounded increasing sequence:

3/2e+1<s_1<…<<s_j+1<1,     j=1,2,…,k-1.

(iv)(Properties of the $\tj$'s)

(v)(Properties of the $\bj$'s)

(vi)(Properties of the $\ej$'s)

Finally, we establish the following bounds for the $\aj$'s, which imply that $\aj\not\in\mathbb N$ for all
$j\ge2$, and thus yield invertibility of the linearized operator.

The solution $(\aj,\dej,\tj)$, $j=1,2,\ldots,k$, obtained in Proposition <ref>–(i) satisfies

for all $j\geq2$.
In particular,
$\alpha_i\not\in\mathbb  N$ for any $i\ge2$.

The remaining part of this section is devoted to the proofs of Proposition <ref>, Proposition <ref>
and Proposition <ref>.
The following lemma establishes the leading profile" of the rescaled approximate solution $\Up(\dej y)$
for $x=\dej y\in\Aj$:

Suppose $\ti$, $\ai$, $\dei$, $i=1,2,\ldots,k$, satisfy system (<ref>).
Then, there holds the expansion:

=   (-1)^j-1p{1+(y)/p+(y)/p^2+(y)/p^3+O(e^-β_jp/p)}
uniformly for $x=\dej y\in\Aj$, for some $\beta_j>0$.
In particular,
uniformly for $x\in\Om$.

The main underlying reason for the following expansions is that

\[
\]

More precisely, we recall the definitions of $\tvi$, $\tai$, $i=1,2,\ldots,k$, in (<ref>)–(<ref>).
In view of Lemma <ref>, we have

\[
P\tUdai(\dej y)=\left\{
\bal
&\tvj(y)+2\taj\ln\frac{1}{\dej}+4\pi\taj h(0)-\ln(2\al_j^2)\\
&&\hbox{for }i=j;\\
&2\tai\ln\frac{1}{|y|}+2\tai\ln\frac{1}{\dej}+4\pi\tai h(0)\\
&&\hbox{for }1\le i<j;\\
&2\tai\ln\frac{1}{\dei}+4\pi\tai h(0)+\frac{\woai(0)}{p}+\frac{\wiai(0)}{p^2}\\
&&\hbox{for }j<i\le k,
\eal
\right.
\]

uniformly for $x=\dej y\in\Aj$.
We deduce that

\[
\bal
\Up(\dej y)=&\tj(-1)^{j-1}\tvj(y)
+2\left(\sum_{1\le i<j}\ti(-1)^{i-1}\tai\right)\,\ln\frac{1}{|y|}
+2\left(\sum_{1\le i\le j}\ti(-1)^{i-1}\tai\right)\,\ln\frac{1}{\dej}\\
&+2\sum_{j<i\le k}\ti(-1)^{i-1}\tai\,\ln\frac{1}{\dei}
\eal
\]

where the bounded constants $\cj$ are defined in (<ref>).
Equivalently, we may write

(y)=   (-1)^j-1{(y)+2(∑_1≤i<j/(-1)^i-j) ln1/|y|
-2(∑_1≤i≤j/(-1)^i-j) ln

-2∑_j<i≤k/(-1)^i-j ln++_j}

\[
\om_j=O\left(\sum_{i=1}^k\frac{\ti}{\tj}\dej^{\ej}\dejj^{1-\ej}+\sum_{i<j}\frac{\ti}{\tj}(\frac{\de_{j-1}}{\dej})^{(1-\veps_{j-1})\ai}
\]

Since $\ti/\tj=O(1)$ in view of (A3), we have

uniformly for y∈A_j,
for some $\beta_j>0$.
We observe that the second equation in (<ref>) implies that

\[
-2\sum_{1\le i<j}\frac{\ti}{\tj}(-1)^{i-j}\tai=\aj-2
\]

and therefore
Consequently, we may rewrite (<ref>)
in the form

Using the fourth equation in (<ref>) and the definition of $\tvj$ in (<ref>),
we derive
(y)=   (-1)^j-1{v_(y)+(y)/p+(y)/p^2+(-2)ln|y|+p+_j}.

Now (<ref>) yields the asserted expansion.

Lemma <ref> and the facts $|w_{\aj}^\ell(y)|\le C\ln(|y|+2)=O(p)$ in $\Aj$, readily implies the following lower-order expansions,
which will be also used in the sequel:

\begin{align}
\label{eq:Upexpfirst}
\Up(x)
=&(-1)^{j-1}\tj p\Big\{1+\frac{\Vaj(y)}{p}+\frac{\woaj(y)}{p^2}+\frac{O(\ln(|y|+2))}{p^3}\Big\}\\
\label{eq:Upexpzero}
\Up(x)
=&(-1)^{j-1}\tj p\Big\{1+\frac{\Vaj(y)+O(1)}{p}\Big\},
\end{align}

uniformly for $x=\dej y\in\Aj$.
Moreover, as a direct consequence of Lemma <ref> and the Taylor expansions, as stated in Lemma <ref>,
we obtain the following expansions.

The following expansions hold true:

((x))=   |x|^-2e^(x)×

uniformly for $x=\dej y\in\Bj$, $j=1,2,\ldots,k$.

Proof of (i).
In view of Lemma <ref>, we have
((x))=   ((y))

=   ((-1)^j-1p{1+(y)/p+(y)/p^2+(y)/p^3+O(e^-β_jp/p)}).
For $\dej y\in\Bj$ we have, by definition
Therefore, for $\dej y\in\Bj$ and $p\gg1$ we have
Hence, we may write
Now, the claim follows by the Taylor expansions, as stated in Lemma <ref>–(i) with $t=|y|$,
$a(t)=\Vaj(y)$, $b(t)=\woaj(y)$, $c(t)=\wiaj(y)$ and the fact
$(\tj p)^p=\tj\dej^{-2}$.
Indeed, for $x=\dej y\in\Bj$ we derive
((x))=   (-1)^j-1^-2 e^(y){1+1/p((y)-((y)))

which, recalling the definition of $\Vaj$, yields (i).
Proof of (ii). Similarly as above, in view of (<ref>), for $x=\dej y\in\Bj$ we have
((x))=   p(p)^p-1{1+(y)/p+(y)/p^2+O(ln(|y|+2))/p^3}^p-1

=   e^(y)/^2{1+1/p[(y)-((y))-(y)]+O(|(y)|^4+1)/p^2},
where we used Lemma <ref>–(ii) with $\kappa=1$ to derive the last equality.
Now (ii) follows by the mass scaling property as stated in Lemma <ref>, and recalling the definition of $\Vaj$.

From (<ref>) we derive

\[
-\Delta P\tUdaj(\dej y)
\left(1+\frac{\woaj(y)-\vphio(\Vaj(y))}{p}+\frac{\wiaj(y)-\vphii(\Vaj(y),\woaj(y))}{p^2}\right).
\]

Therefore, we may write, for $x=\dej y\in\Bj$:

We may now apply Lemma <ref>–(i) to derive

\[
\]

which yields the asserted estimate since $\tj=O(p^{-1})$.

We turn to the proof of Proposition <ref>.

System (<ref>) implies the following system in terms of $\aj$, $j=1,2,\ldots,k$ and $\sj$,

1/2(+2-/p-/p^2)ln+1+=+c_j+1-(1+)/p-1,      j=1,2,…,k-1

(_j+1-2)=+2-/p-/p^2,      j=1,2,…,k-1.

We combine the first and the second equation in (<ref>) into a single formula:

\begin{equation}
\label{s1}
-2\sum\limits_{i=1}^{j-1 }(-1)^{i-j}{\ti}\tai=(\al_j-2)\tj,\quad j=1,\dots,k,
\end{equation}

where we agree that the sum is zero if $j=1$.
Writing (<ref>) for the index $j$ and for the index $j+1$, and adding the two resulting expressions, we obtain
the following recursive formula for $\al_j$ (in terms of the $\ti$'s):

(2-+2)=(_j+1-2)τ_j+1,  j=1,…,k-1,
which yields the third equation in (<ref>).
Similarly, we combine into a single formula the third and the fourth equation in (<ref>):

\begin{equation}
\label{s3}
-2\sum\limits_{i=j+1}^{k}(-1)^{i-j}{\ti }\tai\ln\de_i
+c_j\tj=p\tj,\quad j=1,\dots,k,
\end{equation}

where we agree that the sum is zero if $j=k$.
Writing (<ref>) for the index $j$ and for the index $j+1$, and adding the two resulting expressions,
we obtain

=p(+τ_j+1),  j=1,…,k-1,

\[
\]

Using (<ref>),
we obtain the following recursive formula for $\de_j$ (in terms of the $\sj$'s):

ln_j=ln_j+1-1+s_j/2-+2 p
From (<ref>) we derive $\dej/\de_{j+1}=\sj^{(p-1)/2}$, and inserting into (<ref>)
we deduce
and the second equation in (<ref>) follows.

We shall use the last equation in (<ref>) in the simplified form:

Now we can conclude the proof of Proposition <ref>.

We first observe that

=s_j+1⋯s_i-1,      if i>j

1,      if i=j

(/)^-1=1/s_is_i+1⋯s_j-1,      if i<j.
In particular, the $\cj$'s, as defined in (<ref>), are continuously differentiable
with respect to $(s_1,\ldots,s_{k-1})\in(0,1]^{k-1}$.
We set $t:=p^{-1}$ and we rearrange the parameters $\aj$, $\sj$ in the form
For $i=1,\ldots,k$ we set

_1-2,      if i=1

(-2)s_i-1-(2^(w)_i-1-_i-1+2),      if 2≤i≤k
and for $i=1,\ldots,k-1$ we set

With this notation, system (<ref>) is equivalent to the equation
$\G((\aj,\sj)_{j=1,\ldots,k-1},\ak,t)=\mathbf{0}\in\rr^{2k-1}$, where

=   (

We seek a branch of solutions $((\aj(t),\sj(t))_{j=1,\ldots,k-1},\ak(t),t)$ to equation (<ref>), for small values of $t$,
by an implicit function argument.
We first consider the case $t=0$, namely we consider the equation

corresponding to
the unperturbed system"

+2/2ln+1+=0,      j=1,2,…,k-1

(_j+1-2)=_j+2,      j=1,2,…,k-1.
We note that for any $k\in\mathbb N$,
equation(<ref>) admits a unique solution, denoted
Indeed, let us denote
g_i^0((,)_j=1,…,k-1,):=   g_i((,)_j=1,…,k-1,,0)

_1-2   if i=1

s_i-1(-2)-(_i-1+2)   if 2≤i≤k,

for $i=1,2,\ldots,k$, and
for $i=1,2,\ldots,k-1$.
Then, we are reduced to solving the system

((,   )_j=1,…,k-1,,0)

=   (

Setting $\zeta_\al(s):=\frac{\al+2}{2}\ln s+1+s$, it is elementary to check that for any
fixed $\al\ge2$ there holds $\zeta_\al'(s)=(\al+2)/(2s)+1>0$ for all $s>0$,
$\lim_{s\to0^+}\zeta_\al(s)=-\infty$ and $\zeta_\al(1)=2$.
Therefore, for any $\al\ge2$ the nonlinear equation $\zeta_\al(s)=0$
admits a unique solution $s_\al\in(0,1)$.
Since we may write $h_i^0((\aj,\sj)_{j=1,\ldots,k-1},\ak)=\zeta_{\ai}(s_i)$,
we deduce that system (<ref>) admits a unique solution $((\aj^0,\sj^0)_{j=1,\ldots,k-1},\ak^0)$ defined recursively.
We claim that any solution $((\aj,\sj)_{j=1,\ldots,k-1},\ak)$ to (<ref>) satisfies

Indeed, from the third equation in (<ref>) we have
$\al_{j+1}-2=(\aj+2)/\sj$. Since $\sj\in(0,1)$, we deduce that if
$\aj>0$ then $\al_{j+1}-2>\aj+2$, i.e., $\al_{j+1}>\aj+4>0$.
Since $\al_1=2$, we obtain (<ref>) recursively.
We now check that the $(2k-1)\times(2k-1)$ Jacobian matrix of $\G$ with respect to
the variables $((\aj,\sj)_{j=1,\ldots,k-1},\ak)$ is invertible
at the solution $((\aj^0,\sj^0)_{j=1,\ldots,k-1},\ak^0)$.
To this end, it is readily checked that

_ij,   if i=1,

s_i-1^0_ij-_i-1,j,   if 2≤i≤k;
, j=1,2,…,k;

