# Critical points of arbitrary energy for the Trudinger-Moser embedding in
planar domains

Andrea Malchiodi  Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126
Pisa, Italy<EMAIL_ADDRESS>,  Luca Martinazzi  Dipartimento di
Matematica Guido Castelnuovo, Università La Sapienza, Piazzale Aldo Moro 5,
00185 Roma, Italy<EMAIL_ADDRESS>and  Pierre-Damien Thizy
Institut Camille Jordan, Université Claude Bernard Lyon 1, Bâtiment
Braconnier, 21 avenue Claude Bernard, 69622 Villeurbanne Cedex, France
<EMAIL_ADDRESS>
###### Abstract.

Given a smoothly bounded non-contractible domain
$\Omega\subset\mathbb{R}^{2}$, we prove the existence of positive critical
points of the Trudinger-Moser embedding for arbitrary Dirichlet energies. This
is done via degree theory, sharp compactness estimates and a topological
argument relying on the Poincaré-Hopf theorem.
Keywords: Moser-Trudinger, critical inequalities, Leray-Schauder degree,
Poincaré-Hopf theorem.

##  1\. Introduction

Given a smoothly bounded domain $\Omega\subset\mathbb{R}^{2}$, i.e. a bounded
domain with smooth boundary, let $H^{1}_{0}=H^{1}_{0}(\Omega)$ be the usual
Sobolev space of functions with zero trace on $\partial\Omega$ endowed with
the norm $\|\cdot\|_{H^{1}_{0}}$ given by

$\|u\|_{H^{1}_{0}}^{2}=\int_{\Omega}|\nabla u|^{2}dx.$

Given any positive real number $\beta>0$, our main purpose is to discuss the
existence of a nonnegative function, critical point of the classical [5, 19,
20, 24, 32, 34, 37, 39] Moser-Trudinger functional

$F(u):=\int_{\Omega}\left(e^{u^{2}}-1\right)dx\,,$ (1.1)

constrained to the submanifold

$\mathcal{M}_{\beta}:=\left\\{v\in
H^{1}_{0}~{}:~{}\|v\|^{2}_{H^{1}_{0}}=\beta\right\\}$ (1.2)

of $H^{1}_{0}$. This is equivalent to finding a solution of the following
elliptic problem

$\begin{cases}&\Delta u=2\lambda ue^{u^{2}}\,,\,u>0\,\text{ in }\Omega\,,\\\
&u=0\text{ on }\partial\Omega\,,\\\ &\int_{\Omega}|\nabla
u|^{2}dx=\beta\,,\end{cases}$ ($\mathcal{E}_{\beta}$)

for some real number $\lambda$, where $\Delta=-\partial_{xx}-\partial_{yy}$ is
the (nonnegative) Laplacian. Our main result reads as follows (see also Remark
2.5 below):

######  Theorem 1.1.

Let $\Omega\subset\mathbb{R}^{2}$ be a smoothly bounded non-contractible
domain. Then, given any positive real number $\beta>0$, there exists a
nonnegative function $u$, critical point of $F$ constrained to
$\mathcal{M}_{\beta}\subset H^{1}_{0}$. In particular, $u$ is smooth and
solves ($\mathcal{E}_{\beta}$).

Theorem 1.1 eventually gives a fairly complete answer to a long-standing open
problem. Indeed the existence of critical points of $F|_{\mathcal{M}_{\beta}}$
in the supercritical regime $\beta>4\pi$ is quite subtle, due to the critical
nature of the nonlinearity which, in addition to the failure of the Palais-
Smale condition, makes the blow-up estimates very delicate. For $\beta\leq
4\pi$ maximizers are known to exist thanks to the celebrated results of
Carleson-Chang [5], Struwe [37] and Flucher [19]. Moreover, still in [37],
Struwe proved the existence of $\varepsilon_{0}>0$ such that
$F|_{\mathcal{M}_{\beta}}$ has a positive local maximizer for every
$\beta\in(4\pi,4\pi+\varepsilon_{0})$ and a mountain pass-type positive
critical point for _almost every_ $\beta\in(4\pi,4\pi+\varepsilon_{0})$. The
existence of a second positive critical point for _every_
$\beta\in(4\pi,4\pi+\varepsilon_{0})$ was proven by Lamm-Robert-Struwe [25]
using a flow approach. In these results, there is no condition on the smoothly
bounded domain $\Omega$, but on the other hand, the requirement that $\beta$
is close to $4\pi$ is essential. More recently, via a Lyapunov-Schmidt
construction, Del Pino-Musso-Ruf [13] studied the case in which $\Omega$ is
not simply connected and $\beta$ is close to $8\pi$. That the topology of
$\Omega$ plays an important role in the existence of critical points of
$F|_{\mathcal{M}_{\beta}}$ is also suggested by the result of the first and
second authors [28], showing that when $\Omega$ is a disk
$F|_{\mathcal{M}_{\beta}}$ has no positive critical point for $\beta$
sufficiently large.

The case in which the domain $\Omega$ is replaced by a two-dimensional closed
surface was recently studied in [11] using the _barycenter technique_ of
Djadli-Malchiodi [16] (see also Bahri [2]) taking advantage of the variational
structure, a minmax argument together with Struwe’s monotonicity trick to
obtain solutions of a _subcritical_ approximate problem for almost every
$\beta>0$, and blow-up analysis together with energy estimates at critical
energy levels to conclude. Here instead, in order to prove Theorem 1.1, we
will use the Leray-Schauder degree, together with a deformation argument based
on compactness to link our problem to the _Mean-Field_ equation

$\begin{cases}&\Delta u=\beta\frac{2e^{u}}{\int_{\Omega}e^{u}dx}\,,\\\
&u=0\text{ on }\partial\Omega\,,\end{cases}$ ($\mathcal{E}_{\beta}^{MF}$)

studied e.g. in Ding-Jost-Li-Wang [15], and for which a degree theory approach
has been proposed by Chen and Lin [7, 8]. In particular, the purpose of
Section 3 is to give an argument to get a degree formula for a gradient vector
field, proven by [8] to encode the so-called _finite dimensional reduction_ of
($\mathcal{E}_{\beta}^{MF}$) (see Proposition 2.3 below).

In order to deform Problem ($\mathcal{E}_{\beta}$) into a mean-field type
equation, given $\beta>0$, we consider the following $p$-dependent family of
problems:

$\begin{cases}&\Delta u=p\lambda u^{p-1}e^{u^{p}}\,,\,u>0\,\text{ in
}\Omega\,,\\\ &u=0\text{ on }\partial\Omega\,,\\\ &\frac{\lambda
p^{2}}{2}\left(\int_{\Omega}\left(e^{u^{p}}-1\right)dx\right)^{\frac{2-p}{p}}\left(\int_{\Omega}u^{p}e^{u^{p}}dx\right)^{\frac{2(p-1)}{p}}=\beta\,,\end{cases}$
($\mathcal{E}_{p,\beta}$)

parametrized by $p\in[1,2]$. First, as seen when integrating by parts, solving
($\mathcal{E}_{p,\beta}$) for $p=2$ is equivalent to solving
($\mathcal{E}_{\beta}$). Moreover, ($\mathcal{E}_{p,\beta}$) still has a nice
variational structure for $p\in[1,2)$ and $\beta>0$ (see [11]). At last, any
nonnegative nontrivial weak $H^{1}_{0}$-solution of the PDE in
($\mathcal{E}_{p,\beta}$) is eventually smooth and positive in $\Omega$, as
proven from Trudinger [39] with standard elliptic theory (see for instance
Gilbarg-Trudinger [21]).

Let $\mathbb{N}^{\star}$ be the set of the positive integers and $\mathbb{N}$
the set of the nonnegative ones. The Leray-Schauder degree computation, which
is a by-product of our strategy to solve ($\mathcal{E}_{\beta}$), is given in
the following result:

######  Theorem 1.2.

Let $\Omega\subset\mathbb{R}^{2}$ be a smoothly bounded domain. Let
$p\in(1,2]$, $N\in\mathbb{N}$ and $\beta\in(4\pi N,4\pi(N+1)]$ be given. Then
the total Leray-Schauder degree $d_{p,\beta}(\Omega)$ of the solutions of
($\mathcal{E}_{p,\beta}$) is well-defined and equals the binomial number
$\binom{N-\chi(\Omega)}{N}$, where $\chi(\Omega)$ stands for the Euler
characteristic of $\Omega$. In particular, if either $\Omega$ is not simply
connected or if $N=0$, we have $d_{p,\beta}(\Omega)\neq 0$, so that the set
$\mathcal{C}_{p,\beta}(\Omega)$ of the solutions of ($\mathcal{E}_{p,\beta}$)
is not empty.

We adopt here the usual convention that $\binom{-1}{0}=1$ and
$\binom{N-1}{N}=0$ for $N\in\mathbb{N}^{\star}$. We recall that the genus
$g(\Omega):=1-\chi(\Omega)$ is basically the _“number of holes”_ inside
$\Omega$, so that $\Omega$ is simply connected if and only if
$\chi(\Omega)=1$.

Interestingly enough, given $p\in[1,2)$, our topological assumption on
$\Omega$ is sharp in the following sense: ($\mathcal{E}_{p,\beta}$) does not
have any solution for $\beta$ sufficiently large, as soon as $\Omega$ is
_simply connected_ (see [3, 4]). Yet, for $p=2$, we state the following
delicate question which looks open, except in the aforementioned case of the
disk:

######  Open problem.

Let $\Omega\subset\mathbb{R}^{2}$ be a _simply connected_ domain. Does there
exists $\beta^{\sharp}>4\pi$ such that ($\mathcal{E}_{\beta}$) does not have
any solution for $\beta>\beta^{\sharp}$?

To conclude this introductory part, we mention that the degree formula in
Theorem 1.2 also implies the existence of blowing-up solutions for
($\mathcal{E}_{p,\beta}$) (see Remark 2.4).