0,   if i=1

(^0-2)_i-1,j   if 2≤i≤k-1
, j=1,2,…,k;

h_i^0/=lns_i^0/2_ij,    i=1,2,…,k-1, j=1,2,…,k;

h_i^0/=(^0+2/2s_i^0+1)_ij    i=1,2,…,k-1,     j=1,2,…,k-1,
where the $\de_{ij}$'s denote Kronecker deltas.
Consequently, the Jacobian matrix for the mapping $\mathbf{G}((\aj,\sj)_{j=1,\ldots,k-1},\ak,0)$,
is given by

g_1^0/_1   g_1^0/s_1   g_1^0/_2   g_1^0/s_2
⋯   g_1^0/_k-2   g_1^0/s_k-2
g_1^0/_k-1   g_1^0/s_k-1   g_1^0/_k

h_1^0/_1   h_1^0/s_1   h_1^0/_2   h_1^0/s_2
⋯   h_1^0/_k-2   h_1^0/s_k-2
h_1^0/_k-1   h_1^0/s_k-1   h_1^0/_k

g_2^0/_1   g_2^0/s_1   g_2^0/_2   g_2^0/s_2
⋯   g_2^0/_k-2   g_2^0/s_k-2
g_2^0/_k-1   g_2^0/s_k-1   g_2^0/_k

h_2^0/_1   h_2^0/s_1   h_2^0/_2   h_2^0/s_2
⋯   g_2^0/_k-2   g_2^0/s_k-2
h_2^0/_k-1   h_2^0/s_k-1   h_2^0/_k

⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮

g_k-1^0/_1   g_k-1^0/s_1   g_k-1^0/_2   g_k-1^0/s_2
⋯   g_k-1^0/_k-2   g_k-1^0/s_k-2
g_k-1^0/_k-1   g_k-1^0/s_k-1   g_k-1^0/_k

h_k-1^0/_1   h_k-1^0/s_1   h_k-1^0/_2   h_k-1^0/s_2
⋯   h_k-1^0/_k-2   h_k-1^0/s_k-2
h_k-1^0/_k-1   h_k-1^0/s_k-1   h_k-1^0/_k

g_k^0/_1   g_k^0/s_1   g_k^0/_2   g_k^0/s_2
⋯   g_k^0/_k-2   g_k^0/s_k-2
g_k^0/_k-1   g_k^0/s_k-1   g_k^0/_k

1   0   0   0   ⋯   0   0   0   0   0

lns_1^0/2   _1^0+2/2s_1^0+1   0   0   ⋯   0   0   0   0   0

-1   _2^0-2   s_1^0   0   ⋯   0   0   0   0   0

0   0   lns_2^0/2   _2^0+2/2s_2^0+1   ⋯   0   0   0   0   0

⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮   ⋮

0   0   0   0   ⋯   -1   _k-1^0-2   s_k-2^0   0   0

0   0   0   0   ⋯   0   0   lns_k-1^0/2   _k-1^0+2/2+1   0

0   0   0   0   ⋯   0   0   -1   _k-2^0   s_k-1^0

In particular, it is a lower triangular matrix with positive diagonal entries given by
1, _1^0+2/2s_1^0+1,  s_1^0,  _2^0+2/2s_2^0+1,…, _k-2^0+2/2s_k-2^0+1, s_k-2^0, _k-1^0+2/2s_k-1^0+1,  s_k-1^0.
It follows that $D_{((\aj,\sj)_{j=1,\ldots,k-1},\ak)}\mathbf{G}((\aj,\sj)_{j=1,\ldots,k-1},\ak,0)$
is invertible at $((\aj^0,\sj^0)_{j=1,\ldots,k-1},\ak^0)$.
Now, the implicit function theorem yields a unique branch of solutions
(((p),(p))_j=1,…,k-1,(p),p),    p=t^-1,
to system (<ref>), for all sufficiently large values of $p$, continuously depending on $p$.
We are left to derive a branch of solutions to system (<ref>) from the branch of solutions to
system (<ref>).
To this end, we note that in view of (<ref>) the solution $((\aj,\sj)_{j=1,\ldots,k-1},\ak)$
uniquely determines the constants $\cj$, $j=1,2,\ldots,k$, defined in (<ref>).
Now, we are able to compute the $\de_j$'s.
Indeed, the from the third equation in (<ref>) we derive
From $\de_k$ and the last equation in system (<ref>) we obtain
Recursively, we derive $\tj$, $j=1,2,\ldots,k-1$, using the property
Finally, from the fourth equation in (<ref>) we recursively derive $\de_1,\ldots,\de_{k-1}$.
The existence of the desired branch of solutions $(\aj,\dej,\tj)$ is now completely established.

By continuity, it suffices to check properties (A1)–(A2)–(A3) for solutions
to the unperturbed system (<ref>). Hence, let $((\aj,\sj)_{j=1,2,\ldots,k-1},\ak)$
be a solution to system (<ref>).
Since $s_j\in(0,1)$, and since  (<ref>) implies $\al_{j+1}-2>0$, we deduce from
system (<ref>) that

\[
\aj+2=\sj(\al_{j+1}-2)<\al_{j+1}-2,
\]

for all $j\ge1$. In particular, $\aj+4<\al_{j+1}$, for all $j\ge1$,
and therefore assumption (A1) is satisfied.
We note that (A2) holds true for $j=k$ with
From (<ref>) we derive, recursively:

and consequently:

Hence, assumption (A2) is completely verified.
At this point, it is clear that assumption (A3) is also satisfied.

We note that $b_1\to+\infty$ as $k\to+\infty$, that is, the concentration rate of the fast peaks
increases as the number of peaks increases. The rate of the slowest peak does not change.

By continuity, it suffices to verify (<ref>) for the solution
$((\aj^0,\sj^0)_{j=1,\ldots,k-1},\ak^0)$ to the unperturbed system (<ref>).
To this end, we observe that
h_j+1^0((^0,^0)_j=1,…,k-1,^0)=   ζ__j+1^0(^0)

=   ^0+2/2ln^0+1+^0+^0+2/2(_j+1^0+2/^0+2-1)ln^0

=   ^0+2/2(_j+1^0+2/^0+2-1)ln^0<0.
Since $\zeta_{\al_{j+1}^0}$ is strictly increasing and since $\zeta_{\al_{j+1}^0}(s_{j+1}^0)=0$,
we deduce that $0<\sj^0<s_{j+1}^0<1$, for any $j=1,2,\ldots,k-1$.
We claim that

Indeed, from the second equation in (<ref>) we obtain the nonlinear equation

which readily implies the upper bound for
$\sj$ since $\sj\in(0,1)$. By convexity of the
function $f(s)=e^{-\frac{2(1+s)}{\aj+2}}$ at $s=1$, it is elementary to
check that
Using again $\sj\in(0,1)$ and $\sj=f(\sj)$, we derive the lower bound in  (<ref>).
For $j=1$, we obtain the lower bound in (<ref>).

We are left to establish (<ref>).
It is readily seen that the $\aj$'s increase asymptotically linearly with respect to
$j$, more precisely $\aj=8j+O(1)$ as $j\to+\infty$.
To see this, we note that from (<ref>) we derive
$\sj=1-\frac{2(1+\sj)}{\aj+2}+O(\aj^{-2})$ and therefore
Inserting into the third equation of (<ref>) we deduce that

Let us introduce the Lambert function $W=f^{-1}$, where $f(x)=xe^x$ in $\mathbb R^+$.
We remark that

$$ax\ln x=x+1
\ \Leftrightarrow\   {1\over ax}e^{1\over ax}={1\over a}e^{-{1\over a}}
\ \Leftrightarrow\   {1\over ax}=W\left({1\over a}e^{-{1\over a}}\right)\ \Leftrightarrow\   x=\frac1{a}\frac 1{W\left({1\over a}e^{-{1\over a}}\right)}$$

Therefore, by recurrence

$${1\over\sj}={2\over \alpha_j+2}{1\over W\({2\over \alpha_j+2}e^{-{2\over \alpha_j+2}}\)}$$

and so

\begin{equation}
\label{ai}
\alpha_1=2\ \hbox{and}\ \alpha_{j+1}=2+{2\over W\({2\over \alpha_j+2}e^{-{2\over \alpha_j+2}}\)},\ j=1,\dots,k-1.
\end{equation}

First of all,  straightforward but tedious computations show that

$$0<\frac1{W\(xe^{-x}\)}-2-\frac 1x\le \frac43x^2\ \hbox{for any}\ x\in(0,1/2].$$

Indeed, using the definition of Lambert's function, it is equivalent to prove that

$$\min\limits_{x\in(0,1/2]}\frac1{2x+1}e^{{x\over 2x+1}+x}=\max \limits_{x\in(0,1/2]}\frac3{4x^3+6x+3}e^{{3x\over 4x^3+6x+3}+x}=1.$$

That implies (setting $x=\frac2{\alpha_i+2}$)

$$\alpha_i+8<\alpha_{i+1}\le \alpha_i+8+\frac{32}3{1\over(\alpha_i+2)^2}\ \hbox{for any}\ i\ge1.$$

Therefore, by recurrence we get

$$8i-6<\alpha_i\le 8i-6+\frac{32}3\sum\limits_{j=1}^{i-1} \frac1{(\alpha_i+2)^2}
\ \hbox{for any}\ i\ge2.$$

and also

$$ \frac{32}3\sum\limits_{j=1}^{i-1} \frac1{(\alpha_i+2)^2}
\le  \frac{32}3\sum\limits_{j=1}^{i-1} \frac1{(8j-4)^2}\le  \frac23\sum\limits_{j=1}^{\infty} \frac1{(2j-1)^2}= {\pi^2\over 12}<1,$$

where we used the well-known fact $\sum_{n=1}^\infty n^{-2}=\pi^2/6$ and consequently
$\sum_{j=1}^\infty (2j)^{-2}=4^{-1}\sum_{j=1}^\infty j^{-2}=\pi^2/24$,
$\sum_{n=1}^\infty (2j-1)^{-2}=\pi^2/6-\pi^2/24=\pi^2/8$ to derive the last inequality.
Finally, the asserted estimate (<ref>) follows.

There exist constants $0<R_j'<R_j''$, $j=1,2,\ldots,k$, and $0<r_j''<r_j'$,
$j=2,\ldots,k$, independent of $p$,
such that
{|x|≤R_1'_1 e^p/8}
and, for all $j=2,\ldots,k$,
In particular, there holds $\Bj\subset\Aj$, for all sufficiently large values of $p$.

The specific form (<ref>) of $\ej$ is essential for the proof.

We recall from (<ref>) that
for all $\al\ge2$ and that
={x∈: 𝒱_(x/)≥-p/2}.
Now, the asserted inclusions readily follow.
Hence, we need only check that $\Bj\subset\Aj$.
We claim that
for all sufficiently large values of $p$.
We equivalently check that $e^{\frac{p}{2(\aj+2)}}\le(\de_{j+1}/\dej)^{1-\ej}$.
Recalling the properties of $\dej$ as in (<ref>), we have
so that
Now the result follows since, in view of (<ref>) we have
provided that $p$ is sufficiently large.
Similarly, we claim that
for all sufficiently large values of $p$.
We equivalently check that $e^{-\frac{p}{2(\aj-2)}}\ge(\de_{j-1}/\dej)^{\veps_{j-1}}$.
Recalling the properties of $\dej$ in (<ref>), we have
ln(_j-1/)^_j-1=   C-_j-1(b_j-1-b_j)p
=   C-_j-11+s_j-1/(-2)s_j-1p=C-p/-2,
where we used (<ref>) to derive the last equality.
The asserted inclusions $\Bj\subset\Aj$ are now completely established.

Proof of (iv). We have
as asserted.

§.§.§ The case of one bubble $k=1$

We remark that for $k=1$ we have $\al_k=\al_1=2$, $b_k=b_1=(2\ta_k-\al_k+2)^{-1}=(4+o(1))^{-1}$,
and therefore

\[
\bal
c_k=&8\pi h(0)-\ln8+o(1)\\
\de_k=&\de_1=C_ke^{-\frac{p}{4+o(1)}}\\
\tau_kp=&\tau_1p=\frac{p}{p^{p/(p-1)}\de_k^{2/(p-1)}}=\sqrt{e}(1+O(\frac{\ln p}{p})),
\eal
\]

in agreement with [11].

§ WEIGHTED ESTIMATION OF $\RP$

We recall from (<ref>), (<ref>) and (<ref>) that the error $\Rp$ is defined by
Our aim in this section is to estimate $\Rp$ with respect to a suitable weight function
$\rp(x)$ defined by

(x)=   ∑_j=1^k(x)χ_(x),

(x)=   ^2+η+|x|^2+η/^η,
where $0<\eta<1$.
Then, in view of (<ref>), we may write

We note that upon rescaling we have:

(y)=^2(1+|y|^2+η),  y∈.
We set

h_:= h_L^∞(),
We observe that the choice of $\rj$ ensures uniform weighted boundedness of the $j$-th mass
with logarithmic errors, $j=1,2,\ldots,k$:
for any $q>0$,
see Lemma <ref> below for a more precise statement.
The main result in this section is the following.