##  2\. Proof of Theorems 1.1 and 1.2

As a first ingredient, since the nonlinearities in the right-hand of the PDE
in ($\mathcal{E}_{p,\beta}$) are autonomous, we may use the following by-
product of the moving plane technique, already used for instance by Adimurthi
and Druet [1] in a similar context:

######  Proposition 2.1.

Let $\Omega\subset\mathbb{R}^{2}$ be a smoothly bounded domain. Then there
exists $\delta,\delta^{\prime}>0$ depending only on $\Omega$ such that, for
any nonnegative, nondecreasing $C^{1}$-function $f:[0,+\infty)\to\mathbb{R}$,
for all $v$ solving

$\begin{cases}&\Delta v=f(v),v>0\text{ in }\Omega\,,\\\ &v=0\text{ on
}\partial\Omega\,,\end{cases}$

and all $x\in\partial\Omega$, the function $t\mapsto u(x-t\nu(x))$ has
positive derivative in $[0,\delta]$, where $\nu(x)$ is the unit outward normal
to $\partial\Omega$ at $x$, and we have

$\nabla v(x)=0\quad\implies\quad d(x,\partial\Omega)\geq\delta^{\prime}>0\,.$
(2.1)

###### Proof of Proposition 2.1 (completed).

The existence of $\delta>0$ as in the statement of the proposition is a direct
consequence of the classical moving plane argument in de Figueiredo-Lions-
Nussbaum [10] if $\Omega$ is convex, while it has to be combined with Kelvin’s
transform in our general two-dimensional case, as observed by Han [22]. Then,
since the map $(t,x)\mapsto x-t\nu(x)$ is a diffeomorphism from
$[0,\delta]\times\partial\Omega$ onto a neighborhood of $\partial\Omega$
inside $\bar{\Omega}$, up to reducing $\delta>0$, (2.1) follows by compactness
of $\partial\Omega$. ∎

Proposition 2.1 is an important ingredient to take advantage of the blow-up
analysis developed to get [11, Theorem 4.1] on a closed surface in our present
setting of a bounded domain $\Omega$, ensuring that no new phenomena arise
close to the boundary. Hence we have the following theorem:

######  Theorem 2.1.

Let $(\lambda_{\varepsilon})_{\varepsilon}$ be any sequence of positive real
numbers and $(p_{\varepsilon})_{\varepsilon}$ be any sequence of numbers in
$[1,2]$. Let $(u_{\varepsilon})_{\varepsilon}$ be a sequence of smooth
functions solving

$\begin{cases}&\Delta
u_{\varepsilon}=p_{\varepsilon}\lambda_{\varepsilon}u_{\varepsilon}^{p_{\varepsilon}-1}e^{u_{\varepsilon}^{p_{\varepsilon}}}\,,\quad
u_{\varepsilon}>0\text{ in }\Omega\,,\\\ &u_{\varepsilon}=0\quad\text{ on
}\partial\Omega\,,\end{cases}$ (2.2)

for all $\varepsilon$. Let $\beta_{\varepsilon}$ be given by

$\beta_{\varepsilon}=\frac{\lambda_{\varepsilon}p_{\varepsilon}^{2}}{2}\left(\int_{\Omega}\left(e^{u_{\varepsilon}^{p_{\varepsilon}}}-1\right)~{}dx\right)^{\frac{2-p_{\varepsilon}}{p_{\varepsilon}}}\left(\int_{\Omega}u_{\varepsilon}^{p_{\varepsilon}}e^{u_{\varepsilon}^{p_{\varepsilon}}}~{}dx\right)^{\frac{2(p_{\varepsilon}-1)}{p_{\varepsilon}}}$

for all $\varepsilon$. If we assume the energy bound

$\lim_{\varepsilon\to 0}\beta_{\varepsilon}=\beta\in[0,+\infty)\,,$

but the pointwise blow-up of the $u_{\varepsilon}$’s, namely
$\lim_{\varepsilon\to 0}\max_{\Omega}u_{\varepsilon}=+\infty\,,$ then
$\lambda_{\varepsilon}\to 0^{+}$ and there exists an integer
$N\in\mathbb{N}^{\star}$ such that $\beta=4\pi N\,.$

Independently, multiplying the equation in ($\mathcal{E}_{p,\beta}$) by a
first Dirichlet eigenfunction $v_{1}$ of the Laplacian, integrating by parts
and using $pt^{p-1}e^{t^{p}}\geq 2t$ for all $t\geq 0$ and $p\in[1,2]$, we get

$0<2\lambda\leq\lambda_{1}$ (2.3)

as soon as a (nontrivial) solution $u$ of ($\mathcal{E}_{p,\beta}$) exists.

###### Outline of the proof of Theorem 2.1.

Overall, the proof follows closely [11, Sections 2-4]. It is even simpler in
the present framework where Dirichlet boundary conditions are imposed, since
we no longer need to handle the additional coercivity term $hu_{\varepsilon}$
in the left-hand $\Delta_{g}u_{\varepsilon}+hu_{\varepsilon}$ of the equation
in [11, equation (4.1)] analogue to (2.2) here: indeed, all along the argument
of [11], this additional term is just estimated by error terms which can be
taken now to be zero.

More precisely, as pioneered by Druet [17, Sections 3-4] when
$p_{\varepsilon}=2$ for all $\varepsilon$ (see also [18, Section 3]), we start
with the following proposition giving a first pointwise exhaustion of the
concentration points, as well as weak gradient estimates under the assumptions
of Theorem 2.1:

######  Proposition 2.2.

Up to a subsequence, there exist an integer $n\in\mathbb{N}^{\star}$ and
sequences $(x_{i,\varepsilon})_{\varepsilon}$ of points in $\Omega$ such that
$\nabla u_{\varepsilon}(x_{i,\varepsilon})=0$, such that, setting
$\gamma_{i,\varepsilon}:=u_{\varepsilon}(x_{i,\varepsilon})$,

$\mu_{i,\varepsilon}:=\left(\frac{8}{\lambda_{\varepsilon}p_{\varepsilon}^{2}\gamma_{i,\varepsilon}^{2(p_{\varepsilon}-1)}e^{\gamma_{i,\varepsilon}^{p_{\varepsilon}}}}\right)^{\frac{1}{2}}\to
0\,,$ (2.4)

and such that

$\frac{p_{\varepsilon}}{2}\gamma_{i,\varepsilon}^{p_{\varepsilon}-1}(\gamma_{i,\varepsilon}-u_{\varepsilon}(x_{i,\varepsilon}+\mu_{i,\varepsilon}\cdot))\to
T_{0}:=\ln\left(1+|\cdot|^{2}\right)\text{ in }C^{1}_{loc}(\mathbb{R}^{2})\,,$
(2.5)

as $\varepsilon\to 0$, for all $i\in\\{1,...,n\\}$. Moreover, there exist
$C>0$ such that we have

$\min_{i\in\\{1,...,n\\}}u_{\varepsilon}^{p_{\varepsilon}-1}|x_{i,\varepsilon}-\cdot||\nabla
u_{\varepsilon}|\leq C\text{ in }\Omega$

for all $\varepsilon$. We also have that $\lim_{\varepsilon\to
0}x_{i,\varepsilon}=x_{i}$ for all $i$, and that there exists $u_{0}\in
C^{2}(\bar{\Omega}\backslash\mathcal{S})$ such that

$\lim_{\varepsilon\to 0}u_{\varepsilon}=u_{0}\text{ in
}C^{1}_{loc}(\bar{\Omega}\backslash\mathcal{S})\,,$

where $\mathcal{S}\subset\Omega$ consists of all the $x_{i}$’s.

About the concentration points $x_{i,\varepsilon}$ of this proposition, being
critical points of the $u_{\varepsilon}$’s by construction, (2.1) yields that
they _cannot collapse to the boundary_ so that proving Proposition 2.2 does
not require new arguments with respect to the ones to get [11, Proposition
4.1]. In addition, (2.3) and (2.4) yield that
$\gamma_{i,\varepsilon}\to+\infty$, so that (2.5) really points out a
concentration profile at first order for the $u_{\varepsilon}$’s.

But at that stage, as in [11, Section 2], the strategy is to consider a
radially symmetric much more precise ansatz solving the same equation (i.e.
set now $h_{0}\equiv 0$ in [11, (2.6)]), whose relevance in the non-radial
setting can be quantified (see [11, Section 3] picking now $h\equiv 0$ in [11,
(3.8)]). Thus assumption [11, (2.5)] and the related [11, (3.6),
(3.17)-(3.19), second requirement in (3.21), (4.26), Lemma 4.1] can be now
ignored. The terms $w_{l,\varepsilon}$ in [11, (4.47)] and
$\tilde{w}_{\varepsilon}$ in [11, (4.85)] used to control this additional
linear term are also now useless and we eventually do get Theorem 2.1 by
following the lines of [11, Sections 2-4], since no specific new difficulty
may arise close to the boundary thanks to Propositions 2.1 and 2.2. As a last
important remark, the lower bound on $d(x_{i,\varepsilon},\partial\Omega)$
from Proposition 2.2 is also used to get the key property
$\lambda_{\varepsilon}\to 0^{+}$, arguing as in [11, Step 4.2]. ∎

######  Remark 2.1.

As pointed out in [31], the positivity of the $u_{\varepsilon}$’s is essential
to have energy quantization (i.e. $\beta\in 4\pi\mathbb{N}^{\star}$) in
Theorem 2.1. Moreover, even restricting to nonnegative functions, we stress
that there exist Palais-Smale sequences associated to ($\mathcal{E}_{\beta}$)
with arbitrary limiting energy $\beta>4\pi$ and converging weakly to $0$ in
$H^{1}_{0}$ (see Costa-Tintarev [9]), so that not only the quantization, but
even the quantification proved for solutions by Druet [17] fail for general
Palais-Smale sequences (see also [30] for examples with non-zero weak limit
entering in the framework of [17], but where _quantization fails_). This is in
striking constrast with a large class of otherwise closely related critical
problems for which quantification already holds for Palais-Smale sequences
(see for instance the pioneering work [36] by Struwe). This is maybe the
clearest evidence of the huge difficulty to run directly standard variational
techniques generating Palais-Smale sequences to solve ($\mathcal{E}_{\beta}$),
and motivates our compactness techniques considering only _exact solutions_
instead of _Palais-Smale sequences_.

In the specific case of the mean-field equation, i.e. when $p_{\varepsilon}=1$
for all $\varepsilon$, with the same notation and assumptions as in Theorem
2.1, we obviously get

$\beta_{\varepsilon}^{MF}:=\frac{\lambda_{\varepsilon}}{2}\int_{\Omega}e^{u}dx=\beta_{\varepsilon}+\frac{\lambda_{\varepsilon}|\Omega|}{2}\to\beta=4\pi
N\in 4\pi\mathbb{N}^{\star}$ (2.6)

as $\varepsilon\to 0$. At that stage, another additional ingredient for our
proof is the following result stated here with the same notation and
conventions as in Theorem 1.2:

######  Theorem 2.2.

Let $\Omega\subset\mathbb{R}^{2}$ be a smoothly bounded domain and $h\in
C^{1}(\bar{\Omega})$ be positive. Let $N\in\mathbb{N}$ and $\beta\in(4\pi
N,4\pi(N+1))$ be given. Then the total Leray-Schauder degree
$d^{MF}_{\beta,h}(\Omega)$ of all the solutions of the mean-field equation

$\begin{cases}&\Delta u=\beta\frac{2he^{u}}{\int_{\Omega}he^{u}dx}\,,\\\
&u=0\text{ on }\partial\Omega\,,\end{cases}$ ($\mathcal{E}_{\beta,h}^{MF}$)

is well-defined and equals $\binom{N-\chi(\Omega)}{N}$.

As an obvious remark, a solution of ($\mathcal{E}_{\beta,h}^{MF}$) has to be
positive in $\Omega$. Theorem 2.2, which is only needed here for a constant
function $h>0$, has been stated first in Chen-Lin [8]. Indeed, these authors
show in their beautiful work:

######  Proposition 2.3 (Chen-Lin [8]).

For all $N\in\mathbb{N}$ and all $\beta\in(4\pi N,4\pi(N+1))$,
$d^{MF}_{\beta,h}(\Omega)$ is equal to a constant $d_{N}^{MF}(\Omega)$, which
is $1$ for $N=0$. Moreover, we have the following degree jump formula:

$d_{N+1}^{MF}(\Omega)-d_{N}^{MF}(\Omega)=\frac{(-1)^{N}}{N!}\deg(\nabla\Phi_{N,h},\Omega^{N}\setminus
D_{N},0),$ (2.7)

where

$\Phi_{N,h}(x_{1},\dots,x_{N})=8\pi\sum_{i\neq
j}\mathcal{G}(x_{i},x_{j})+4\pi\sum_{1\leq i\leq
N}\mathcal{H}(x_{i},x_{i})+\sum_{1\leq i\leq N}\ln h(x_{i})$ (2.8)

is defined in $\Omega^{N}\setminus D_{N}$ with
$D_{N}:=\\{(x_{1},\dots,x_{N})\in\Omega^{N}:\exists i\neq
j\,,~{}x_{i}=x_{j}\\}$, where $\mathcal{G}$ is the Green’s function of
$\Delta$ with zero Dirichlet boundary conditions on $\Omega$, where
$\mathcal{H}$ is the regular part of $\mathcal{G}$ normalized here as

$\mathcal{G}(x,y)=\frac{1}{2\pi}\ln\frac{1}{|x-y|}+\mathcal{H}(x,y),$ (2.9)

and where $\deg(\nabla\Phi_{N,h},\Omega^{N}\setminus D_{N},0)$ is the Brouwer
degree (or total index) of the vector field $\nabla\Phi_{N,h}$.