The following estimate holds true
for some $C>0$ independent of $p$.

We devote the remaining part of this section to the proof of Proposition <ref>.
The estimates contained in the following lemma will be used systematically in the sequel.

The following estimates hold true:

|(y)|=O(p),    (y)=O(p),
uniformly for $x=\dej y\in\Aj$, $j=1,2,\ldots,k$,
where $\foaj(y)=\varphi^0(\Vaj(y))$ and $\fiai(y)=\varphi^1(\Vaj(y),\woaj(y))$.

It suffices to observe that for $x=\dej y$ in $\Aj$ we have:
|ln|y||≤   Cln(_j-1/)^_j-1=O(p),
for |y|≤1, j=2,…,k;

ln(|y|+2)≤   Cln(2+(/)^1-)=O(p),
for |y|≥1, j=2,…,k-1;

ln(|y|+2)≤   Cln(2+(/)^1-_k)=O(p),
for |y|≥1, j=k.

We observe that in view of (<ref>) we have

In view of Lemma <ref> and Proposition <ref>, we deduce that

=O(p |x|^-2e^),
uniformly for $x\in\Aj$.
Therefore, using (<ref>) and (<ref>) we may decompose the error estimate as follows:

_≤   ∑_j=1^k_L^∞()

We estimate the right hand sides in (<ref>)–(<ref>) term by term in the following lemmas.

There holds

In view of (<ref>) we have, uniformly for $x=\dej y\in\Bj$:
(x)(x)=   (x) |x|^-2e^(x)

Now, the asserted estimate readily follows from (<ref>) with $q=6$.

Suppose $i\neq j$.
The following estimates hold true, uniformly for $x=\dej y\in\Aj$:
if i<j;

if i=j;

if i>j,

where $(1-\veps_{j-1})\ai-2\veps_{j-1}>0$ and $\ej\ai-(1-\ej)\eta>0$.
In particular, for any $q>0$ we have
O(p e^-[(1-_j-1)-2_j-1] (b_j-1-b_j) p),      if i<j;

O(1),      if i=j;

O(p e^-[-(1-)η] (b_j-b_j+1) p),      if i>j.

Suppose $i<j$.
For $x=\dej y\in\Aj$ we have, using Lemma <ref>:
where we used (<ref>) to deduce that $\dei/|\dej y|=o(1)$ for $\dej y\in\Aj$.
Therefore, for $x=\dej y\in\Aj$:
For $y\in\Aj$, $|y|\ge1$, we estimate:
For $y\in\Aj$, $|y|\le1$, we estimate
(x)|x|^-2e^(x)=   O(/)^1/|y|^+2

≤   O(/)^(/_j-1)^_j-1(+2)
because |y|≥(_j-1/)^_j-1 in

≤   O(_j-1/)^(/_j-1)^_j-1(+2)
because i≤j-1

≤   (_j-1/)^(1-_j-1)-2_j-1.
We observe that
Hence, the asserted estimate is established for $i<j$.
Suppose $i>j$.
In view of Lemma <ref>,
we have
where we used again (<ref>) to deduce that $|\frac{\dej y}{\dei}|=o(1)$ for $\dej y\in\Aj$.
Therefore, for $x=\dej y\in\Aj$, we estimate:
(x)|x|^-2e^(x)=   (^2+η+|x|^2+η)/^η|x|^-2e^(x)

=   ^2(1+|y|^2+η) O(/)^|y|^-2/^2

=   O(/)^(1+|y|^2+η)|y|^-2
Since for $y\in\Aj/\dej$ we have $|y|\le(\de_{j+1}/\dej)^{1-\ej}$, we deduce that
where we used the fact $i\ge j+1$ to derive the last inequality.
We observe that
The proof for $i=j$ follows by straightforward rescaling.

There holds:
uniformly for $x\in\Aj\setminus\Bj$.

We recall from Lemma <ref> that
Recalling (<ref>), it follows that
We recall from Lemma <ref> that
∪   {R_j'e^p/2(+2)≤|y|≤(_j+1/)^1-},

For $0<|y|\le1$, $j=1$, there is nothing to prove. For $0<|y|\le1$, $j=2,\ldots,k$, we estimate:
Hence, for $x=\dej y\in\Aj\setminus\Bj$, $|y|\le1$, we have
For $x=\dej y\in\Aj\setminus\Bj$, $|y|\ge1$, we estimate:
The asserted estimate follows, since $(\aj-\eta)/(\aj+2)<1$.

The following expansion holds true:
where $\|\rj\omp\|_{L^\infty(\Aj\setminus\Bj)}=O(e^{-\veps_0p}/p)$, where $\veps_0=\frac{\aj-\eta}{2(\aj+2)}$.

We estimate separately in the sets $\Bj$, $(\Aj\setminus\Bj)\cap\{\Up>0\}$ and $(\Aj\setminus\Bj)\cap\{\Up<0\}$,
Claim 1. (Estimation of $\gp(\Up)$ in $(\Aj\setminus\Bj)\cap\{\Up>0\}$).
There holds:

≤C/p e^--η/2(+2)p,
uniformly for $x\in\Aj\setminus\Bj$, $\Up(x)>0$.
Indeed we have, using Lemma <ref>–(i) and (<ref>), for $x=\dej y\in\Aj\setminus\Bj$,
For $x=\dej y\in\Aj\setminus\Bj$, $|y|\le1$, $j=2,\ldots,k$, we estimate:
For $x=\dej y\in\Aj\setminus\Bj$, $|y|\ge1$, $j=1,2,\ldots,k$, we estimate:
Now the claim follows recalling that $\tj=O(p^{-1})$.
Claim 2. (Estimation of $\gp(\Up)$ in $\Aj\cap\{\Up<0\}$).
The following decay estimate holds true in $\{\Up<0\}\cap\Aj$:

uniformly for $x\in\Aj$.
Indeed, we recall from (<ref>) that
where $\Vaj$ is defined in (<ref>).
We observe that $\Vaj(y)<0$ for $0<|y|\ll1$ and for $|y|\gg1$.
Moreover, we have for $x=\dej y\in\Aj$
where we used the last equation in (<ref>) to derive the last equality.
We have, using Lemma <ref>–(i),
and therefore, in view of (<ref>),

For $x=\dej y\in\Aj$, $|y|\le1$, $j\ge2$ (for $j=1$ there is nothing to prove) we estimate, recalling that $|y|\ge(\de_{j-1}/\dej)^{\veps_{j-1}}$:
≤   Ce^-2p/|y|^-2

=   Ce^_j-1(b_j-1-b_j)(-2)pe^-2p.
In view of (<ref>) we may simplify the exponent above:
so that for $x=\dej y\in\Aj$, $|y|\le1$, $j\ge2$, we finally obtain that
For $x=\dej y\in\Aj$, $|y|\ge1$, we estimate using (<ref>) and $|y|\le(\de_{j+1}/\dej)^{1-\ej}$:
≤   C(1+|y|)^+4+ηe^-2p

≤   Ce^(1-)(b_j-b_j+1)(+4+η)pe^-2p.
In view of Proposition <ref>–(v) we may simplify the exponent above:
(1-)(b_j-b_j+1)(+4+η)=   (1-)1+/+2+O(p^-1)(+4+η)

\[
\]

we derive Claim 2.
Finally, from Claim 1 and Claim 2 we deduce that

Finally, we conclude the proof of the main error estimate.

In view of the decompositions of $\Rp$ as in (<ref>)–(<ref>)
and the estimates in Lemma <ref>, Lemma <ref>,
Lemma <ref> and Lemma <ref>, we finally derive:
for $\eps_0=\frac{\aj-\eta}{2(\aj+2)}$.

§ THE LINEARIZED PROBLEM: ESTIMATES FOR $\WP$ AND CHOICE OF THE $\EJ$'S

We define the linearized operator

where the potential" $\Wp$ is defined by

and $\gp'(t)=p|t|^{p-1}$.
(y):=   (y)-(y)-^2(y)/2

(x):=   (x/).
Note that
uniformly for $y\in\Aj/\dej$.
Our aim in this section is to establish the following fact.

The following estimate holds true uniformly for $x\in\Om$:

(x)≤ ∑_i=1^k|x|^-2e^(x)χ_(x),
for some $\ovC>0$ independent of $p\to+\infty$.
Moreover, for any $j=1,2,\ldots,k$ there holds the expansion
where $\|\omp\|_{\rp}\le Ce^{-\beta_0p}$ for some $\beta_0>0$,
uniformly for $x\in\Om$.

We beign by establishing some auxiliary results.
The following result justifies the choice of the $\ej$'s as in (<ref>), namely
=/1+,     j=1,2,…,k-1.

Let $\aj$, $\dej$, $j=1,2,\ldots,k$, and $\sj$, $j=1,2,\ldots,k-1$ be the parameters defined in Proposition <ref>.
Let $0<\veps<1$.
The following implications hold true:

(i)For all $j=2,\ldots,k,$ if
(-2)ln|y|≥-p-C  for all  (_j-1/)^≤|y|≤1.

(ii) For all $j=1,2,\ldots,k-1$, if
(+2)ln1/|y|≥-p-C  for all  1≤|y|≤(/)^1-.

Consequently, if $\ej$, $j=1,2,\ldots,k-1$, is given by (<ref>), then

(y)≥-p-C  uniformly for  y∈/, j=1,2,…,k.

Proof of (i).
Assuming that $\veps\le s_{j-1}/(1+s_{j-1})$, it suffices to show that

We have:
(-2)ln(_j-1/)^=   -(-2)(b_j-1-) p-C
=-(-2)1+s_j-1/s_j-1(-2) p-C

=   -1+s_j-1/s_j-1p-C,
where we used Proposition <ref>-(v) to derive the last equality.
Hence, (<ref>) holds true if $\veps(1+s_{j-1})/s_{j-1}\le1$ and therefore Part (i)
is established.
Proof of (ii).
Assuming that $\veps\ge\sj/(1+\sj)$, it suffices to show that

We have, using Proposition <ref>-(v):
(+2)ln(/)^1-=   -(+2)(1-)(-) p-C

=   -(+2)(1-)1+/+2+O(1/p) p-C
=-(1-)(1+) p-C.
Hence, (<ref>) holds true if $(1-\veps)(1+\sj)\le1$, that is
if $\veps\ge\sj/(1+\sj)$.
Thus, Part (ii) is established.
Finally, we assume that $\ej$ is given by (<ref>).
For $j=1$, the asserted inequality (<ref>) follows from Part (ii), since $\mathcal V_{\al_1}$
is bounded at $y=0$.
For $j=2,\ldots,k-1$ the asserted inequality (<ref>) follows from Part (i) and Part (ii).
For $j=k$ we need only consider the case $|y|\ge1$.
Recalling from Proposition <ref>–(v) that $b_k=(\ak+2+O(p^{-1}))^{-1}$,
we compute
(y)≥ln1/|y|^+2-C≥   ln_k^+2-C=-b_k(+2) p-C=-+2/+2+O(1/p) p-C

≥   -p-C.

Estimate (<ref>) is a direct consequence of Lemma <ref>–(ii).

§ LINEARIZED PROBLEM: EVALUATION OF INTEGRALS

(y):=   (y)-(y)-^2(y)/2

(x):=   (x/),
where $\Vaj$ is defined in (<ref>) and $\zajo$ is the radial eigenfunction defined by

which satisfies the linearized equation

in ^2.
Observing that in view of equation (<ref>) we may write  $|y|^{\aj-2}e^{\vj(y)}\zajo(y)=-\Delta\zajo$,
and integrating by parts,  we obtain the following integrals to be used below:

∫_^2|y|^-2e^(y)(y) dy=0

∫_^2|y|^-2e^(y)(y)(y) dy=4π

∫_^2|y|^-2e^(y)(y)ln|y| dy=-2π,
see also (4.30)–(4.31) in [16].
We consider the linear operator $\Lpone$ defined for $\phi\in C^2(\Om)$ by:

Our aim in this section is to show the following.

There exists $C>0$ such that for any solution $\phi\in C^2(\Om)\cap C(\overline\Om)$ to the problem

ϕ=h   in

ϕ=0   on ∂
there holds

We derive the proof of Proposition <ref> by a contradiction argument.
Suppose that $\phin\in C^2(\Om)\cap C(\overline\Om)$, $n\in\mathbb N$, is such that
$\mathcal L_{p_n}\phin=h_n$ with $\pn\to+\infty$, $\pn\|h_n\|_{\rho_{p_n}}\to0$ and $\|\phi_n\|_{L^\infty(\Om)}=1$.

Let $\phij(y)=\phin(\dej y)$, $j=1,2,\ldots,k$.
Then, for every $j=1,2,\ldots,k$ there exists $\gj\in\rr$ such that $\phij(y)\to\gj\zajo(y)$ in $C_{\mathrm{loc}}^2(\rr^2\setminus\{0\})$,
weakly in $\mathbb D^{1,2}(\rr^2)$ and almost everywhere.

Let $j\in\{1,2,\ldots,k\}$ be fixed.
The rescaled function $\phij$ satisfies
-Δ=^2 𝒲_(y)-^2(y)    in /,
and $\|\phij\|_{L^\infty(\Om/\dej)}=1$.
Let $\mathcal K\subset\rr^2\setminus\{0\}$ be compact.
Since $\Bj$ invades $\rr^2$, we may assume that
$\mathcal K\subset\Bj$. In view of Lemma <ref>–(ii) we have
^2 𝒲_(x)=|y|^-2e^(y)(1+O(1/)),
uniformly for $x=\dej y\in\mathcal K$.
In view of Lemma <ref>–(ii) and the decay assumption on $\|\hn\|_{\rho_{\pn}}$ we have
By elliptic regularity there exists $\phijo\in C^2(\rr^2\setminus\{0\})$, satisfying

-Δ=|y|^-2e^(y)in $\rr^2\setminus\{0\}$,
such that $\phij\to\phijo$ in $\mathcal C_{\mathrm{loc}}^{1,\al}(\rr^2\setminus\{0\})$,
$\al\in(0,1)$. Since $\|\phij\|_{L^\infty(\Om/\dej)}=1$, the right hand side in (<ref>) is uniformly bounded in
$\Om/\dej$, and thus we deduce that $\phijo$ satisfies (<ref>)
in whole space $\rr^2$. Now we recall that in view of Proposition <ref> we have $\aj\not\in\mathbb N$
for all $j\ge2$. Therefore, by the characterization of bounded solutions to (<ref>) as established in
[9], we deduce that $\phijo=\gj\zajo$ for some $\gj\in\rr$.

Our aim is to show that $\gj=0$ for all $j=1,2,\ldots,k$. To this end,
extending and approach in [16],
:=p_n∫_|x|^-2e^(x){1+(x)/p_n}(x) dx.
We begin by obtaining two linear relations between the $\sjs$'s and the $\gj$'s.
It is convenient to set

=∫_^2|y|^-2e^((y))^2(y) dy
=2^2∫_^2|y|^-2(1-|y|^)^2/(1+|y|^)^4(y) dy.
With this notation, we have:

For $j=1,2,\ldots,k$, the following linear relations hold true:

as $n\to\infty$.

We observe that in view of Proposition <ref> we may write
where $\|\widetilde\om_n\|_{\rho_{\pn}}=O(\pn^{-2})$.
Therefore, setting $\thn=\hn-\widetilde\om_n\phin$, we obtain from the contradiction assumption that
ℒ_^1=,     _ρ_=o(1),     _L^∞()=1,
where $\mathcal L_{\pn}^1$ is the operator defined in (<ref>) with $p=\pn$.
Hence, we consider problem

-Δ=   ∑_i=1^k|x|^-2e^(x){1+/}χ_(x)-      in Ω

=   0,
on Ω,
and, for any fixed $j=1,2,\ldots,k$, we consider the problems

-ΔP=   |x|^-2e^,
in Ω

P=   0,
on Ω.

-ΔP=   |x|^-2e^,      in Ω,

P=   0      on Ω.
Proof of $(S_j^1)$. Testing problem (<ref>) by $P\zajod$, integrating by parts
and using (<ref>) we obtain the relation:

We may write
∫_   |x|^-2e^{1+/}P-∫_|x|^-2e^

=   ∫_|x|^-2e^{1+/}(P-)

Therefore, multiplying  (<ref>) by $\pn$, we derive that
∫_|x|^-2   e^{1+/}(P-)

=   ∫_P-∫_∖|x|^-2e^.

Multiplying by $\pn$ and observing that

∫_∖|x|^-2e^ dx=O(_j-1/+(/_j+1)^/2)
we derive

∫_|x|^-2e^   {1+/}(P-)

In view of Lemma <ref>, the first term in (<ref>) takes the form
∫_|x|^-2   e^{1+/}(P-)

=   ∫_|x|^-2e^{1+/}(1+O(^))
The second term in (<ref>) satisfies
∫_|x|^-2e^ dx=+o(1).
In order to estimate the third term in (<ref>) we set $x=\dei y$:
∫_|x|^-2e^   {1+/}P dx

=   ∫_/|y|^-2e^(y){1+(y)/}(y)P(y) dy.
For $i<j$ (fast scaling) we estimate, using (<ref>):
∫_/|y|^-2e^(y){1+(y)/}(y)P(y) dy

=   ∫_/|y|^-2e^(y){1+(y)/}(y)(2+O(|y|/)^) dy
For $i>j$ (slow scaling) we estimate, using (<ref>):
∫_/|y|^-2e^(y){1+(y)/}(y) P(y) dy

=   ∫_/|y|^-2e^(y){1+(y)/}(y)O((/|y|)^+^) dy=o(1).
Hence, ($S_j^1$) is established.
Proof of ($S_j^2$).
Testing problem (<ref>) by $P\Udaj$, integrating by parts and recalling problem (<ref>), we obtain the identity:

We note that $\int_\Om|x|^{\ai-2}e^{\Udai}\phin=o(1)$ by the first equation in (<ref>), and $\int_\Om\thn P\Udai=o(1)$
by decay rate of $\thn$.
By the change of variables $x=\dei y\in\Bi$, we obtain
∫_|x|^-2e^   {1+/}P(x) dx

=   ∫_/|y|^-2e^(y){1+(y)/}(y)P(y) dy.
We estimate the integral above, using Lemma <ref>.
For $i=j$ (natural scaling), we estimate:
∫_/   |y|^-2e^(y){1+(y)/}(y)P(y) dy

=   ∫_/|y|^-2e^(y){1+(y)/}(y)×

×{(y)-ln(2^2)-2ln+4πh(0)+O(|y|)+O(^)} dy

=   ∫_/|y|^-2e^(y){1+(y)/}(y)(y) dy

-2ln∫_/|y|^-2e^(y){1+(y)/}(y) dy+o(1).

In view of the second equation in (<ref>)
we have
∫_/   |y|^-2e^(y){1+(y)/}(y)(y) dy

=   ∫_^2|y|^-2e^(y)(y)(y) dy+o(1)
We deduce that if $i=j$, then
∫_/|y|^-2e^(y){1+(y)/}(y)P(y) dy
For $i<j$ (fast scaling):
∫_/   |y|^-2e^(y){1+(y)/}(y)P(y) dy

=   ∫_/|y|^-2e^(y){1+(y)/}(y)×

×   {-2ln+4πh(0)+O((|y|/)^+|y|+^)} dy
For $i>j$ (slow scaling) we have
∫_/   |y|^-2e^(y){1+(y)/}(y)P(y) dy

=   ∫_/|y|^-2e^(y){1+(y)/}(y)×

×   {2ln1/|y|-2ln+4πh(0)
+O((/|y|)^+|y|+^)} dy

=   2∫_/|y|^-2e^(y){1+(y)/}(y)ln1/|y| dy
Using the third equation in (<ref>)

we derive:
∫_/|y|^-2e^(y){1+(y)/}(y)ln1/|y| dy

It follows that for $i>j$:
∫_/|y|^-2e^(y){1+(y)/}(y)P(y) dy
Inserting into (<ref>) we obtain
that is,
Hence, ($S_j^2$) is established.

§ LINEARIZED OPERATOR: UNIFORM VANISHING ON SHRINKING RINGS

§.§ Evaluation of $\Ia$

An easy computation shows that $w$  is a radial solution to

\begin{equation}\label{l1}
\Delta w+2\alpha^2{|y|^{\alpha-2}\over(1+|y|^\alpha)^2}w=2\alpha^2{|y|^{\alpha-2}\over(1+|y|^\alpha)^2}f(|y|)\ \hbox{in}\ \mathbb R^2
\end{equation}

if and only if the function $\tilde w(y)=w\(|y|^{2\over\alpha}\)$ is a radial  solution to

\begin{equation}\label{l2}
\Delta \tilde w+ {8\over(1+|y|^2)^2}\tilde w= {8\over(1+|y|^2)^2}\tilde f(|y|)\ \hbox{in}\ \mathbb R^2
\end{equation}

$\tilde f(|y|):=f\(|y|^{2\over\alpha}\)$.
Now let

\begin{equation}\label{effe}
\fa(y)={1\over2}\(\underbrace{\ln {2\alpha^2\over (1+|y|^\alpha)^2}+(\alpha-2)\ln|y|}_{:=\mVa} |\)^2.
\end{equation}

$$\tVa(y)=  \ln {2\alpha^2\over (1+|y|^2)^2}+  {2(\alpha-2)\over\alpha}\ln|y|$$

Our aim in this section is to evaluate the integrals

:=∫_^2e^(y)z_^2(y)(y) dy
=∫_ℝ^22α^2|y|^α-2(1-|y|^α)^2(1+|y|^α)^4(y) dy.
Indeed, we establish the following.

There holds

=8π\(- \ln(2\alpha^2) +\frac{3\alpha-2}{\alpha }\)<0
for all $\al\ge2$.

By change of variables, we have

\begin{equation}\label{I}
\begin{aligned}
\Ia=&\int\limits_{\mathbb R^2}2\alpha^2{|y|^{\alpha-2}(1-|y|^\alpha)^2\over(1+|y|^\alpha)^4}\Da(y)\,dy\\
=&                                                                                2\pi\int\limits_0^\infty 2\alpha^2{r^{\alpha-1}(1-r^\alpha)^2\over(1+r^\alpha)^4}\Da(r)dr\\
=&                                                                                8\pi\alpha\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4}\Da\(s^{2\over\alpha}\)ds\\
=&                                                                                8\pi\alpha\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4} \tDa\(s \)ds\\
=&                                                                                8\pi\alpha\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4} \(\tilde w\(s \)-\tVa(s)-\frac12\tVa^2(s)\)ds\\
=:&\mathcal I_\al'+\mathcal I_\al''.
\end{aligned}\end{equation}

We compute

\begin{equation}\label{I1}
\begin{aligned}
\mathcal I_\al'&=-8\pi\alpha\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4}  \tilde v (s) ds\\
&=-8\pi\alpha\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4} \( \ln {2\alpha^2\over (1+s^2)^2}+  {2(\alpha-2)\over\alpha}\ln s\)ds\\
&=-8\pi\alpha\(\frac16\ln(2\alpha^2)-\frac 49\)
\end{aligned}
\end{equation}

$$\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4} ds=\frac16$$

$$\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4} \ln (1+s^2)ds=\frac29$$

$$\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4} \ln sds=0.$$

Moreover, taking into account that $\tilde w$ solves

$$\tilde w''+{1\over r}\tilde w'+{8\over(1+r^2)^2}\tilde w={4\over(1+r^2)^2}\(\ln {2\alpha^2\over (1+r^2)^2}+  {2(\alpha-2)\over\alpha}\ln r\)^2=:\Fa(s)$$

In view of Lemma <ref>,
we have the representation

$$\tilde w(r)={1-r^2\over 1+r^2}\(\phi_{\Fa}(1) {r\over r-1}+\int\limits_0^r  {\phi_{\Fa}(s)-\phi_{\Fa}(1)\over (s-1)^2}ds\)$$

$$\phi_{\Fa}(s)=\(1+s^2\over 1-s^2\)^2{(s-1)^2\over s}\int\limits_0^s t{1-t^2\over 1+t^2}{\Fa}(t)dt=
\(1+s^2\over 1+s\)^2{1\over s}\int\limits_0^s t{1-t^2\over 1+t^2}{\Fa}(t)dt.$$