Contrary to what occurs on a closed manifold, it turns out that, in order to
compute the degree of $\nabla\Phi_{N,h}$, the classical Poincaré-Hopf formula
does _not_ directly apply in the present setting of a bounded domain since
$\nabla\Phi_{N,h}$ points outwards only on a strict subset of the boundary of
its domain, non-empty as soon as $N>1$. Another goal of this paper is to
provide an argument showing that, for topological reasons relying strongly on
the fact that every connected component of $\partial\Omega$ has zero Euler
characteristic in our two-dimensional case (crucial point in Lemma 3.5, see
also Remark 2.2 just below), the following formula for the degree of
$\nabla\Phi_{N,h}$ holds in the present setting of a smoothly bounded domain
$\Omega$, so that the degree formula in Theorem 2.2 follows from Propositions
2.3 and 2.4.

######  Remark 2.2.

It is known that any closed odd-dimensional manifold has zero Euler-
characteristic, so that this aforementioned property for every connected
boundary component of $\partial\Omega$ holds more generally for any smoothly
bounded $\Omega\subset\mathbb{R}^{n}$ with $n$ _even_. Indeed, with minimal
changes, our proof extends to this case which is relevant for instance for the
fourth-order mean-field equation in Lin-Wei-Wang [27].

######  Proposition 2.4.

The degree of $\nabla\Phi_{N,h}$ is well-defined on $\Omega^{N}\setminus
D_{N}$ and one has

$\deg(\nabla\Phi_{N,h},\Omega^{N}\setminus
D_{N},0)=\chi(\Omega)(\chi(\Omega)-1)...(\chi(\Omega)-N+1).$ (2.10)

Proving Proposition 2.4 is the purpose of Section 3.

We are now in position to prove Theorems 1.1 and 1.2, starting with the
specific case where $\beta>0$ is out of the set $4\pi\mathbb{N}^{\star}$ of
the critical energy levels.

###### Proof of the degree formula of Theorem 1.2 for $\beta\not\in
4\pi\mathbb{N}^{\star}$.

During this whole first part, we fix $N\in\mathbb{N}$ and $\beta\in(4\pi
N,4\pi(N+1))$. For $t\in[0,1]$ and $p\in[1,2]$, let $T_{p,\beta}$ and
$T_{t,\beta}^{MF}$ be the Fredholm-type nonlinear operators given for any non-
negative non-zero function $v\in C^{1}(\bar{\Omega})$ by

$T_{p,\beta}(v)=v-\Delta^{-1}\left(\frac{2\beta
v^{p-1}e^{v^{p}}}{p\left(\int_{\Omega}\left(e^{v^{p}}-1\right)dx\right)^{\frac{2-p}{p}}\left(\int_{\Omega}v^{p}e^{v^{p}}dx\right)^{\frac{2(p-1)}{p}}}\right)$

and

$T_{t,\beta}^{MF}(v)=v-\Delta^{-1}\left(\frac{2\beta
e^{v}}{\int_{\Omega}(e^{u}-t)dx}\right)\,,$

where $\Delta^{-1}f$ is the solution $w$ of $\Delta w=f$ with zero Dirichlet
boundary conditions. By Theorem 2.1 and formula (2.6) with standard elliptic
theory and (2.3), there exist constants $C>c>0$ such that

$\|u\|_{C^{1}(\bar{\Omega})}<C$ and $u>cv_{1}$ in $\Omega$

for all $u$ positive in $\Omega$ solving $T_{t,\beta}^{MF}(u)=0$ for some
$t\in[0,1]$, and all $u$ solving $T_{p,\beta}(u)=0$, i.e.
($\mathcal{E}_{p,\beta}$), for some $p\in[1,2]$, where $v_{1}$ is still a
given first Dirichlet eigenfunction of $\Delta$ chosen positive in $\Omega$.
Let $C^{1}_{0}(\bar{\Omega})$ be the closed subspace of $C^{1}(\bar{\Omega})$
consisting of the functions vanishing on $\partial\Omega$. Let $\mathcal{V}$
be the open subset of $C^{1}_{0}(\bar{\Omega})$ given by

$\mathcal{V}=\left\\{\quad v\in C^{1}_{0}(\bar{\Omega})\quad\text{ s.t.
}\begin{array}[]{l}v>cv_{1}\text{ in }\Omega\,,\\\
\|v\|_{C^{1}(\bar{\Omega})}<C\end{array}\qquad\right\\}\,.$

Using Hopf’s lemma, observe that only positive functions in $\Omega$ are in
the closure of $\mathcal{V}$ for the $C^{1}(\bar{\Omega})$-topology, in
particular neither the zero nor any sign-changing function is in this closure.
Then,

$(p,v)\mapsto T_{p,\beta}(v)$ and $(t,v)\mapsto T_{t,\beta}^{MF}(v)$

are continuous from $[1,2]\times\mathcal{V}$ and $[0,1]\times\mathcal{V}$
respectively into $C^{1}_{0}(\bar{\Omega})$, by standard elliptic theory.
Observe in particular that this continuity does hold for $T_{p,\beta}$ up to
$p=1$ despite the jump of $(p,t)\in[1,2]\times(0+\infty)\mapsto
pt^{p-1}e^{t^{p}}$ at $t=0^{+}$ as $p\to 1$. Using also that these operators
do not vanish on the boundary of $\mathcal{V}$ by construction, the total
Leray-Schauder degree
$d_{p,\beta}(\Omega):=\mathrm{deg}_{LS}(T_{p,\beta},\mathcal{V},0)$ (resp.
$d_{t,\beta}^{MF}(\Omega):=\mathrm{deg}_{LS}(T_{t,\beta}^{MF},\mathcal{V},0)$)
of all the solutions of ($\mathcal{E}_{p,\beta}$) (resp. of all the functions
$v>0$ in $\Omega$ such that $T_{t,\beta}^{MF}(v)=0$) is well-defined, see for
instance Nirenberg [33].

######  Remark 2.3.

It is convenient to work here within the space $C^{1}_{0}(\bar{\Omega})$, but
one could ask whether the degree thus defined coincides with the one that one
could define within the natural variational space $H^{1}_{0}$. The arguments
in Li [26, Appendix B] slightly transposed to the present situation show that
it is actually the case.

Now, for $t=0$, the formula for the total degree $d_{0,\beta}^{MF}(\Omega)$ of
all the solutions of ($\mathcal{E}_{\beta}^{MF}$) is given by Theorem 2.2.
Moreover, we clearly have $T_{1,\beta}=T_{1,\beta}^{MF}$, so that
$d_{1,\beta}(\Omega)=d_{1,\beta}^{MF}(\Omega)$. Using then our definition of
$\mathcal{V}$, Theorem 2.2 and the homotopy invariance of the total Leray-
Schauder degree as far as compactness holds, we have
$d_{p,\beta}(\Omega)=d_{1,\beta}(\Omega)$ for all $p\in[1,2]$ on the one hand,
while
$d_{t,\beta}^{MF}(\Omega)=d_{0,\beta}^{MF}(\Omega)=d_{1,\beta}^{MF}(\Omega)$
for all $t\in[0,1]$ on the other hand, which concludes the proof of Theorem
1.2, and thus that of Theorem 1.1, in this first case where $\beta\not\in
4\pi\mathbb{N}^{\star}$. ∎

######  Remark 2.4.

Let $p\in[1,2]$ be fixed. A posteriori, by contraposing the homotopy
invariance of the Leray-Schauder degree as far as compactness holds, observe
that the effective degree jump of $d_{p,\beta}(\Omega)$ at the levels
$\beta\in 4\pi\mathbb{N}^{\star}$, as soon as the genus $g(\Omega)$ is greater
than $1$, imposes the existence of a sequence of blowing-up solutions with
limiting energy $\beta=4\pi N$ for all $N\in\mathbb{N}^{\star}$ as in Theorem
2.1 with $p_{\varepsilon}=p$ (see also Deng-Musso [14] or del Pino-Musso-Ruf
[12, 13] for specific constructions of such blowing-up solutions for
($\mathcal{E}_{p,\beta}$)).

In order to complete the proof of Theorems 1.1 and 1.2, it remains to treat
the case $\beta\in 4\pi\mathbb{N}^{\star}$. This will be based on the estimate
in the following theorem, saying roughly that for $p\in(1,2]$ the amount of
Dirichlet energy near each blow-up point approaches $4\pi$ _from above_. This
estimate was first proven in the radial case and for $p=2$ by Mancini and the
second author [29], then extended to the case of a bounded domain in
$\mathbb{R}^{2}$ by the third author [38] (see also Ibrahim-Masmoudi-
Nakanishi-Sani [23] for related results), finally on a closed two-dimensional
surface [11, Theorem 5.1]. Taking Proposition 2.1 into account, the proof of
[11, Theorem 5.1] can be easily adapted to yield:

######  Theorem 2.3.

Assume $p_{\varepsilon}=p\in(1,2]$ for all $\varepsilon$ and let
$u_{\varepsilon}$, $\lambda_{\varepsilon}$, $\beta_{\varepsilon}\to\beta=4\pi
N\in 4\pi\mathbb{N}^{\star}$ be as in Theorem 2.1. Then

$\beta_{\varepsilon}\geq 4\pi\left(N+\frac{4(p-1)(1+o(1))}{p^{2}}\sum_{1\leq
i\leq N}\frac{1}{\gamma_{i,\varepsilon}^{2p}}\right)\quad\text{as
}\varepsilon\to 0$ (2.11)

where the sequence $(\gamma_{i,\varepsilon})_{\varepsilon}$ is given by
Proposition 2.2 for all $i$.