We know that

$$Z={1-|y|^2\over 1+|y|^2}$$

$$-\Delta Z=  {8\over (1+|y|^2)^2} Z= 8{1-|y|^2\over (1+|y|^2)^3} $$

Then $Z^2$ solves

$$\nabla Z={-2y\over 1+|y|^2}-2y{1-|y|^2\over (1+|y|^2)^2}={-4y\over (1+|y|^2)^2}$$

$$|\nabla Z|^2= {16|y|^2\over (1+|y|^2)^4} $$

$$\Delta Z^2=2Z\Delta Z+2|\nabla Z |^2=-{16(1-|y|^2)^2\over (1+|y|^2)^4} +{32|y|^2\over (1+|y|^2)^4}= -16{ 1+|y|^4-4|y|^2\over (1+|y|^2)^4}$$

\begin{equation}\label{I3}
\begin{aligned}
\Ia''&=8\pi\alpha\int\limits_0^\infty s{(1-s^2)^2\over(1+s^2)^4} \(\tilde w\(s\)-\frac12 \tVa^2 (s)\)\,ds\\
&=4 \alpha\int\limits_{\mathbb R^2}  {(1-|y|^2)^2\over(1+|y|^2)^4} \(\tilde w\(y\)-\frac12 \tVa^2 (y)\)\,dy\\
&=-\frac12 \alpha\int\limits_{\mathbb R^2}  {Z^2(y)} \Delta \tilde w(y)\,dy\\
&=-\frac12 \alpha\int\limits_{\mathbb R^2}  \Delta {Z^2(y)}  \tilde w(y)\,dy\\
&=8 \alpha\int\limits_{\mathbb R^2} { 1+|y|^4-4|y|^2\over (1+|y|^2)^4}  \tilde w(y)\,dy\\
&=16\pi \alpha\int\limits_0^\infty r{ 1+r^4-4r^2\over (1+r^2)^4}  \tilde w(r) dr\\
&=16\pi \alpha\int\limits_0^\infty r{ (1-r^2)(1+r^4-4r^2)\over (1+r^2)^5} \(\phi_{\Fa}(1) {r\over r-1}+\int\limits_0^r  {\phi_{\Fa}(s)-\phi_{\Fa}(1)\over (s-1)^2}ds\)
\\
&=16\pi \alpha\left[-\phi_{\Fa}(1)\underbrace{\int\limits_0^\infty r^2{(1+r)(1+r^4-4r^2)\over  (1+r^2)^5} dr}_{= {\pi\over 128}}
+\int\limits_0^\infty r{ (1-r^2)(1+r^4-4r^2)\over (1+r^2)^5}  \int\limits_0^r  {\phi_{\Fa}(s)-\phi_{\Fa}(1)\over (s-1)^2}ds \right]\\
&=16\pi \alpha \left[-{\pi\over 128}\phi_{\Fa}(1)+\int\limits_0^\infty r{ (1-r^2)(1+r^4-4r^2)\over (1+r^2)^5}  \int\limits_0^r  {\phi_{\Fa}(s)-\phi_{\Fa}(1)\over (s-1)^2}ds \right]\\
&=  16\pi \alpha\left[-{\pi\over 128}\phi_{\Fa}(1)-\int\limits_0^\infty {(r^2 - r^4 + r^6)\over  2 (1 + r^2)^4 }{\phi_{\Fa}(r)-\phi_{\Fa}(1)\over (r-1)^2}dr\right]\\
&\\ &=16\pi \alpha\left[-{\pi\over 128}\phi_{\Fa}(1)\right]\\
&+16\pi \alpha\Big\{-{1 - 2 r^2\over  8 (-1 + r^4)}\int\limits_0^r t{1-t^2\over 1+t^2}{\Fa}(t)dt-\int\limits_0^r
t{1 - 2 t^2\over  8  (1+t^2)^2}{\Fa}(t)dt \\
&\qquad\qquad+\phi_{\Fa}(1)\left[ {r (1 + r)^3 (1 - 4r + r^2) \over 64 (-1 + r) (1 +r^2)^3} + {1\over 64} \arctan r\right]\Big\}_{r=0}^{r=\infty}\\
&=16\pi \alpha[-{\pi\over 128}\phi_{{\Fa}}(1)-\int\limits_0^\infty
t{1 - 2 t^2\over  8  (1+t^2)^2}{\Fa}(t)dt+ \phi_{\Fa}(1) {\pi\over128} ]\\
&=-2\pi \alpha \int\limits_0^\infty
t{1 - 2 t^2\over    (1+t^2)^2}{\Fa}(t)dt
\end{aligned}\end{equation}

&  \int   {(r^2 - r^4 + r^6)\over  2 (r-1)^2(1 + r^2)^4 } dr= {r (1 + r)^3 (1 - 4r + r^2) \over 64 (-1 + r) (1 +r^2)^3} + {1\over 64} \arctan r
\end{aligned}$$

&\int\limits  {(r^2 - r^4 + r^6)\over  2 (r-1)^2 (1 + r^2)^4 } \phi_{\Fa}(r) dr \\
&=\int\limits {(r^2 - r^4 + r^6)\over  2 (r-1)^2 (1 + r^2)^4 }\(1+r^2\over 1-r^2\)^2{(r-1)^2\over r}\int\limits_0^r t{1-t^2\over 1+t^2}{\Fa}(t)dt\\
&=\int\limits  {(r^2 - r^4 + r^6)\over  2r  (1 - r^4)^2  } \int\limits_0^r t{1-t^2\over 1+t^2}\Fa(t)dt\\
&={1 - 2 r^2\over  8 (-1 + r^4)}\int\limits_0^r t{1-t^2\over 1+t^2}\Fa(t)dt-\int
{1 - 2 r^2\over  8 (-1 + r^4)} r  {1-r^2\over 1+r^2}\Fa(r)dr \\
&={1 - 2 r^2\over  8 (-1 + r^4)}\int\limits_0^r t{1-t^2\over 1+t^2}\Fa(t)dt+\int
r{1 - 2 r^2\over  8  (1+r^2)^2}\Fa(r)dr \end{aligned}$$

We have to compute

t{1 - 2 t^2\over     (1+t^2)^2}\Fa(t)dt=4\int\limits_0^\infty
t{1 - 2 t^2\over    (1+t^2)^4} \(\ln {2\alpha^2\over (1+t^2)^2}+  {2(\alpha-2)\over\alpha}\ln t
\)^2dt\\
t{1 - 2 t^2\over    (1+t^2)^4} dt}_{=0}+16\underbrace{\int\limits_0^\infty
t{1 - 2 t^2\over    (1+t^2)^4} (\ln(1+t^2))^2dt}_{=-\frac5{36}}+ 16{(\alpha-2)^2\over\alpha^2}\underbrace{\int\limits_0^\infty
t{1 - 2 t^2\over    (1+t^2)^4}(\ln t)^2 dt}_{= \frac1{8} }\\&
-16 (\ln2\alpha^2) \underbrace{\int\limits_0^\infty
t{1 - 2 t^2\over    (1+t^2)^4} \ln(1+t^2) dt}_{=-\frac1{12}}+ 16{(\alpha-2) \over\alpha } (\ln2\alpha^2) \underbrace{\int\limits_0^\infty
t{1 - 2 t^2\over    (1+t^2)^4}(\ln t)  dt}_{=-\frac1{8}}\\ &-32{(\alpha-2) \over\alpha } \underbrace{  \int\limits_0^\infty
t{1 - 2 t^2\over    (1+t^2)^4} (\ln t)\ln(1+t^2)}_{=-\frac1{16}}  dt\\
\end{aligned}$$

I(\alpha) &=8\pi\alpha\( -\frac16\ln(2\alpha^2)+\frac 49+\frac59-\(\frac13- \frac{\alpha-2}{2\alpha}\)\ln(2\alpha^2)- \frac{(\alpha-2)^2}{2\alpha^2}-\frac{\alpha-2}{2\alpha}\)\\
8\pi \(- \ln(2\alpha^2) +\frac{3\alpha-2}{\alpha }\)
\end{aligned}$$

Finally, we have

$${I'(\alpha)}=8\pi \(-\frac2\alpha+\frac2{\alpha^2}\)<0\ \hbox{for any}\ \alpha\ge2$$

$$ $$

and so

$$I(\alpha)<I(2)=8\pi(-\ln8 +2)<0\ \hbox{for any}\ \alpha\ge2,$$

as asserted.

In view of Lemma <ref> we set
Then, system $(S_j^1)-(S_j^2)$ may be written in the form
where $\mathcal M_{k,n}$ is the $2k\times2k$ matrix defined by
1   ℐ__1   0   0   0   0   …   …   0   0

b_1    2π   b_2   4π   b_3   4π   …   …   b_k   4π

2   0   1   ℐ__2   0   0   …   …   0   0

b_2   0   b_2    2π   b_3   4π   …   …   b_k   4π

2   0   2   0   1   ℐ__3   …   …   0   0

b_3   0   b_3     0   b_3   2π   …   …   b_k   4π

⋮   ⋮   ⋮    ⋮   ⋮    ⋮   ⋱   ⋱   ⋮   ⋮

⋮   ⋮   ⋮    ⋮   ⋮    ⋮   ⋱   ⋱   ⋮   ⋮

2   0   2   0   0   0   …   …   1   ℐ_

b_k   0   b_k     0   b_k   0   …   …   b_k   2π).

There exists $c_0>0$ such that $|\mathcal M_{k,n}|\ge c_0>0$ for all $n\in\mathbb N$.

It is equivalent to prove that
$|\widetilde{\mathcal M}_{k,n}|\ge c_0>0$ for all $n\in\mathbb N$,
where $\widetilde{\mathcal M}_{k,n}$ is the $2k\times2k$ matrix  defined by

$$\widetilde{\mathcal M}_{k,n}=\left(\begin{matrix}
b_1& 1&b_2&2&b_3&2&\dots&\dots& b_k&2\\
b_2&0&b_2 & 1& b_3&2&\dots&\dots&b_k&2\\
b_3&0&b_3 & 0&b_3&1&\dots&\dots& b_k&2\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
b_k&0& b_k & 0& b_k&0&\dots&\dots&b_k&1\\
\end{matrix}\right),$$

whose entries satisfy $a_i,b_i>0$, for any $i=1,2,\ldots,k$, and $b_i\ge b_{i+1}$, $i=1,2,\ldots,k-1$.
In $\widetilde{\mathcal M}_{k,n}$ we replace row $2j$ by the difference between row $2j$ and row $(2j+2)$, $j=1,2,\ldots,k-1$.
Thus, we obtain the matrix $\widetilde{\widetilde{\mathcal M}}_{k,n}$:

$$\widetilde{\widetilde{\mathcal M}}_{k,n}=\left(\begin{matrix}
c_1& 1&0&1&0&0&\dots&\dots&0&0\\
c_2&0&c_2 & 1& 0&1&\dots&\dots&0&0\\
c_3&0&c_3 & 0&c_3&1&\dots&\dots& 0&0\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
c_k&0& c_k & 0& c_k&0&\dots&\dots&c_k&1\\
\end{matrix}\right),$$

where $c_i=b_i-b_{i+1}$, $i=1,2,\ldots,k-1$, $c_k=b_k$ satisfy $c_i>0$ for any $i$.
In $\widetilde{\widetilde{\mathcal M}}_{k,n}$
we replace
column $(2j-1)$ by the difference between column $(2j-1)$ and column $(2j+1)$, $j=1,2,\ldots,k-1$.
Thus, we obtain the matrix

$$\mathcal A_k:=\left(\begin{matrix}
c_1& 1&0&1&0&0&\dots&\dots&0&0\\
0&0&c_2 & 1&0&1&\dots&\dots&0&0\\
0&0&0 & 0&c_3&1&\dots&\dots&0&0\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
0&0&0& 0&0&0&\dots&\dots&c_n&1\\
\end{matrix}\right),$$

and $|\widetilde{\widetilde{\mathcal M}}_{k,n}|=|\widetilde{\mathcal M}_{k,n}|=|\mathcal A_k|$.
A recurrence argument shows that

$$|\mathcal A_k|>0\ \hbox{for any}\ n\ge1.$$

Indeed, let us denote by $\mathcal A_j$ the submatrix of $\mathcal A_k$ obtained by deleting the
first $2(j-1)$ rows and the first $2(j-1)$ columns, namely

$$\mathcal A_j:=\left(\begin{matrix}
c_j& 1&0&1\dots&\dots&0&0\\
0&0&c_{j+1} & 1\dots&\dots&0&0\\
\vdots&\vdots&\vdots & \vdots&\ddots&\vdots&\vdots\\
\vdots& \vdots&\vdots & \vdots&\ddots&\vdots&\vdots\\
\end{matrix}\right).$$

Similarly, we denote by $\mathcal B_j$ the submatrix of $\mathcal A_k$ obtained by deleting the first $2j-1$ rows,
the first $2(j-1)$ columns and the $2j$th column,

$$\mathcal B_j:=\left(\begin{matrix}
1& 1&-a_{j+1}&0\dots&\dots&0&0\\
0&1&0 & \dots&\dots&0&0\\
\vdots&\vdots&\vdots & \vdots&\ddots&\vdots&\vdots\\
\vdots& \vdots&\vdots & \vdots&\ddots&\vdots&\vdots\\
\end{matrix}\right).$$

With this notation, we readily check that

|𝒜_j|=   |𝒜_j+1|+a_j|ℬ_j|

|ℬ_j|=   c_j|𝒜_j+1|+|ℬ_j+1|.

Indeed we have

$$\begin{aligned}&|\mathcal M_n|=\left|\begin{matrix}
c_1& 1&0&1&0&0&\dots&\dots&0&0\\
0&0&c_2 & 1&0&1&\dots&\dots&0&0\\
0&0&0 & 0&c_3&1&\dots&\dots&0&0\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots&\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
0&0&0& 0&0&0&\dots&\dots&c_n&1\\
\end{matrix}\right|\\
c_2 & 1&0&1&\dots&\dots&0&0\\
0 & 0&c_3&1&\dots&\dots&0&0\\
\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
0& 0&0&0&\dots&\dots&c_n&1\\
\end{matrix}\right|+a_1
\left|\begin{matrix}
c_1& 0&1&0&0&\dots&\dots&0&0\\
0&c_2 & 1&0&1&\dots&\dots&0&0\\
0&0 & 0&c_3&1&\dots&\dots&0&0\\
\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots&\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
0&0& 0&0&0&\dots&\dots&c_n&1\\
\end{matrix}\right|\\
c_2 & 1&0&1&\dots&\dots&0&0\\
0 & 0&c_3&1&\dots&\dots&0&0\\
\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
0& 0&0&0&\dots&\dots&c_n&1\\
\end{matrix}\right|\\ &+a_1\(c_1
\left|\begin{matrix}
c_2 & 1&0&1&\dots&\dots&0&0\\
0 & 0&c_3&1&\dots&\dots&0&0\\
\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots & \vdots&\vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
0& 0&0&0&\dots&\dots&c_n&1\\
\end{matrix}\right|  +
\left|\begin{matrix}
c_2 & 0&1&\dots&\dots&0&0\\
0 &  c_3&1&\dots&\dots&0&0\\
\vdots &  \vdots & \vdots&\ddots&\ddots&\vdots&\vdots\\
\vdots&  \vdots&\vdots & \ddots&\ddots&\vdots&\vdots\\
0& 0&0&\dots&\dots&1&-a_n\\
0&  0&0&\dots&\dots&c_n&1\\
\end{matrix}\right|\)\\
&=|\mathcal M_{n-1}|+a_1\(c_1|\mathcal M_{n-1}|+c_2|\mathcal M_{n-2}|+c_3|\mathcal M_{n-3}|+\dots+\left|\begin{matrix}
c_{n-1} & 0&1\\
0&   c_n&1\\
\end{matrix}\right|\)\\
&=|\mathcal M_{n-1}|+a_1\(c_1|\mathcal M_{n-1}|+c_2|\mathcal M_{n-2}|+c_3|\mathcal M_{n-3}|+\dots+c_{n-1}|\mathcal M_{1}|+c_n\)\end{aligned}$$

§ BARRIER ESTIMATE

Recall from (<ref>) that $\Lp$ is the operator defined by
For every $i=1,2,\ldots,k$ let $\tRi\gg1$ be a fixed large constant and for every $i=2,\ldots,k$
let $0<\tri\ll1$ be a fixed small constant.
Let $\tAjj$ denote the shrinking annulus defined by

B_∖B_   if j=1,2,…,k-1,

∖B_   if j=k.

The aim of this section is to establish the following result:

Suppose that $\Lp\phi=h$ in $\Om$, $\phi=0$ on $\pl\Om$.
Then, there exist suitable constants $0<\trj\ll1$, $\tRj\gg1$,
and $C>0$ independent of $p$ such that

We begin by showing that the 0-order operator" $\Lpo$ defined by
ϕ=Δϕ+∑_i=1^k|x|^-2e^(x)ϕsatifies a maximum principle in $\tAjj$.

To this end, we recall that
and we define the functions:

(x):=   -(x/),          for 0<≪1,

(x):=   (x/),          for ≫1.

Fix $0<\co<1$ and $\uDj>0$.
Suppose that $\tRj>0$ satisfies

Δ   ≤-|x|^-2e^

in ^2∖B_.

Claim 1. There holds $\Psuj(x)\ge\co>0$ if and only if

Indeed, by a straightforward computation, we have
(x)=|x/|^-1/|x/|^+1≥if and only if
(1-)|x/|^≥1+and the asserted necessary and sufficient condition (<ref>) readily follows.
Now, we assume that (<ref>) is satisfied.
Claim 2. Suppose that $\Psuj\ge\co$. There holds

Indeed, in view of (<ref>) we have
Δ=   -(/)^2(Δ)(/)

=   -(/)^2^2|x|^-2/(|x|^+(/)^)^2.
Thus, we estimate
It follows that a sufficient condition for (<ref>) to hold true is that
and finally
from which (<ref>) follows.
Claim 1 and Claim 2 yield the statement of the asserted lemma.

Fix $0<\co<1$ and $\oDj>0$.
Suppose that $\trj>0$ is such that
Δ   ≤-|x|^-2e^

in B_.

Similarly as in the proof of Lemma <ref>, we first establish the following.
Claim 1. There holds $\Psoj(x)\ge\co$ if and only if

Indeed, in view of (<ref>) we have
(x)=1-|x/|^/1+|x/|^≥if and only if
(1+)|x/|^≤1-and Claim 1 follows.
Claim 2. Suppose that $\Psoj\ge\co$. Then, there holds

Indeed, in view of (<ref>) we have
Δ=   (/)^2(Δ)(x/)

=   -(/)^2^2|x|^-2/(1+|x/|^)^2(x)
Therefore, we have the estimate
We deduce the sufficient condition
from which we derive
This establishes the sufficient condition (<ref>).
Claim 1 and Claim 2 imply the statement of Lemma <ref>.

For any given $\ovC>0$ and $0<\co<1$ there exists a function $\Psij$
and constants $0<\trj\ll1$ and $\tRj\gg1$
such that

}    in .

where $\Psui$, and $\Psoi$
are the functions defined in (<ref>).
We observe that
⋂_i≤j{|x|≥}={|x|≥},        ⋂_i≥j+1{|x|≤}={|x|≤r_j+1}
and therefore
provided that $p$ is sufficiently large.
Consequently, in view of Lemma <ref> and Lemma <ref>,
we may find $\tRj\gg1$ and $0<\trj\ll1$ such that

}    in .
It follows that
≤   -∑_i=1^j|x|^-2e^-∑_i=j+1^k|x|^-2e^+∑_i=1^k|x|^-2e^

≤   -∑_i=1^j(-k)|x|^-2e^-∑_i=j+1^k(-k)|x|^-2e^.
Hence, the asserted supersolution property (<ref>) readily follows by suitable choice of
$\uDi$, $\oDi$.

The following estimates hold for any $i=1,2,\ldots,k$:

\begin{align}
\tag{i}
\qquad\hbox{for all }|x|\ge\tRi\dei\\
\tag{ii}
\qquad\hbox{for all }|x|\le\tri\dei.
\end{align}

Proof of (i). We compute, for $|x|\ge\tRi\dei$:
≤2^2^-η/|x|^-η ^η/|x|^2+η
≤2^2/^-η η/|x|^2+η,
where we used $\dei/|x|\le\tRi^{-1}$ in order to derive the last inequality.
Proof of (ii).
We compute, for $|x|\le\tri\dei$:
where we used $|x|/\dei\le\tri$ in order to derive the last inequality.

In order to control the inhomogeneous term $h$, we define functions $\psij$, $\tpsij$
as follows. Let $M>2\diam\Om$.
Let $\psij$ be defined by

-Δ=^η/|x|^2+η   in B_M∖B_

=0   on (B_M∖B_)
and let $\tpsij$ be defined by

-Δ=1/^2   in B_

=0   on B_.

There holds

where $C_{1,j}$, $C_{2,j}$ are given by
Moreover, the following uniform bounds hold true

By straightforward computations we find that $\psij$ is of the form (<ref>).
The boundary conditions imply that

from which the asserted form of $\psij$ follows.
We observe that

C_1,j<0,    and    C_1,j=o(1)    as →0

C_2,j>0,    and    C_2,j=o(1)    as →0.
We have $\psij\ge0$ by the maximum principle. In order to establish the upper bound, we observe that
and therefore $\psij$ attains its maximum value for $r^\eta=-\dej^\eta/(\eta C_{1,j})$.
We compute:
The second boundary condition in (<ref>) and (<ref>)–(<ref>) yield
so that the first upper bound in (<ref>) is established.
The remaining bounds in (<ref>) are straightforward.

The following estimates hold true in $\tAjj$:

\begin{align}
\tag{i}
\tag{ii}
\end{align}

where $o(1)$ vanishes as $p\to+\infty$.

Proof of (i).
We compute, using Lemma <ref>:
=   Δ+∑_i=1^k|x|^-2e^(x)≤-^η/|x|^2+η+(/η^2^η+o(1))∑_i=1^k|x|^-2e^(x)

≤   -^η/|x|^2+η+(/η^2^η+o(1))∑_i=1^j-12^2/^-η^η/|x|^2+η

≤   (-1+2^2/η^2^

=   (-1+2^2/η^2^
as asserted.
Proof of (ii).
Similarly, we compute:
=   Δ+∑_i=1^k|x|^-2e^(x)

≤   -1/^2+^2/4∑_i=1^k|x|^-2e^(x)

≤   -1/^2+^2/4∑_i=1^j-12^2/^-η^η/|x|^2+η

≤   ^2/4(2^2/^-η

=   ^2/4(2^2/^-η
as asserted.

It is useful to observe that

provided that $p$ is sufficiently large.

The following estimates hold true:

\begin{align}
\tag{i}
&|h(x)|\le\frac{\|h\|_{\rp}\dej^\eta}{|x|^{2+\eta}},\qquad\hbox{for all }x\in\Aj\cap\tAjj\\
\tag{ii}
&|h(x)|\le\frac{\|h\|_{\rp}}{\dejj^{2}},\qquad\hbox{for all }x\in\Ajj\cap\tAjj
\end{align}

In particular, there holds
for all x∈.

We recall from (<ref>) that
$\|h\|_{\rp}=\sup_{1\le i\le k}\|\rho_i h\|_{L^\infty(\Ai)}$, where $\rho_i(x)=(\dei^{2+\eta}+|x|^{2+\eta})/\dei^\eta$.
Proof of (i).
For any $x\in\Aj\cap\tAjj$ we have
Proof of (ii). Similarly, we compute, for all $x\in\Ajj\cap\tAjj$:
as asserted.
Proof of (iii). The asserted estimate readily follows from (i)–(ii)
and (<ref>).

Finally, we provide the proof of the main result in this section, namely Proposition <ref>.

We define the barrier function
Since $\Psij\ge k\co>0$, $\psij\ge0$, $\tpsij\ge0$ in $\tAjj$, we readily have the boundary estimate

Φ≥ϕ_L^∞()    on .
We claim that

Φ≤ϕ=h    in .
Indeed, we have:
We recall from Lemma <ref> and Lemma <ref> with $\uDi=\oDi=D$, that

≤   -(D-k)∑_i=1^k|x|^-2e^<0,
provided that the $D>0$ is sufficiently large.
We recall from Lemma <ref> that

\begin{align*}
\end{align*}

uniformly in $\tAjj$, where $o(1)$ vanishes as $p\to+\infty$,
so that by possibly choosing a larger $\tRj$ and a smaller $\trjj$ we obtain te estimate

In conclusion, from (<ref>), (<ref>), Lemma <ref>–(iii) and the above we derive

≤   -h(^η/|x|^2+η+1/^2)
and (<ref>) is established. Now, the barrier estimate follows by the maximum principle.

§ THE FIXED POINT ARGUMENT

We recall from (<ref>) that the error $\Rp$ is defined by
and that the operator $\Lp$ is defined by
ϕ=Δϕ+(x)ϕfor all $\phi\in C_0(\overline\Om)$.
Therefore, setting

we see that $\up=\Up+\phip$ is a solution for (<ref>)
if and only if $-\Lp\phip=\Rp+\Np(\phip)$.
Defining the operator
we may rewrite (<ref>) with Ansatz $u=\up=\Up+\phip$
in the form

In other words, we are reduced to seek a fixed point $\phip\in C_0(\overline\Om)$ for $\Tp$.

:={ϕ∈C_0(): ϕ_∞<γ/p^3},
where $\ga>0$.

There exist $\ga>0$ and $p_0>0$ such that for every $p\ge p_0$
the operator $\Tp:\mathcal F_\gamma\to\mathcal F_\gamma$
is a contraction.

We first establish the following estimate.