###### Conclusion of the proof of Theorems 1.1 and 1.2.

We fix now $p\in(1,2]$. Then, from Theorem 2.3 we get that
$\beta_{\varepsilon}>\beta$ for all $\varepsilon$ small enough in Theorem 2.1.
As a consequence, for any given $N\in\mathbb{N}$ and small $\eta\in(0,1]$, we
have compactness of all the solutions of ($\mathcal{E}_{p,\beta}$) for all
$\beta\in[4\pi(N+\eta),4\pi(N+1)]$. In particular, we may define as above an
open neighborhood $\mathcal{V}^{\prime}\subset C^{1}_{0}(\bar{\Omega})$
containing all these solutions and whose closure contains only positive
functions in $\Omega$. Then, the map $(\beta,v)\mapsto T_{p,\beta}(v)$ is
continuous from $[4\pi(N+\eta),4\pi(N+1)]\times\mathcal{V}^{\prime}$ to
$C^{1}_{0}(\bar{\Omega})$ and, by a deformation argument as above, we get that
the total Leray-Schauder degree
$\mathrm{deg}_{LS}(T_{p,\beta},\mathcal{V}^{\prime},0)$ of all the solutions
of ($\mathcal{E}_{p,\beta}$) is well-defined and does not depend on
$\beta\in[4\pi(N+\eta),4\pi(N+1)]$: Theorems 1.1 and 1.2 are proven. ∎

######  Remark 2.5.

Relying now on [18], we may get a more general non-homogeneous version of
Theorem 1.1, which can be stated as follows:

######  Theorem 2.4.

Let $\Omega\subset\mathbb{R}^{2}$ be a smoothly bounded domain and $h$ be a
smooth positive $C^{2}$-function in $\bar{\Omega}$. Let $F_{h}$ be given in
$H^{1}_{0}$ by

$F_{h}(v):=\int_{\Omega}\left(e^{v^{2}}-1\right)h~{}dx$

and $\mathcal{M}_{\beta}\subset H^{1}_{0}$ be as in (1.2). Assume that
$\Omega$ is not contractible. Then, given any positive real number $\beta>0$,
there exists a nonnegative function $u$, critical point of $F_{h}$ constrained
to $\mathcal{M}_{\beta}$. In particular, $u$ is smooth and solves
($\mathcal{E}_{p,\beta,h}$) below for $p=2$.

###### Proof of Theorem 2.4.

We fix $p=2$. Given $\beta\in(4\pi N,4\pi(N+1)]$, we claim that the total
Leray-Schauder degree $d_{p,\beta,h}(\Omega)$ of the solutions of

$\begin{cases}&\Delta u=p\lambda hu^{p-1}e^{u^{p}}\,,\,u>0\,\text{ in
}\Omega\,,\\\ &u=0\text{ on }\partial\Omega\,,\\\ &\frac{\lambda
p^{2}}{2}\left(\int_{\Omega}\left(e^{u^{p}}-1\right)h~{}dx\right)^{\frac{2-p}{p}}\left(\int_{\Omega}u^{p}e^{u^{p}}h~{}dx\right)^{\frac{2(p-1)}{p}}=\beta\,,\end{cases}$
($\mathcal{E}_{p,\beta,h}$)

is well-defined and does not depend of $h$, positive $C^{2}$-function in
$\bar{\Omega}$, so that it always equals $\binom{N-\chi(\Omega)}{N}$, formula
given by Theorem 1.2 for $h\equiv 1$. Indeed, first given $\beta>0$ out of
$4\pi\mathbb{N}^{\star}$, it follows from the quantization and the associated
compactness result in [18] that $d_{p,\beta,h_{r}}(\Omega)$ is well-defined
and does not depend of $r\in[0,1]$, where $h_{r}=(1-r)+rh$. Fixing now such an
$h$, $N\in\mathbb{N}$ and $\eta\in(0,1]$ and using that boundary blow-up
cannot either occur for the solutions of ($\mathcal{E}_{p,\beta,h}$) (see
Remark 2.6 just below), the energy estimate in Theorem 2.3 from [11, Section
5] still holds here and gives the compactness of all the solutions of
($\mathcal{E}_{p,\beta,h}$) for all $\beta\in[4\pi(N+\eta),4\pi(N+1)]$, so
that $d_{p,\beta,h}(\Omega)$ does not depend on $\beta$ in this range by
following the second part of the above proof: Theorem 2.4 is proven. ∎

######  Remark 2.6.

Excluding boundary blow-up is actually the most delicate part in [18] where
$p$ is fixed equal to $2$. The analogue property appears to be open for
$p\in(1,2)$.

##  3\. Degree formula for the vector field associated to
$\mathcal{E}_{\beta,h}^{MF}$ on a bounded domain $\Omega$

The purpose of this section is to prove Proposition 2.4. Let $\mathcal{G}$ be
the Green’s function of $\Delta$ with zero Dirichlet boundary conditions and
$\mathcal{H}$ be its regular part with the convention in (2.9). Given
$y\in\Omega$, notice that

$\mathcal{G}(x,y)\to+\infty\quad\hbox{ as }x\to y\qquad\text{ and
that}\qquad\mathcal{H}(x,x)\to-\infty\quad\hbox{ as }x\to\partial\Omega.$

Given a positive function $h\in C^{1}(\overline{\Omega})$, let $\Phi_{N,h}$ be
given by (2.8). To fix the notation, we will write

$\nabla\Phi_{N,h}(x_{1},\dots,x_{N})=(\nabla_{x_{1}}\Phi_{N,h}(x_{1},\dots,x_{N}),\dots,\nabla_{x_{N}}\Phi_{N,h}(x_{1},\dots,x_{N})),$

with

$\begin{split}\nabla_{x_{j}}\Phi_{N,h}(x_{1},\dots,x_{N})&=16\pi\sum_{\scriptstyle
1\leq\ell\leq N\atop\scriptstyle\ell\neq
j}\nabla\mathcal{G}(x_{j},x_{\ell})+8\pi\nabla\mathcal{H}(x_{j},x_{j})+\frac{\nabla
h(x_{j})}{h(x_{j})},\end{split}$

where we denote by $\nabla\mathcal{G}(x,y)$ and $\nabla\mathcal{H}(x,y)$ the
gradient with respect to the first variable.

In order to get Proposition 2.4, we first need to analyse the compactness of
the critical points of $\Phi_{N,h}$.

###  3.1. Compactness of critical points of $\Phi_{N,h}$

In this subsection we prove that, uniformly with respect to $t\in[0,1]$,
critical the points of

$\Phi_{N,h}^{t}(x_{1},\dots,x_{N}):=8\pi\sum_{1\leq i,j\leq N\atop i\neq
j}\mathcal{G}(x_{i},x_{j})+4\pi\sum_{1\leq i\leq
N}\mathcal{H}(x_{i},x_{i})+t\sum_{1\leq i\leq N}\ln h(x_{i})$

cannot be arbitrarily close to the boundary of $\Omega^{N}\setminus D_{N}$,
implying that the degree in Proposition 2.4 is well-defined. More precisely,
define

$\begin{split}(\Omega^{N}\setminus
D_{N})_{\delta}:=\\{(x_{1},\dots,x_{N})\in\Omega^{N}\setminus
D_{N}:&\,|x_{i}-x_{j}|>\delta\text{ for }1\leq i<j\leq N,\\\
&d(x_{i},\partial\Omega)>\delta\text{ for }1\leq i\leq N\\}.\end{split}$ (3.1)

######  Lemma 3.1.

For every $M>0$, there exists $\delta>0$ such that

$|\nabla\Phi_{N,h}^{t}(x_{1},\dots,x_{N})|\geq M,\quad\text{for
}(x_{1},\dots,x_{N})\in(\Omega^{N}\setminus
D_{N})\setminus(\Omega^{N}\setminus D_{N})_{\delta},\;t\in[0,1].$

As a consequence, the degree of $\nabla\Phi_{N,h}^{t}$ is well-defined on
$\Omega^{N}\setminus D_{N}$ as

$\deg(\nabla\Phi_{N,h},\Omega^{N}\setminus
D_{N},0):=\deg(\nabla\Phi_{N,h},A,0),$

for any open set $A$ with $(\Omega^{N}\setminus D_{N})_{\delta}\subset
A\subset\Omega^{N}\setminus D_{N}$.

###### Proof.

Consider sequences

$(t_{k})_{k\in\mathbb{N}}\subset[0,1],\quad(x_{1,k},\dots,x_{N,k})_{k\in\mathbb{N}}\subset\Omega^{N}\setminus
D_{N}$

such that

$\nabla\Phi_{N,h}^{t_{k}}(x_{1,k},\dots,x_{N,k})=O(1),\quad\text{as
}k\to\infty.$ (3.2)

Set

$r_{i,k}:=\min\left\\{d_{i,k},\min_{j\neq i}|x_{i,k}-x_{j,k}|\right\\},\quad
d_{i,k}:=d(x_{i,k},\partial\Omega).$ (3.3)

Up to extracting a subsequence and reordering the indices we can assume that

$r_{1,k}\leq r_{j,k},\quad\text{for }1\leq j\leq N,$ (3.4)

and set

$r_{k}:=r_{1,k},\quad x_{k}:=x_{1,k},\quad d_{k}:=d_{1,k}.$ (3.5)

It suffices to prove that $\liminf_{k\to\infty}r_{k}>0$. Then we assume, by
contradiction, that

$\lim_{k\to\infty}r_{k}=0.$ (3.6)

We now scale the domain $\Omega$ and the functional $\Phi_{N,h}^{t_{k}}$ at
$x_{k}$ by a factor $r_{k}$. More precisely, we set
$\Omega_{k}:=r_{k}^{-1}(\Omega-x_{k})$ and, for
$(z_{1},\dots,z_{N})\in\Omega_{k}^{N}\setminus D_{N}$, we define

$\begin{split}\Xi_{k}(z_{1},\dots,z_{N})&=\Phi_{N,h}^{t_{k}}(x_{k}+r_{k}z_{1},\dots,x_{k}+r_{k}z_{N})\\\
&=8\pi\sum_{\scriptstyle 1\leq i,j\leq N\atop\scriptstyle i\neq
j}\mathcal{G}_{k}(z_{i},z_{j})+4\pi\sum_{1\leq i\leq
N}\mathcal{H}_{k}(z_{i},z_{i})+t_{k}\sum_{1\leq i\leq N}\ln
h_{k}(z_{i}),\end{split}$

where $\mathcal{G}_{k}$ and $\mathcal{H}_{k}$ are the Green’s function of
$\Delta$ on $\Omega_{k}$ and its regular part, and
$h_{k}(z):=h(x_{k}+r_{k}z)$. Then (3.2) gives

$\nabla\Xi_{k}(z_{1,k},\dots,z_{N,k})=O(r_{k}),\quad\text{as }k\to\infty,$
(3.7)

where

$(z_{1,k},\dots,z_{N,k}):=(r_{k}^{-1}(x_{1,k}-x_{k}),\dots,r_{k}^{-1}(x_{N,k}-x_{k})).$

Up to a further subsequence we set

$\mathcal{D}:=\\{i\in\\{1,\dots,N\\}:|x_{i,k}-x_{k}|=O(r_{k})\text{ as
}k\to\infty\\},$ (3.8)

and have

$\lim_{k\to\infty}\frac{|x_{i,k}-x_{k}|}{r_{k}}=+\infty\quad\text{for
}i\in\\{1,\dots,N\\}\setminus\mathcal{D}.$ (3.9)

Clearly $\mathcal{D}\neq\emptyset$, since $1\in\mathcal{D}$. Define also, up
to a subsequence,

$Z^{\prime}=(z_{i})_{i\in\mathcal{D}},\quad
Z^{\prime}_{k}=(z_{i,k})_{i\in\mathcal{D}}\to
Z^{\prime}_{\infty}=(z_{i,\infty})_{i\in\mathcal{D}},\quad\text{as
}k\to\infty.$

Notice that (3.3) and (3.8) imply

$\frac{1}{C}r_{k}\leq|x_{i,k}-x_{j,k}|\leq Cr_{k}\quad\text{for
}i,j\in\mathcal{D},\quad i\neq j,$ (3.10)

and in particular $z_{i,\infty}\neq z_{j,\infty}$ for $i,j\in\mathcal{D}$,
$i\neq j$. We now define the “reduced” functional

$\begin{split}\Xi_{\mathcal{D},k}(z_{1},\dots,z_{N}):=&8\pi\sum_{\scriptstyle
i,j\in\mathcal{D}\atop\scriptstyle i\neq
j}\mathcal{G}_{k}(z_{i},z_{j})+4\pi\sum_{i\in\mathcal{D}}\mathcal{H}_{k}(z_{i},z_{i})+t_{k}\sum_{i\in\mathcal{D}}\ln
h_{k}(z_{i})\\\ &+16\pi\sum_{i\in\mathcal{D}}\sum_{\scriptstyle 1\leq j\leq
N\atop\scriptstyle
j\not\in\mathcal{D}}\mathcal{G}_{k}(z_{i},z_{j}).\end{split}$ (3.11)

From (3.7) we get

$\nabla_{Z^{\prime}}\Xi_{\mathcal{D},k}(z_{1,k},\dots,z_{N,k})=O(r_{k}).$
(3.12)

Taking (3.6) into account, we now consider 2 cases.