There holds

We have
(x)   |(+O(1/p^3))|^(p-2)/p

=   ∑_j=1^k(x)||x|^-2e^(x){1+(x/)+O(1)/p}

≤   ∑_j=1^k(x)(|x|^-2e^(x)|{1+(x/)+O(1)/p}|^p-2/pχ_

Claim 1. $\Tp:\Fg\to\Fg$.
By applying the mean value theorem twice, we may write
for some $0\le\theta_p',\theta_p''\le1$.
Consequently, for $\phi\in\Fg$,
|(ϕ)|≤   |(+O(1/p^3))| |ϕ|^2
=p(p-1)|(+O(1/p^3))|^(p-2)/p |ϕ|^2,
and, in view of Lemma <ref>:
(ϕ)_≤   p(p-1)(+O(1/p^3))_ϕ_∞^2

≤   Cp(/p^3)^2=O(1/p^5).
It follows that, for any $\phi\in\Fg$ we have
Now Claim 1 follows by choosing $\ga$ suitably large.
Claim 2. $\Tp$ is a contraction.
Indeed, similarly as above, by two applications of the mean value theorem we obtain
for some $0\le\eta_p',\eta_p''\le1$.
|(ϕ_1)-(ϕ_2)|≤   |(+O(1/p^3))|max_i=1,2ϕ_i_∞|ϕ_1-ϕ_2|

≤   p(p-1)|(+O(1/p^3))|^(p-2)/p/p^3
≤   p(p-1)|(+O(1/p^3))|^(p-2)/p_/p^3ϕ_1-ϕ_2_∞

≤   C/p^2ϕ_1-ϕ_2_∞Therefore, we deduce that
This establishes Claim 2.

§ APPENDIX: USEFUL FACTS

In this Appendix we collect some useful properties and estimates.

§.§ Estimates in the annuli $\Aj$

We recall that

|x|<C_1^_1C_2^1-_1e^-(_1b_1+(1-_1)b_2)p    for all x∈A_1;

for all x∈, j=2,…,k-1;

|x|≥C_k-1^_k-1C_k^1-_k-1e^-(_k-1b_k-1+(1-_k-1)b_k)p,    for all x∈.
Consequently, we have

for all x∈, i<j, j=2,…,k;

for all x∈, i>j, j=1,2,…,k-1.
The next lemma clarifies the leading term of the quantity
$\ln(\dei^{\ai}+|\dej y|^{\ai})^{-2}$ for $\dej y\in\Aj$, $i,j=1,2,\ldots,k$.

There holds:
ln1/(1+|y|^)^2      i=j (natural scaling)

-2ln+O(/)^      i>j, i=j+1,…,k (fast scaling)

2ln1/|y|-2ln+O(_j-1/)^(1-_j-1)      i<j, i=1,…,j-1, (slow scaling),
uniformly for $\dej y\in\Aj$.

It suffices to establish the following:
^(1+|y|^)      i=j (natural scaling)

^(1+O(/)^)      i>j, i=j+1,…,k (fast scaling)

i<j, i=1,…,j-1, (slow scaling),
uniformly for $\dej y\in\Aj$.
To this end, suppose $i=j$. Then, we readily have
Suppose $i>j$.
Then, $\de_{j+1}=O(\dei)$, $|x|=|\dej y|<\dej^{\ej}\de_{j+1}^{1-\ej}$, and therefore

as asserted.
Suppose $i<j$.
Then, $\dei=O(\de_{j-1})$, $|x|=|\dej y|\ge\de_{j-1}^{\veps_{j-1}}\dej^{1-\veps_{j-1}}$,
and therefore

It follows that
as asserted.

§.§ Expansions and scalings in the $\Aj$'s.

We recall from (<ref>) that
where we recall that $\Coai$, $\Ciai$ are defined by the property (<ref>).
Moreover, we set

The following expansions hold true:
in C^1()

in C_loc^1(∖{0})
in C^1()

in C_loc^1(∖{0})

The proof is well-known, see, e.g., [11].
We outline the proof for the sake of completeness.
For any fixed $r>0$ we have
uniformly in $C^1(\Om\setminus B_r(0))$. Therefore, we may write
uniformly in $C^1(\Om\setminus B_r(0))$.
It follows that
Δ((x)-ln(2^2^)+4πH(x,0))=Δ(x)   in

(x)-ln(2^2^)+4πH(x,0)=O(^)   on ∂,
so that $P\Udai=\Udai(x)-\ln(2\ai^2\dei^{\ai})+4\pi\ai H(x,0)+O(\dei^{\ai})$ in $C^1(\overline\Om)$,
as asserted.
By properties of $\wlai$, $\ell=0,1$, we have
Hence, for any $r>0$ we may write
uniformly in $\Om\setminus B_r(0)$.
The statement follows observing that

Δ(-2πH(x,0)+ln)=Δ   in

-2πH(x,0)+ln=O()   on ∂,

so that $P\wldai=\wldai-2\pi\Clai H(x,0)+\Clai\ln\dei+O(\dei)$ in $C^1(\overline\Om)$,
as asserted.

The aim of the next lemma is to establish the profile of the $i$-th bubble observed in the shrinking ring $\Aj$.
It will be useful to note that we may write

Let $i,j=1,2,\ldots,k$.
The following expansions hold, uniformly for $x=\dej y\in\Aj$:

\[
\left\{
\bal
+4\pi\aj h(0)
+O(|\dej y|)+O(\dej^{\aj}),\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{if }i=j\ \hbox{(natural scaling)};\\
&-2\ai\ln\dei+4\pi\ai h(0)+O(\frac{\dej}{\dejj})^{\ej\ai}+O(\dej|y|)+O(\dei^{\ai}),\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{if }i>j\ \hbox{(slow bubble)};\\
-2\ai\ln\dej+4\pi\ai h(0)
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{if }i<j\ \hbox{(fast bubble)};
\eal
\right.
\]

\[
\left\{
\bal
&\wlaj(y)+\Claj\ln\dej-2\pi\Claj h(0)
+O(|\dej y|)+O(\dej),\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{if }i=j\ \hbox{(natural scaling)};\\
&\wlai(0)+\Clai\ln\dei-2\pi\Clai h(0)
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\hbox{if }i>j\ \hbox{(slow bubble)};\\
&\Clai\ln|y|+\Clai\ln\dej-2\pi\Clai h(0)
&\qquad\qquad\qquad\qquad\qquad\qquad\hbox{if }i<j\ \hbox{(fast bubble)}.
\eal
\right.
\]

Suppose $i=j$.
In view of Lemma <ref> and (<ref>) we readily derive the expansion.
Suppose $i>j$.
In view of Lemma <ref> and Lemma <ref> we have
P(y)=   ln1/(^+|y|^)^2+4πH(y,0)+O(^)

=   -2ln+O(/)^+4πh(0)+O(|y|)+O(^),
as asserted.
Suppose $i<j$ (fast bubble). In view of Lemma <ref> and Lemma <ref> we have
P(y)=   ln1/(^+|y|^)^2+4πH(y,0)+O(^)

=   2ln1/|y|-2ln+O(_j-1/)^(1-_j-1)+4πh(0)+O(|y|)+O(^),
and the asserted expansion follows.
Now, we consider the correction term $\wldai$.
Suppose $i<j$ (fast bubble).
In view of Lemma <ref> we have
Using (<ref>) we obtain
|y|/≥C^-1(/_j-1)^1-_j-1→+∞and therefore
It follows that
P(y)=   ln|y|+ln/+O(/|y|)+ln-2πh(0)+O(|y|)+O()

=   ln|y|+ln-2πh(0)+O(_j-1/)^1-_j-1+O(|y|)+O(),
as asserted.

The following expansions holds true, uniformly for $x=\dej y\in\Aj$:

if i=j

if i>j

if i<j,
where $\ej\ai+2(1-\ej)>0$ and $(1-\veps_{j-1})\ai-2\veps_{j-1}>0$.

For $i=j$ the proof follows by the change of variables $x=\dej y$.
For $i<j$, $x=\dej y\in\Aj$, we obtain by change of variables that
Recalling the definition of $\Aj$, we observe that
Finally, we observe that
and, by definition of $\veps_{j-1}$,
Similarly, for $i>j$ we obtain by change of variables that
Recalling the definition of $\Aj$ we have
In order to conclude the proof, we observe that

There holds:

uniformly in $\Om$.

if i=j (natural scaling);

if i>j (fast scaling);

if i<j slow scaling,
uniformly for $x=\dej y\in\Aj$.

The proof is straightforward.

§.§ Choice of the parameters $\de,\tau$ in the case $k=1$.

We recall from Section <ref> that

\[
\Ud(x)=\ln\frac{8\de^2}{(\de^2+|x|^2)^2},
\]

for all $\de>0$.
Then, in view of Lemma <ref> with $\ai=2$ and $\de=\dei$, we have

\[
P\Ud(x)=\ln\frac{1}{(\de^2+|x|^2)^2}+8\pi H(x,0)+O(\de^2)=v(\frac{x}{\de})+\ln\frac{1}{\de^4}+O(1).
\]

Therefore, taking $\ln\de^{-4}=p$, we may expand:

\[
\bal
\eal
\]

On the other hand, we have

\[
-\Delta(\tau P\Ud)=\tau e^{\Ud}.
\]

Choosing $\tau^{p-1}p^p\de^2=1$, we obtain

and therefore $\tau P\Ud$ is indeed an approximate solution for (<ref>).
We have obtained the following necessary conditions for the parameters:

=e^-p/4,         τ=e^p/[2(p-1)]/p^p/(p-1)=√(e)/p(1+O(lnp/p)).

§.§ Properties of the weighted norm

The following properties hold true:

$\|h\|_{L^1(\Om)}\le C\|h\|_{\rp}$,
for some $C>0$ independent of $h$ and $p$;

$\|\dej^2 h\|_{L^\infty(\Aj)}\le C\|h\|_{\rp}$; (relevant)

$\|\rj h\|_{L^\infty(\Aj)}\le\|h\|_{\rp}$;(obvious)

If $\eta\le2s_1$, then $\|\rp\|_{L^\infty(\Om)}\le C$.

In particular, for any $q>0$ we have the weighted mass estimate

Proof of (i).
Recall that
(x) dx=   ∑_j=1^k(x)χ_(x) dx

(x) dx=   ^2+η+|x|^2+η/^η dx=^2(1+|y|^2+η) dy,
and $\|h\|_{\rp}=\|\rp h\|_{L^\infty(\Om)}$.
We readily check that:
∫_|h(x)| dx=∫_(x)|h(x)|/(x) dx
Proof of (ii).
For all $x\in\Aj$ we compute:
^2 h(x)≤^2h_L^∞()/(x)=h_L^∞()/1+|x/|^2+η≤h_.
Proof of (iii).
By definition of $\|\cdot\|_{\rp}$, we have
Proof of (iv). For any fixed $j=1,2,\ldots,k$ and $x\in\Aj$,
Proof of (v).
Let $j=1,2,\ldots,k$. Recalling that $x=\dej y\in\Aj$ implies that $|y|\le(\dejj/\dej)^{1-\ej}$,
we estimate
Thus, a sufficient condition for boundedness of $\rj$ is given by $\eta\le2\ej/(1-\ej)=2\sj$.
In view of Proposition <ref>–(iii), by choosing $\eta\le2s_1\le\sj$, we obtain
the asserted uniform boundedness for $\rp$.

For $r>1$ the above argument yields:
∫_|h(x)|^r dx≤h_∫_^r/(^2+|x|^2)^3r/2(/|x|)^r dx
which does not yield a uniform embedding constant.

The following elementary inequalities hold true:
if s≥-p

if s<-p;
if s≥-p

if s<-p;
if s≥-p

if s<-p.

The proof of (i) follows by concavity of the logarithmic function and reflection properties.
The proof of (ii)–(iii) follow by the identities $\gpp(t)=p|t|^{p-1}=p|\gp(t)|^{\frac{p-1}{p}}$
and $|\gp''(t)|=p(p-1)|t|^{p-2}=p(p-1)|\gp(t)|^{\frac{p-2}{p}}$.

Let $a(t)$, $b(t)$, $c(t)$, $t>0$, be smooth, real-valued functions
for some $0\le\si<\tau$ and $C>0$.
Let $\Eap\subset(0,+\infty)$ be defined by
:={t>0: a(t)>-p/2}.
Then, the following expansions hold as $p\to+\infty$, uniformly for $t\in\Eap$:

\[
\tag{i}
\bal
&\ =e^{a(t)}\left\{1+\frac{b(t)-\vphio(a(t))}{p}
\eal
\]

where $\vphio$, $\vphii$ are defined in (<ref>).
Moreover, for any fixed $0<\kappa<p$, there holds

=e^a(t){1+1/p(b(t)-(a(t))-κ a(t))
uniformly with respect to $t\in\Eap$.

We shall repeatedly use the following properties:

$C^{-1}e^{-p/(2\si)}\le t\le Ce^{\frac{p}{2(\tau-\si)}}$, for some $C>0$ independent of $t\in\Eap$;

$-\frac{p}{2}\le a(t)\le C$, in particular $a(t)=O(p)$ and $|b(t)|+|c(t)|=O(|a(t)|+1)=O(p)$,
uniformly for $t\in\Eap$.

Proof of (i).
Let $\xip=\xip(t)$ be defined by

\[
\xip(t)=\frac{a(t)}{p}+\frac{b(t)}{p^2}+\frac{c(t)+o(1)}{p^3}=\frac{a(t)+O(1)}{p}.
\]

Since $a(t)$ is bounded from above, by taking $p$ sufficiently large we may assume that
$|\xip(t)|\le3/4$ in $\Eap$.
Therefore, by Taylor expansion of the logarithmic function up to the third order, we may write
log(1+)=   -^2/2+^3/3-^2/4(1+θ_p)^4

=   a/p+1/p^2(b-a^2/2)+1/p^3(c-ab+a^3/3)-^2/4(1+θ_p)^4+O(|a|^3+1)/p^4+o(1/p^3)
for some $0\le\theta_p(t)\le1$, uniformly with respect to $p\to+\infty$ and $t\in\Eap$.
It follows that
plog(1+)=a+1/p(b-a^2/2)+1/p^2(c-ab+a^3/3)-p ^2/4(1+θ_p)^4+O(|a|^3+1)/p^3+o(1/p^2),
uniformly with respect to $p\to+\infty$ and $t\in\Eap$.
We set
:=1/p(b-a^2/2)+1/p^2(c-ab+a^3/3)-p ^2/4(1+θ_p)^4+O(|a|^3+1)/p^3+o(1/p^2),
and we observe that $\txip(t)\le C$, namely $\txip$ is uniformly bounded from above.
Therefore, by Taylor expansion of the exponential function to the second order we may write
e^=+^2/2+e^θ_p /6^3,
for some $0\le\widetilde\theta_p\le1$, so that $e^{\widetilde\theta_p\txip}\le C$,
namely $e^{\widetilde\theta_p\txip}$ is bounded uniformly
with respect to $p\to+\infty$ and $t\in\Eap$.
Observing that
p ^4=O(|a|^4+1)/p^3,
we obtain the expansions
=   1/p(b-a^2/2)+1/p^2(c-ab+a^3/3)+O(|a|^4+1)/p^3+o(1/p^2),

^2=   1/p^2(b-a^2/2)^2+O(|a|^6+1)/p^4+o(1/p^2)

^3=   O(|a|^6+1)/p^3,
from which we finally derive
uniformly with respect to $p\to+\infty$ and $t\in\Eap$.
This established the asserted expansion (i).

Proof of (ii).
Similarly as above, let
By taking $p$ sufficiently large, we may assume that $|\zep(t)|\le3/4$
for all $t\in\Eap$.
Expansion of the logarithmic function to the second oder yields
log(1+)=   -^2/2+^3/3(1+θ_p)^3

=   a/p+1/p^2(b-a^2/2)+O(|a|^2+1)/p^3+^3/3(1+θ_p)^3+o(1/p^2),
uniformly with respect to $p\to+\infty$ and $t\in\Eap$.
We deduce that
We set
and we observe that $\tzep(t)$ is uniformly bounded from above with respect
to $p\to+\infty$ and $t\in\Eap$.
Therefore, expansion of the exponential function to the first order yields
for some $0\le\widetilde\theta_p(t)\le1$, so that $e^{\widetilde\theta_p(t)\tzep(t)}\le C$
is uniformly bounded with respect
to $p\to+\infty$ and $t\in\Eap$.
We deduce that
and the asserted expansion (ii) follows.

By adapting the arguments in [11] we have the following result.

Fix $\al\ge2$.
Let $F\in C^1([0,+\infty))$ be such that

as t→+∞.
Then, there exists a $C^2$ radial solution $w_F(y)$ to the equation

as |y|→+∞where
C_F=∫_0^+∞t(t)F(t) dt.

In view of Lemma 2.1 in [5] it is known that there esists a $C^2$ radial solution $w(r)$
to (<ref>) of the form
w(r)=(r){∫_0^rϕ_F(s)-ϕ_F(1)/(1-s)^2 ds+ϕ_F(1)r/1-r}

ϕ_F(s)=(1-s)^2/s(s)^2∫_0^st(t)F(t) dt,
|w(r)|≤C(ln^+r+1)    as r→+∞,
where $\phiF(1)$ and $w(1)$ are defined as limits of $\phiF(r)$ and $w(r)$ as $r\to1$.
In order to derive the exact logarithmic growth factor, we write for $r\ge2$:
∫_0^r(s)-(1)/(s-1)^2 ds
=   ∫_0^2(s)-(1)/(s-1)^2 ds+∫_2^r(s)/(s-1)^2 ds

=   ∫_2^r(s)/(s-1)^2 ds+∫_0^2(s)-(1)/(s-1)^2 ds

=   ∫_2^r(s)/(s-1)^2 ds+D_F,1+O(1/r),

D_F,1=∫_0^2(s)-(1)/(s-1)^2 ds-(1).
In turn, we write, recalling (<ref>) and the fact $\phoa^{-1}(s)=-1+2/(1-s^\al)$,

∫_2^r(s)/(s-1)^2 ds
=   ∫_2^rds/s^2(s)∫_0^st(t)F(t) dt
=∫_2^rds/s(-1+2/1-s^)^2∫_0^st(t)F(t) dt

=   ∫_2^rds/s∫_0^st(t)F(t) dt
-4∫_2^rds/s(1-s^)∫_0^st(t)F(t) dt

+4∫_2^rds/s(1-s^)^2∫_0^st(t)F(t) dt.
Integration by parts yields
∫_2^rds/s   ∫_0^st(t)F(t) dt
=lnr∫_0^rt(t)F(t) dt+D_F,2
+∫_r^+∞slns(s)F(s) ds

=   C_Flnr+D_F,2-lnr∫_r^+∞t(t)F(t) dt+∫_r^+∞slns(s)F(s) ds
D_F,2=   -ln2∫_0^2t(t)F(t) dt-∫_2^+∞slns(s)F(s) ds

C_F=   ∫_0^+∞t(t)F(t) dt.

It is straightforward to check that in view of (<ref>) there holds:
|∫_r^+∞t(t)F(t) dt|≤C∫_r^+∞t(|lnt|+1)^4/t^+2 dt

|∫_r^+∞slns(s)F(s) ds|≤C∫_r^+∞s|lns|(|lns|+1)^4/s^+2 ds
so that the first term on the right hand side of (<ref>) takes the form
∫_2^rds/s∫_0^st(t)F(t) dt
Similarly, we write
∫_2^rds/s(1-s^)∫_0^st(t)F(t) dt
=   D_F,3
-∫_r^+∞ds/s(1-s^)∫_0^st(t)F(t) dt

=   D_F,3+O(1/r^)
D_F,3=∫_2^+∞ds/s(1-s^)∫_0^st(t)F(t) dt.

∫_2^rds/s(1-s^)^2∫_0^st(t)F(t) dt
=   D_F,4
-∫_r^+∞ds/s(1-s^)^2∫_0^st(t)F(t) dt

=   D_F,4+O(1/r^2)
D_F,4=∫_2^+∞ds/s(1-s^)^2∫_0^st(t)F(t) dt.
Observing that e may write $\phoa(r)=-1+O(\frac{1}{r^\al})$, we deduce from the above that
where $D_F=D_{F,1}+D_{F,2}-4D_{F,3}+4D_{F_4}$.
The desired solution is given by

[1]
Adimurthi and M. Grossi.
Asymptotic estimates for a two-dimensional problem with polynomial
Proc. Amer. Math. Soc., 132(4):1013–1019, 2004.

[2]
T. Bartsch, A. M. Micheletti, and A. Pistoia.
The Morse property for functions of Kirchhoff-Routh path type.
Discrete Contin. Dyn. Syst. Ser. S, 12(7):1867–1877, 2019.

[3]
T. Bartsch, A. Pistoia, and T. Weth.
$N$-vortex equilibria for ideal fluids in bounded planar domains
and new nodal solutions of the sinh-Poisson and the Lane-Emden-Fowler
Comm. Math. Phys., 297(3):653–686, 2010.

[4]
T. Bartsch, A. Pistoia, and T. Weth.
Erratum to: $N$-vortex equilibria for ideal fluids in bounded
planar domains and new nodal solutions of the sinh-Poisson and the
Lane-Emden-Fowler equations [mr2653899].
Comm. Math. Phys., 333(2):1107, 2015.

[5]
D. Chae and O. Y. Imanuvilov.
The existence of non-topological multivortex solutions in the
relativistic self-dual Chern-Simons theory.
Comm. Math. Phys., 215(1):119–142, 2000.

[6]
F. De Marchis, M. Grossi, I. Ianni, and F. Pacella.
$L^\infty$-norm and energy quantization for the planar
Lane-Emden problem with large exponent.
Arch. Math. (Basel), 111(4):421–429, 2018.

[7]
F. De Marchis, I. Ianni, and F. Pacella.
Asymptotic analysis and sign-changing bubble towers for
Lane-Emden problems.
J. Eur. Math. Soc. (JEMS), 17(8):2037–2068, 2015.

[8]
F. De Marchis, I. Ianni, and F. Pacella.
Asymptotic profile of positive solutions of Lane-Emden problems
in dimension two.
J. Fixed Point Theory Appl., 19(1):889–916, 2017.

[9]
M. del Pino, P. Esposito, and M. Musso.
Nondegeneracy of entire solutions of a singular Liouville equation.
Proc. Amer. Math. Soc., 140(2):581–588, 2012.

[10]
K. El Mehdi and M. Grossi.
Asymptotic estimates and qualitative properties of an elliptic
problem in dimension two.
Adv. Nonlinear Stud., 4(1):15–36, 2004.

[11]
P. Esposito, M. Musso, and A. Pistoia.
Concentrating solutions for a planar elliptic problem involving
nonlinearities with large exponent.
J. Differential Equations, 227(1):29–68, 2006.

[12]
P. Esposito, M. Musso, and A. Pistoia.
On the existence and profile of nodal solutions for a two-dimensional
elliptic problem with large exponent in nonlinearity.
Proc. Lond. Math. Soc. (3), 94(2):497–519, 2007.

[13]
M. Grossi, C. Grumiau, and F. Pacella.
Lane-Emden problems: asymptotic behavior of low energy nodal
Ann. Inst. H. Poincaré C Anal. Non Linéaire,
30(1):121–140, 2013.

[14]
M. Grossi, C. Grumiau, and F. Pacella.
Lane Emden problems with large exponents and singular Liouville
J. Math. Pures Appl. (9), 101(6):735–754, 2014.

[15]
M. Grossi, I. Ianni, P. Luo, and S. Yan.
Non-degeneracy and local uniqueness of positive solutions to the
Lane-Emden problem in dimension two.
J. Math. Pures Appl. (9), 157:145–210, 2022.

[16]
M. Grossi and A. Pistoia.
Multiple blow-up phenomena for the sinh-Poisson equation.
Arch. Ration. Mech. Anal., 209(1):287–320, 2013.

[17]
M. Grossi and F. Takahashi.
Nonexistence of multi-bubble solutions to some elliptic equations on
convex domains.
J. Funct. Anal., 259(4):904–917, 2010.

[18]
I. Ianni and A. Saldaña.
Sharp asymptotic behavior of radial solutions of some planar
semilinear elliptic problems.
J. Differential Equations, 304:102–164, 2021.

[19]
R. Kajikiya.
Sobolev norms of radially symmetric oscillatory solutions for
superlinear elliptic equations.
Hiroshima Math. J., 20(2):259–276, 1990.

[20]
N. Kamburov and B. Sirakov.
Uniform a priori estimates for positive solutions of the
Lane-Emden equation in the plane.
Calc. Var. Partial Differential Equations, 57(6):Paper No. 164,
8, 2018.

[21]
A. M. Micheletti and A. Pistoia.
Non degeneracy of critical points of the Robin function with
respect to deformations of the domain.
Potential Anal., 40(2):103–116, 2014.

[22]
X. Ren and J. Wei.
On a two-dimensional elliptic problem with large exponent in
Trans. Amer. Math. Soc., 343(2):749–763, 1994.

[23]
X. Ren and J. Wei.
Single-point condensation and least-energy solutions.
Proc. Amer. Math. Soc., 124(1):111–120, 1996.

[24]
P.-D. Thizy.
Sharp quantization for Lane-Emden problems in dimension two.
Pacific J. Math., 300(2):491–497, 2019.