Case 1. Assume that, up to a subsequence,

$r_{k}=o(d_{k})\quad\text{as }k\to\infty.$ (3.13)

Then $\Omega_{k}$ invades $\mathbb{R}^{2}$ as $k\to\infty$ and, choosing
$K\subset\mathbb{R}^{2}$ compact containing $z_{i,k}$ for $i\in\mathcal{D}$
and $k$ large, we get from Lemma A.1

$\displaystyle\nabla\mathcal{G}_{k}(z_{i,k},z_{j,k})$
$\displaystyle=\frac{1}{2\pi}\nabla\ln\frac{1}{|z_{i,k}-z_{j,k}|}+O\left(\frac{r_{k}}{d_{k}}\right),\quad\text{for
}i,j\in\mathcal{D},$ $\displaystyle\nabla\mathcal{G}_{k}(z_{i,k},z_{j,k})$
$\displaystyle=\frac{1}{2\pi}\nabla\ln\frac{1}{|z_{i,k}-z_{j,k}|}+O\left(\frac{r_{k}}{d_{k}}\right)=o(1),\quad\text{for
}i\in\mathcal{D},\,j\not\in\mathcal{D},$
$\displaystyle\nabla\mathcal{H}_{k}(z_{i,k},z_{i,k})$
$\displaystyle=O\left(\frac{r_{k}}{d_{k}}\right)\quad\text{for
}i\in\mathcal{D},$

as $k\to\infty$. Moreover $\nabla\ln h_{k}(z_{i,k})=O(r_{k})$ as $k\to\infty$
for $i\in\mathcal{D}$. Then from (3.11) and (3.12) we infer

$\nabla\Xi_{\mathcal{D},\mathbb{R}^{2}}(Z^{\prime}_{k})=o(1),\quad\text{as
}k\to\infty,$ (3.14)

where

$\Xi_{\mathcal{D},\mathbb{R}^{2}}(Z^{\prime}):=4\sum_{\scriptstyle
i,j\in\mathcal{D}\atop\scriptstyle i\neq j}\ln\frac{1}{|z_{i}-z_{j}|}.$ (3.15)

Notice that, thanks to (3.3), (3.8) and (3.13), we have
$\mathcal{D}\cap\\{2,\dots,N\\}\neq\emptyset$. Since also $1\in\mathcal{D}$,
the sum in (3.15) is non-empty. Letting now $k\to\infty$ in (3.14) it follows
that $\nabla\Xi_{\mathcal{D},\mathbb{R}^{2}}(Z^{\prime}_{\infty})=0$. This
contradicts Lemma A.3.

Case 2. Assume that we are not in Case 1. Then since $r_{k}\leq d_{k}$, we
have up to a subsequence

$\frac{d_{k}}{r_{k}}\to L\geq 1,\quad\text{as }k\to\infty.$

Then, up to a rotation, $\Omega_{k}$ converges to the half-space
$H=\\{(x,y)\in\mathbb{R}^{2}:y<L$}. Let $\mathcal{G}_{H}$ and
$\mathcal{H}_{H}$ be the Green’s function on $H$ and its regular part, as
given explicitly in the appendix. Choosing $K\subset H$ compact containing
$z_{i,k}$ for $i\in\mathcal{D}$ and $k$ large, by Lemma A.2 we estimate

$\displaystyle\nabla\mathcal{G}_{k}(z_{i,k},z_{j,k})$
$\displaystyle=\nabla\mathcal{G}_{H}(z_{i,k},z_{j,k})+O\left(r_{k}\right),\quad\text{for
}i,j\in\mathcal{D}$ $\displaystyle\nabla\mathcal{G}_{k}(z_{i,k},z_{j,k})$
$\displaystyle=o(1),\quad\text{for }i\in\mathcal{D},\,j\not\in\mathcal{D}$
$\displaystyle\nabla\mathcal{H}_{k}(z_{i,k},z_{i,k})$
$\displaystyle=\nabla\mathcal{H}_{H}(z_{i,k},z_{i,k})+O\left(r_{k}\right)\quad\text{for
}i\in\mathcal{D},$

and $\nabla\ln h_{k}(z_{i,k})=O(r_{k})$ as $k\to\infty$ for $i\in\mathcal{D}$,
hence, from (3.11) and (3.12) we obtain

$\nabla\Xi_{\mathcal{D},H}(Z^{\prime}_{k})=o(1),\quad\text{as }k\to\infty,$
(3.16)

where

$\Xi_{\mathcal{D},H}(Z^{\prime}):=8\pi\sum_{\scriptstyle
i,j\in\mathcal{D}\atop\scriptstyle i\neq
j}\mathcal{G}_{H}(z_{i},z_{j})+4\pi\sum_{i\in\mathcal{D}}\mathcal{H}_{H}(z_{i},z_{i}).$

Letting $k\to\infty$ in (3.16) we get
$\nabla\Xi_{\mathcal{D},H}(Z^{\prime}_{\infty})=0,$ contradicting Lemma A.5.

These contradictions, arising from assumptions (3.2) and (3.6), complete the
proof of the lemma. ∎

A first consequence of Lemma 3.1 is that the degree of $\Phi_{N,h}$ does not
depend on $h$, so that in the following we will work with $h\equiv 1$.

######  Corollary 3.1.

We have

$\deg(\nabla\Phi_{N,h},\Omega^{N}\setminus
D_{N},0)=\deg(\nabla\Phi_{N},\Omega^{N}\setminus D_{N},0),$

where, for $(x_{1},\dots,x_{N})\in\Omega^{N}\setminus D_{N}$,

$\Phi_{N}(x_{1},\dots,x_{N}):=8\pi\sum_{\scriptstyle 1\leq i,j\leq
N\atop\scriptstyle i\neq j}\mathcal{G}(x_{i},x_{j})+4\pi\sum_{1\leq i\leq
N}\mathcal{H}(x_{i},x_{i}).$

###### Proof.

We can deform $\Phi_{N,h}=\Phi^{1}_{N,h}$ into $\Phi_{N}=\Phi^{0}_{N,h}$ and
use the invariance of the degree under homotopy, since for $\delta$
sufficiently small and $t\in[0,1]$, $\Phi_{N,h}^{t}$ has no critical points on
$\partial(\Omega^{N}\setminus D_{N})_{\delta}$. ∎

###  3.2. Bending the Robin function to apply the Poincaré-Hopf theorem

Since we will not be able to apply the theorem of Poincaré-Hopf directly to
$\nabla\Phi_{N}$ to compute $\deg(\nabla\Phi_{N},\Omega^{N}\setminus
D_{N},0)$, as $\nabla\Phi_{N}$ does not always point outwards near
$\partial(\Omega^{N}\setminus D_{N})$, we now introduce a modified function
$\tilde{\Phi}_{N}$ by bending the function $\mathcal{H}(y,y)$ upwards near the
boundary of $\Omega$, in order to obtain a function that tends to $+\infty$ on
$\partial(\Omega^{N}\setminus D_{N})$. More precisely, let $\tau\in
C^{\infty}([0,\infty))$ be a non-decreasing function such that

$\tau(t)=1\quad\text{for }t\geq\frac{1}{2},\quad\tau(t)=-1\quad\text{for
}t\leq\frac{1}{4},$

and set

$\tilde{\mathcal{H}}(x):=\mathcal{H}(x,x)\tau\left(\frac{d(y,\partial\Omega)}{\tilde{\delta}}\right),$
(3.17)

see Figure 1. Let then $\tilde{\Phi}_{N}$ be given in $\Omega^{N}\setminus
D_{N}$ by

Figure 1. Heuristic sketch of $\tilde{\mathcal{H}}$ near $\partial\Omega$

$\tilde{\Phi}_{N}(x_{1},\dots,x_{N})=8\pi\sum_{\scriptstyle 1\leq i,j\leq
N\atop\scriptstyle i\neq j}\mathcal{G}(x_{i},x_{j})+4\pi\sum_{1\leq i\leq
N}\tilde{\mathcal{H}}(x_{i}).$ (3.18)

We next show that for $0<\delta^{\prime}<\delta/2$ and for $\tilde{\delta}$
sufficiently small (depending on $\delta^{\prime}$), the topological strips
determined by $\delta^{\prime}<d(y,\partial\Omega)<\delta$ is a forbidden zone
for critical points of $\tilde{\Phi}$, as made precise in the following lemma.

######  Lemma 3.2.

For every $M^{\prime}>0$ there exist $\delta>0$, such that for any
$\delta^{\prime}\in(0,\delta/2)$ there exists
$\tilde{\delta}\in(0,\delta^{\prime}/2)$ such that, setting $\tilde{\Phi}_{N}$
depending on $\tilde{\delta}$ as in (3.17)-(3.18), we have

$|\nabla\tilde{\Phi}_{N}(x_{1},\dots,x_{N})|\geq M^{\prime}$

if $(x_{1},\dots,x_{N})\in\Omega^{N}\setminus D_{N}$ is such that
$\delta^{\prime}\leq d(x_{i},\partial\Omega)\leq\delta$ for some $1\leq i\leq
N$. In particular, if $(x_{1},\dots,x_{N})$ is a critical point of
$\tilde{\Phi}_{N}$, then either $d(x_{i},\partial\Omega)>\delta$ or
$d(x_{i},\partial\Omega)<\delta^{\prime}$ for every $i=1,\dots,N$.

###### Proof.

Fix $M^{\prime}>0$ and let $\delta:=\min_{1\leq I\leq N}\delta_{I}$, where
$\delta_{I}$ is the constant $\delta$ appearing in Lemma 3.1 applied with $N$
replaced by $I$ and with $M=M^{\prime}+1>0$.

Assuming by contradiction that the lemma is false, we can find
$\delta^{\prime}\in(0,\delta/2)$, and sequences $\tilde{\delta}_{k}\to 0$ and
$(x_{1,k},\dots,x_{N,k})\subset\Omega^{N}\setminus D_{N}$ such that, for
$\tilde{\Phi}_{N,k}$ defined using $\tilde{\delta}_{k}$, we have

$|\nabla\tilde{\Phi}_{N,k}(x_{1,k},\dots,x_{N,k})|<M^{\prime}\quad\text{as
}k\to\infty,$ (3.19)

and such that, up to relabelling,

$\delta^{\prime}\leq d(x_{1,k},\partial\Omega)\leq\delta.$ (3.20)

Up to passing to subsequence and a further relabelling, let
$I\in\\{1,\dots,N\\}$ such that

$\lim_{k\to\infty}d(x_{i,k},\partial\Omega)>0\quad\text{for }1\leq i\leq I,$
(3.21)

and

$\lim_{k\to\infty}d(x_{j,k},\partial\Omega)=0,\quad I+1\leq j\leq N.$

By (3.20) and Lemma 3.1 we have

$|\nabla\Phi_{I}(x_{1,k},\dots,x_{I,k})|\geq M^{\prime}+1,$ (3.22)

where

$\Phi_{I}(x_{1},\dots,x_{I}):=8\pi\sum_{\scriptstyle 1\leq i,j\leq
I\atop\scriptstyle i\neq j}\mathcal{G}(x_{i},x_{j})+4\pi\sum_{1\leq i\leq
I}\mathcal{H}(x_{i},x_{i}).$

Observe, on the other, hand that

$\nabla\mathcal{G}(x_{i,k},x_{j,k})=o(1),\quad 1\leq i\leq I,\quad I+1\leq
j\leq N,$ (3.23)

with $o(1)\to 0$ as $k\to\infty$ (since $x_{j,k}$ is approaching
$\partial\Omega$, while $x_{i,k}$ is not as $k\to\infty$). Then, by (3.21),
for $k$ large enough (hence $\tilde{\delta}_{k}$ sufficiently small), we have

$\begin{split}\tilde{\Phi}_{N,k}(x_{1,k},\dots,x_{N,k})=&\Phi_{I}(x_{1,k},\dots,x_{I,k})+16\pi\sum_{\scriptstyle
1\leq i\leq I\atop\scriptstyle I+1\leq j\leq N}\mathcal{G}(x_{i,k},x_{j,k})\\\
&+8\pi\sum_{I+1\leq i,j\leq N}\mathcal{G}(x_{i,k},x_{j,k})+4\pi\sum_{I+1\leq
J\leq N}\tilde{\mathcal{H}}_{k}(x_{j,k}).\end{split}$

Then, setting $X^{\prime}:=(x_{1},\dots,x_{I})$, from (3.19), (3.22) and
(3.23) we get

$\begin{split}M^{\prime}>&|\nabla_{X^{\prime}}\tilde{\Phi}_{N,k}(x_{1,k},\dots,x_{N,k})|=|\nabla\Phi_{I}(x_{1,k},\dots,x_{I,k})|+o(1)\\\
\geq&M^{\prime}+1+o(1),\end{split}$

giving a contradiction for $k$ large enough. ∎

We now have the analog of Lemma 3.1 for $\tilde{\Phi}_{N}.$

######  Lemma 3.3.

For every $M^{\prime\prime}>0$ there exists
$\delta^{\prime\prime}\leq\tfrac{\tilde{\delta}}{2}$ depending on
$\tilde{\delta}$ and $M^{\prime\prime}$ such that

$|\nabla\tilde{\Phi}_{N}(x_{1},\dots,x_{N})|\geq M^{\prime\prime}$

for every $(x_{1},\dots,x_{N})\in(\Omega^{N}\setminus
D_{N})\setminus(\Omega^{N}\setminus D_{N})_{\delta^{\prime\prime}}$. In
particular, all the critical points of $\nabla\tilde{\Phi}_{N}$ are in
$(\Omega^{N}\setminus D_{N})_{\delta^{\prime\prime}}$ and we can define

$\deg(\nabla\tilde{\Phi}_{N},\Omega^{N}\setminus
D_{N},0):=\deg(\nabla\tilde{\Phi}_{N},A,0),$

for any open set $A$ such that $(\Omega^{N}\setminus
D_{N})_{\delta^{\prime\prime}}\subset A\subset\Omega^{N}\setminus D_{N}.$

###### Proof.

The proof by contradiction is essentially identical to the proof of Lemma 3.1,
upon noticing the following facts. In Case $1$, namely $r_{k}=o(d_{k})$, if
(up to a subsequence) $d_{k}\to 0$ as $k\to\infty$, then

$\tilde{\mathcal{H}}(x_{i,k})=-\mathcal{H}(x_{i,k},x_{i,k})\quad\text{for
}i\in\mathcal{D},\quad k\text{ large},$

hence

$\nabla\tilde{\mathcal{H}}_{k}(z_{i,k})=-\nabla\mathcal{H}_{k}(z_{i,k},z_{i,k})=O(r_{k}),\quad\text{for
}i\in\mathcal{D},\quad k\text{ large}.$

If $d_{k}\not\to 0$, from (3.17), we bound

$|\nabla\tilde{\mathcal{H}}(x_{i,k})|=O(\nabla\mathcal{H}(x_{i,k},x_{i,k})+C_{\tilde{\delta}}|\mathcal{H}(x_{i,k},x_{i,k})|)=O(1),\quad\text{as
}k\to\infty,\quad i\in\mathcal{D},$

hence again $\nabla\tilde{\mathcal{H}}_{k}(z_{i,k})=O(r_{k})$ as $k\to\infty$
for $i\in\mathcal{D}$, where
$\tilde{\mathcal{H}}_{k}(z):=\tilde{\mathcal{H}}(x_{k}+r_{k}z)$. Then, up to a
subsequence, $Z_{k}^{\prime}$ converges to $Z_{\infty}^{\prime}$, a critical
point of $\Xi_{\mathcal{D},\mathbb{R}^{2}}$, contradicting Lemma A.3.

In Case $2$, since $d(x_{1,k},\partial\Omega)\to 0$, we also have
$d(x_{i,k},\partial\Omega)\to 0$ as $k\to\infty$ for $i\in\mathcal{D}$, hence
$\tilde{\mathcal{H}}(x_{i,k})=-\mathcal{H}(x_{i,k},x_{i,k})$ for
$i\in\mathcal{D}$ and $k$ large enough. Then, setting
$\Xi^{-}_{\mathcal{D},k}$ as in (3.11) with $h_{k}\equiv 1$ and
$\mathcal{H}_{k}(z_{i},z_{i})$ replaced by $-\mathcal{H}_{k}(z_{i},z_{i})$, we
obtain $\nabla\Xi^{-}_{\mathcal{D},H}(Z^{\prime}_{\infty})=0,$ where

$\Xi^{-}_{\mathcal{D},H}(Z^{\prime}):=8\pi\sum_{\scriptstyle
i,j\in\mathcal{D}\atop\scriptstyle i\neq
j}\mathcal{G}_{H}(z_{i},z_{j})-4\pi\sum_{i\in\mathcal{D}}\mathcal{H}_{H}(z_{i},z_{i}),$

contradicting Lemma A.5. ∎

We recall next a variant of the classical Poincaré-Hopf index theorem, which
can be found e.g. in [6], pages 99-104 (here we adapt the statement to our
purposes).

######  Proposition 3.1.

Let $U\subset\mathbb{R}^{n}$ be an open set and consider $f\in
C^{2}(U,\mathbb{R})$. Assume also that, for some $b\in\mathbb{R}$,
$f^{b}:=\\{x\in U:f(x)<b\\}$ has compact closure, and that $f$ has no critical
points in $f^{-1}(b)$. Then we have

$\deg(\nabla f,f^{b},0)=\chi(f^{b}).$ (3.24)

######  Lemma 3.4.

We have

$\deg(\nabla\tilde{\Phi}_{N},\Omega^{N}\setminus
D_{N},0)=\chi(\Omega)(\chi(\Omega)-1)...(\chi(\Omega)-N+1).$ (3.25)

###### Proof.

Since

$\lim_{(x_{1},\dots,x_{N})\to\partial(\Omega^{N}\setminus
D_{N})}\tilde{\Phi}_{N}(x_{1},\dots,x_{N})=+\infty,$

we can take $b>0$ large enough such that

$(\Omega^{N}\setminus
D_{N})_{\delta^{\prime\prime}}\subset(\tilde{\Phi}_{N})^{b},$

where $\delta^{\prime\prime}$ is as in Lemma 3.3 with $M^{\prime\prime}=1$.
Then, since $(\Omega^{N}\setminus D_{N})\setminus(\tilde{\Phi}_{N})^{b}$
contains no critical points of $\tilde{\Phi}_{N}$ by Lemma 3.3, we can apply
Proposition 3.1 to get

$\deg(\nabla\tilde{\Phi}_{N},\Omega^{N}\setminus
D_{N},0)=\deg(\nabla\tilde{\Phi}_{N},(\tilde{\Phi}_{N})^{b},0)=\chi((\tilde{\Phi}_{N})^{b}).$

Since $(\Omega^{N}\setminus D_{N})\setminus(\tilde{\Phi}_{N})^{b}$ contains no
critical points of $\tilde{\Phi}_{N}$, we can use a gradient flow to retract
$(\Omega^{N}\setminus D_{N})$ to $(\tilde{\Phi}_{N})^{b}$. Therefore
$\chi((\tilde{\Phi}_{N})^{b})=\chi(\Omega^{N}\setminus D_{N}),$ and we
conclude with Proposition B.1 below. ∎

###  3.3. Counting the new critical points

By bending the Robin function we have in general created new critical points
near the boundary. In this section we will show that these new critical points
do not change the total degree. To this purpose, we now define the following
sets (see Figure 2):

$\Omega_{\delta}:=\left\\{y\in\Omega\;:\;d(y,\partial\Omega)>\delta\right\\};\quad\Sigma_{\delta^{\prime}}:=\Omega\setminus\overline{\Omega}_{\delta^{\prime}},$

and for $0\leq I\leq N$

$\begin{split}&\Theta_{\delta,\delta^{\prime},I}^{*}=\left\\{(x_{1},\dots,x_{N})\in\Omega^{N}\setminus
D_{N}\;:\;x_{1},\dots,x_{I}\in{\Omega}_{\delta},\;x_{I+1},\dots,x_{N}\in{\Sigma}_{\delta^{\prime}}\right\\}\\\
&\qquad\quad=((\Omega_{\delta})^{I}\setminus
D_{I})\times((\Sigma_{\delta^{\prime}})^{N-I}\setminus D_{N-I});\\\
&\Theta_{\delta,\delta^{\prime},I}=\big{\\{}(x_{1},\dots,x_{N})\in\Omega^{N}\setminus
D_{N}\;:\;(x_{\sigma(1)},\dots,x_{\sigma(N)})\in\Theta_{\delta,\delta^{\prime},I}^{*}\\\
&\qquad\qquad\text{ for a permutation }\sigma\in S_{N}\big{\\}};\\\
&\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*}=\Theta_{\delta,\delta^{\prime},I}^{*}\cap(\Omega^{N}\setminus
D_{N})_{\delta^{\prime\prime}},\quad\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}=\Theta_{\delta,\delta^{\prime},I}\cap(\Omega^{N}\setminus
D_{N})_{\delta^{\prime\prime}}.\end{split}$

Figure 2.
$(x_{1},\dots,x_{N})\in\Theta^{*}_{\delta,\delta^{\prime},I}=((\Omega_{\delta})^{I}\setminus
D_{I})\times((\Sigma_{\delta^{\prime}})^{N-I}\setminus D_{N-I})$

By Lemmata 3.2 and 3.3, we can define for $0\leq I\leq N-1$

$\deg(\nabla\tilde{\Phi}_{N},\Theta_{\delta,\delta^{\prime},I},0):=\deg(\nabla\tilde{\Phi}_{N},A,0)$

for any open set $A$ such that
$\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}\subset
A\subset\Theta_{\delta,\delta^{\prime},I}.$ Moreover, there are no critical
points of $\tilde{\Phi}_{N}$ in $(\Omega^{N}\setminus
D_{N})\setminus\cup_{I=0}^{N}\Theta_{\delta,\delta^{\prime},I}$ by Lemma 3.2;
therefore by the excision property and noting that
$\Theta_{\delta,\delta^{\prime},N}=(\Omega_{\delta})^{N}\setminus D_{N}$ and
that $\tilde{\Phi}_{N}=\Phi_{N}$ inside $(\Omega_{\delta})^{N}\setminus
D_{N}$, we have

$\begin{split}\deg(\nabla\tilde{\Phi}_{N},\Omega^{N}\setminus
D_{N},0)&=\deg(\nabla\Phi_{N},(\Omega_{\delta})^{N}\setminus
D_{N},0)+\\!\sum_{I=0}^{N-1}\\!\deg(\nabla\tilde{\Phi}_{N},\Theta_{\delta,\delta^{\prime},I},0)\\\
&\\!\\!\\!\\!\\!\\!\\!=\deg(\nabla\Phi_{N},(\Omega_{\delta})^{N}\setminus
D_{N},0)+\sum_{I=0}^{N-1}\binom{N}{I}\deg(\nabla\tilde{\Phi}_{N},\Theta_{\delta,\delta^{\prime},I}^{*},0).\end{split}$
(3.26)

######  Lemma 3.5.

Up to choosing $\delta^{\prime}$ sufficiently small in Lemma 3.2 (and
$\tilde{\delta}$ and $\delta^{\prime\prime}$ correspondingly), we have that

$\begin{split}\deg(\nabla\tilde{\Phi}_{N},\Theta_{\delta,\delta^{\prime},I}^{*},0)=&\deg(\nabla_{X^{\prime}}\Phi_{I},(\Omega_{\delta})^{I}\setminus
D_{I},0)\\\
&\qquad\qquad\times\deg(\nabla_{X^{\prime\prime}}\tilde{\Phi}_{N-I},(\Sigma_{\delta^{\prime}})^{N-I}\setminus
D_{N-I},0),\end{split}$

for $0\leq I\leq N-1$, where

$X^{\prime}=(x_{1},\dots,x_{I}),\quad
X^{\prime\prime}=(x_{I+1},\dots,x_{N}),\quad X=(X^{\prime},X^{\prime\prime}),$

and the degrees are well-defined intersecting the domains with
$(\Omega^{I}\setminus D_{I})_{\delta^{\prime\prime}}$ and
$(\Omega^{N-I}\setminus D_{N-I})_{\delta^{\prime\prime}}$.

###### Proof.

For $X\in\Theta_{\delta,\delta^{\prime},I}^{*}$ we have

$x_{i}\in\Omega_{\delta}\quad\hbox{for }1\leq i\leq I,\qquad
x_{a}\in\Sigma_{\delta^{\prime}}\quad\hbox{for }I+1\leq a\leq N.$ (3.27)

Interestingly, the gradient almost decouples into the first $I$ components and
the last $N-I$. In fact, since $\tilde{\mathcal{H}}(x)$ coincides with
$\mathcal{H}(x,x)$ for $x\in\Omega_{\delta}$, we have

$\tilde{\Phi}_{N}(X)=\Phi_{I}(X^{\prime})+\tilde{\Phi}_{N-I}(X^{\prime\prime})+\Xi_{I,N}(X),$

where

$\begin{split}\tilde{\Phi}_{N-I}(X^{\prime\prime})&=8\pi\sum_{\scriptstyle
I+1\leq a,b\leq N\atop\scriptstyle a\neq
b}\mathcal{G}(x_{a},x_{b})+4\pi\sum_{I+1\leq a\leq
N}\tilde{\mathcal{H}}(x_{a});\\\ \Xi_{I,N}(X)&=16\pi\sum_{1\leq i\leq I<a\leq
N}\mathcal{G}(x_{i},x_{a}).\end{split}$

Notice that

$\mathcal{G}(x_{i},x_{a})=o_{\delta^{\prime}}(1),\quad\nabla\mathcal{G}(x_{i},x_{a})=o_{\delta^{\prime}}(1),$

with $o_{\delta^{\prime}}(1)\to 0$ as $\delta^{\prime}\to 0$ (for $\delta$
fixed), uniformly with respect to $x_{i}$ and $x_{a}$ as in (3.27). Now
observe that
$X=(x_{1},\dots,x_{N})\in\partial\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*}$
(with $\delta^{\prime\prime}<\tfrac{\tilde{\delta}}{2}$ to be fixed) implies
that one of the following holds

  1. (1)

$|x_{i}-x_{j}|=\delta^{\prime\prime}$ for some $1\leq i<j\leq I$;

  2. (2)

$|x_{a}-x_{b}|=\delta^{\prime\prime}$ for some $I+1\leq a<b\leq N$;

  3. (3)

$d(x_{i},\partial\Omega)=\delta$ for some $1\leq i\leq I$;

  4. (4)

$d(x_{a},\partial\Omega)=\delta^{\prime}$ for some $I+1\leq a\leq N$;

  5. (5)

$d(x_{a},\partial\Omega)=\delta^{\prime\prime}$ for some $I+1\leq a\leq N$.

If we choose $M=M^{\prime}=M^{\prime\prime}=1$ in Lemmata 3.1, 3.2 and 3.3 and
$\delta,\delta^{\prime},\delta^{\prime\prime}$ sufficiently small, we obtain
that in all the above cases

$\max\\{|\nabla_{X^{\prime}}\Phi_{I}(X^{\prime})|,|\nabla_{X^{\prime\prime}}\tilde{\Phi}_{N-I}(X^{\prime\prime})|\\}\geq
1,\quad\text{for
}X\in\partial\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*}.$
(3.28)

Then, by the analog of (3.23), we can choose $\delta^{\prime}$ even smaller
(changing also $\tilde{\delta}$ accordingly) so that in addition to (3.28) we
also have

$|\nabla\Xi_{I,N}(X)|\leq\frac{1}{2}\quad\text{for
}X\in\partial\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*}.$
(3.29)

Consider now for $t\in[0,1]$

$\tilde{\Phi}_{N}^{t}(X):=\Phi_{I}(X^{\prime})+\tilde{\Phi}_{N-I}(X^{\prime\prime})+t\Xi_{I,N}(X).$

Then (3.28)-(3.29) imply at once

$|\nabla\tilde{\Phi}_{N}^{t}(X)|\geq|(\nabla_{X^{\prime}}\Phi_{I}(X^{\prime}),\nabla_{X^{\prime\prime}}\tilde{\Phi}_{N-I}(X^{\prime\prime}))|-|\nabla\Xi_{I,N}(X)|>0\text{
for }X\in\partial\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*}.$

By the invariance of the degree under homotopy, we obtain

$\begin{split}\deg(\nabla\tilde{\Phi}_{N},\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*},0)&=\deg(\nabla\tilde{\Phi}_{N}^{0},\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*},0)\\\
&=\deg((\nabla_{X^{\prime}}\Phi_{I}(X^{\prime}),\nabla_{X^{\prime\prime}}\tilde{\Phi}_{N-I}(X^{\prime\prime})),\Theta_{\delta,\delta^{\prime},\delta^{\prime\prime},I}^{*},0),\end{split}$
(3.30)

and we conclude by the product formula of the degree. ∎

######  Lemma 3.6.

Up to choosing $\delta^{\prime}$ sufficiently small (reducing
$\tilde{\delta},\delta^{\prime\prime}$ accordingly), for $1\leq J\leq N$ we
have

$\deg(\nabla\tilde{\Phi}_{J},(\Sigma_{\delta^{\prime}})^{J}\setminus
D_{J},0)=0.$

###### Proof.

We will work on the the larger domain $\Sigma_{2\delta^{\prime}}$, and modify
$\tilde{\mathcal{H}}$ in the strip
$\Sigma_{2\delta^{\prime}}\setminus\Sigma_{\delta^{\prime}}=\bar{\Omega}_{\delta^{\prime}}\setminus\bar{\Omega}_{2\delta^{\prime}}$
by defining

$\hat{\mathcal{H}}(x):=\tilde{\mathcal{H}}(x)+\sigma\left(\frac{d(x,\partial\Omega)}{\delta^{\prime}}\right),$

where $\sigma\in C^{\infty}([0,2))$ is non-negative, non-decreasing, and

$\sigma(t)=\begin{cases}0&\text{for }0\leq t\leq 1,\\\
\ln\left(\frac{1}{2-t}\right)&\text{for }\frac{3}{2}\leq t<2.\end{cases}$

Clearly $\hat{\mathcal{H}}=\tilde{\mathcal{H}}$ in $\Sigma_{\delta^{\prime}}$,
so that

$\deg(\nabla\tilde{\Phi}_{J},(\Sigma_{\delta^{\prime}})^{J}\setminus
D_{J},0)=\deg(\nabla\hat{\Phi}_{J},(\Sigma_{\delta^{\prime}})^{J}\setminus
D_{J},0),$ (3.31)

where, for $(x_{1},\dots,x_{J})\in(\Sigma_{2\delta^{\prime}})^{J}\setminus
D_{J}$,

$\begin{split}\hat{\Phi}_{J}(x_{1},\dots,x_{J})&=\tilde{\Phi}_{J}(x_{1},\dots,x_{J})+\sum_{1\leq
i\leq J}\sigma\left(\frac{d(x_{i},\partial\Omega)}{\delta^{\prime}}\right)\\\
&=8\pi\sum_{\scriptstyle 1\leq i,j\leq J\atop\scriptstyle i\neq
j}\mathcal{G}(x_{i},x_{j})+4\pi\sum_{1\leq i\leq
J}\hat{\mathcal{H}}(x_{i}).\end{split}$ (3.32)

Thanks to Lemmata 3.7 and 3.8 below, we can choose $\delta^{\prime}>0$ even
smaller than before, and $\tilde{\delta}$ accordingly small (without affecting
the previous results, which are valid for $\delta^{\prime}$, $\tilde{\delta}$
and $\delta^{\prime\prime}$ sufficiently small), so that $\hat{\Phi}_{J}$ has
no critical points in
$\left((\Sigma_{2\delta^{\prime}})^{J}\setminus(\Sigma_{\delta^{\prime}})^{J}\right)\setminus
D_{J}\,.$ Then

$\deg(\nabla\hat{\Phi}_{J},(\Sigma_{\delta^{\prime}})^{J}\setminus
D_{J},0)=\deg(\nabla\hat{\Phi}_{J},(\Sigma_{2\delta^{\prime}})^{J}\setminus
D_{J},0).$ (3.33)

Since now

$\lim_{(x_{1},\dots,x_{J})\to\partial((\Sigma_{2\delta^{\prime}})^{J}\setminus
D_{J})}\hat{\Phi}_{J}(x_{1},\dots,x_{J})=+\infty,$

we can apply Proposition 3.1, and get

$\deg(\nabla\hat{\Phi}_{J},(\Sigma_{2\delta^{\prime}})^{J}\setminus
D_{J},0)=\chi((\Sigma_{2\delta^{\prime}})^{J}\setminus D_{J}).$ (3.34)

On the other hand, $\chi(\Sigma_{2\delta^{\prime}})=0$, since
$\Sigma_{2\delta^{\prime}}$ retracts to a union of circles (up to choosing
$\delta^{\prime}$ sufficiently small) and $\chi(\mathbb{S}^{1})=0$. Then it
follows from Proposition B.1

$\chi((\Sigma_{2\delta^{\prime}})^{J}\setminus D_{J})=0.$

This, together with (3.31), (3.33) and (3.34) yields
$\,\deg(\nabla\tilde{\Phi}_{J},(\Sigma_{\delta^{\prime}})^{J}\setminus
D_{J},0)=\chi((\Sigma_{2\delta^{\prime}})^{J}\setminus D_{J})=0.$ ∎

######  Lemma 3.7.

For $1\leq J\leq N$ and $M^{\prime}>0$ and $\delta>0$ there exists
$\delta^{\prime}\in(0,\delta/2)$ such that

$|\nabla\Psi_{J}|\geq M^{\prime}\quad\text{in
}(\Sigma_{2\delta^{\prime}})^{J}\setminus D_{J},$

where, for
$x_{1},\dots,x_{J}\in(\Omega\setminus\overline{\Omega}_{2\delta^{\prime}})^{J}\setminus
D_{J}$,

$\begin{split}\Psi_{J}(x_{1},\dots,x_{J})&:=\Phi_{J}(x_{1},\dots,x_{J})+\sum_{1\leq
i\leq J}\sigma\left(\frac{d(x_{i},\partial\Omega)}{\delta^{\prime}}\right)\\\
&=8\pi\sum_{\scriptstyle 1\leq i,j\leq J\atop\scriptstyle i\neq
j}\mathcal{G}(x_{i},x_{j})+4\pi\sum_{1\leq i\leq
J}\mathcal{H}(x_{i})+\sum_{1\leq i\leq
J}\sigma\left(\frac{d(x_{i},\partial\Omega)}{\delta^{\prime}}\right),\end{split}$
(3.35)

and $\mathcal{G}$ and $\mathcal{H}$ are still the Green’s function of $\Delta$
in $\Omega$ (not in $\Omega\setminus\overline{\Omega}_{2\delta^{\prime}}$) and
its regular part.

###### Proof.

We assume by contradiction that there exist sequences $\delta^{\prime}_{k}\to
0^{+}$ and

$(x_{1,k},\dots,x_{J,k})\in(\Sigma_{2\delta_{k}^{\prime}})^{J}\setminus D_{J}$

such that

$\nabla\Psi_{J}^{k}(x_{1,k},\dots,x_{J,k})=O(1)\quad\text{as }k\to\infty,$
(3.36)

where the notation $\Psi_{J}^{k}$ is meant to emphasize that $\Psi_{J}$
depends on $\delta_{k}^{\prime}$, due to the last term in (3.35). We proceed
as in the proof of Lemma 3.1. Set

$r_{i,k}=\min\left\\{d_{i,k},\hat{d}_{i,k},\min_{j\neq
i}|x_{i,k}-x_{j,k}|\right\\},$

where

$d_{i,k}:=d(x_{i,k},\partial\Omega),\quad\hat{d}_{i,k}:=d(x_{i,k},\partial\Omega_{2\delta_{k}^{\prime}})=2\delta_{k}^{\prime}-d(x_{i,k},\partial\Omega).$

Up to reordering we assume $r_{1,k}\leq r_{i,k}$ for every $1\leq i\leq J$,
and set

$r_{k}:=r_{1,k},\quad d_{k}:=d_{1,k},\quad\hat{d}_{k}:=\hat{d}_{1,k},\quad
x_{k}:=x_{1,k}.$

Define $\mathcal{D}$, $Z_{k}$, $Z^{\prime}_{k}\to Z^{\prime}_{\infty}$,
$\Omega_{k}$, as in the proof of Lemma 3.1, and similarly

$(\Sigma_{2\delta_{k}^{\prime}})_{k}:=r_{k}^{-1}(\Sigma_{2\delta_{k}^{\prime}}-x_{k}).$

Then we set, for
$Z=(z_{1},\dots,z_{J})\in((\Sigma_{2\delta_{k}^{\prime}})_{k})^{J}\setminus
D_{J}$,

$\begin{split}\hat{\Xi}_{\mathcal{D},k}(z_{1},\dots,z_{N}):=&8\pi\sum_{\scriptstyle
i,j\in\mathcal{D}\atop\scriptstyle i\neq
j}\mathcal{G}_{k}(z_{i},z_{j})+16\pi\sum_{i\in\mathcal{D}}\sum_{\scriptstyle
1\leq j\leq N\atop\scriptstyle
j\not\in\mathcal{D}}\mathcal{G}_{k}(z_{i},z_{j})\\\
&+4\pi\sum_{i\in\mathcal{D}}\mathcal{H}_{k}(z_{i},z_{i})+\sum_{i\in\mathcal{D}}\sigma\left(\frac{r_{k}d(z_{i},\partial\Omega_{k})}{\delta^{\prime}_{k}}\right),\end{split}$
(3.37)

where $\mathcal{G}_{k}$ and $\mathcal{H}_{k}$ are the Green’s function of
$\Delta$ on $\Omega_{k}$ and its regular part, respectively. Observe that
$r_{k}\leq\delta_{k}^{\prime}\to 0$ as $k\to\infty$. We consider several
cases.

Case 1 Assume that, up to a subsequence, $r_{k}=o(d_{k})$,
$r_{k}=o(\hat{d}_{k})$ as $k\to\infty$. Since

$\nabla_{z}\sigma\left(\frac{r_{k}d(z,\partial\Omega_{k})}{\delta^{\prime}_{k}}\right)=\nabla_{z}\sigma\left(\frac{d(x_{k}+r_{k}z,\partial\Omega)}{\delta^{\prime}_{k}}\right)=r_{k}\nabla_{x}\sigma\left(\frac{d(x,\partial\Omega)}{\delta^{\prime}_{k}}\right)\bigg{|}_{x=x_{k}+r_{k}z},$

and

$\sigma^{\prime}\left(\frac{d(x_{i,k},\partial\Omega)}{\delta^{\prime}_{k}}\right)=O\left(\frac{\delta_{k}^{\prime}}{\hat{d}_{k}}\right),\quad|\nabla
d(\cdot,\partial\Omega)|=1,\quad\text{for }i\in\mathcal{D}$

we infer

$\nabla_{z}\sigma\left(\frac{r_{k}d(z_{i,k},\partial\Omega_{k})}{\delta^{\prime}_{k}}\right)=O\left(\frac{r_{k}}{\hat{d}_{k}}\right)=o(1)\quad\text{as
}k\to\infty,\quad\text{for }i\in\mathcal{D}.$

Then we can proceed as in Case 1 of the proof of Lemma 3.1 to get that
$Z_{\infty}^{\prime}$ is critical point of $\Xi_{\mathcal{D},\mathbb{R}^{2}}$,
as defined in (3.15), contradicting Lemma A.3.

Case 2 Assume that, up to a subsequence, $r_{k}=o(\hat{d}_{k})$,
$\frac{d_{k}}{r_{k}}\to L\geq 1$ as $k\to\infty$. This implies that
$r_{k}=o(\delta_{k}^{\prime})$ as $k\to\infty$ and, up to a rotation,
$(\Sigma_{\delta_{k}^{\prime}})_{k}$ converges to a half-space
$H=\\{(x,y)\in\mathbb{R}^{2}:y<L\\}$. We proceed as in Case 1, except that now

$\sigma\left(\frac{r_{k}d(z_{i,k},\partial\Omega_{k})}{\delta^{\prime}_{k}}\right)=0,\quad\text{for
}i\in\mathcal{D},\quad k\text{ large enough}.$

Then $\hat{\Xi}_{\mathcal{D},k}(Z_{k})=\Xi_{\mathcal{D},k}(Z_{k})$ for $k$
large enough, where $\Xi_{\mathcal{D},k}$ is as in (3.11) (with $t_{k}=0$).
Then we are in the same situation as Case 2 of the proof of Lemma 3.1, leading
to the same contradiction, i.e. $Z_{\infty}^{\prime}$ is a critical point of
$\Xi_{\mathcal{D},H}$, violating Lemma A.5.

Case 3 Assume that, up to a subsequence, $r_{k}=o(d_{k})$,
$\frac{\hat{d}_{k}}{r_{k}}\to\hat{L}_{1}\geq 1$. This implies in particular
that, up to a subsequence,

$r_{k}=o(\delta_{k}^{\prime}),\quad\frac{\hat{d}_{i,k}}{r_{k}}\to\hat{L}_{i}\geq
1\quad\text{as }k\to\infty,\text{ for }i\in\mathcal{D}.$ (3.38)

Then, up to a rotation $(\Sigma_{2\delta_{k}^{\prime}})_{k}$ converges to
$\hat{H}=\\{(x,y)\in\mathbb{R}^{2}:y>-\hat{L}\\}$. As before, $r_{k}=o(d_{k})$
implies $r_{k}=o(d_{i,k})$ for $i\in\mathcal{D}$ and it follows
$\nabla\mathcal{H}_{k}(z_{i,k},z_{i,k})=O(r_{k})$ as $k\to\infty$ for
$i\in\mathcal{D}$. Considering that

$\frac{r_{k}d(z_{i,k},\partial\Omega_{k})}{\delta_{k}^{\prime}}=\frac{d(x_{i,k},\partial\Omega)}{\delta_{k}^{\prime}}\to
2,\quad\text{as }k\to\infty,\quad\text{for }i\in\mathcal{D},$

and $\sigma(t)=-\ln(2-t)$ for $\frac{3}{2}\leq t<2$, we have that

$\begin{split}\nabla_{z}\sigma\left(\frac{r_{k}d(z_{i,k},\partial\Omega_{k})}{\delta_{k}^{\prime}}\right)&=\frac{r_{k}\nabla
d(z_{i,k},\partial\Omega_{k})}{2\delta_{k}^{\prime}-r_{k}d(z_{i,k},\partial\Omega_{k})}=\frac{-r_{k}\nabla
d(z_{i,k},\partial(\Omega\setminus\Omega_{2\delta_{k}^{\prime}})_{k})}{\hat{d}_{i,k}}\\\
&\to-\frac{\nabla_{z}d(z_{i,\infty},\partial\hat{H})}{\hat{L}_{i}}=\nabla_{z}\ln\frac{1}{d(z_{i,\infty},\partial\hat{H})},\end{split}$
(3.39)

since $\hat{L}_{i}=d(z_{i,\infty},\partial\hat{H}).$ Then
$Z_{\infty}^{\prime}$ is a critical point of

$\hat{\Xi}_{\mathcal{D},\hat{H}}(Z)=4\sum_{\scriptstyle
i,j\in\mathcal{D}\atop\scriptstyle i\neq
j}\ln\frac{1}{|z_{i}-z_{j}|}+\sum_{i\in\mathcal{D}}\ln\frac{1}{d(z_{i},\partial\hat{H})},$

contradicting Lemma A.4.

Case 4 If we are not in any of the above cases, then, up to a subsequence,

$\frac{d_{k}}{r_{k}}\to L_{1}\geq
1,\quad\frac{\hat{d}_{k}}{r_{k}}\to\hat{L}_{1}\geq 1,\quad\text{as
}k\to\infty.$

Then,

$\frac{r_{k}}{\delta_{k}^{\prime}}\to\gamma>0,\quad\text{as }k\to\infty,$

and,

$\frac{d_{i,k}}{r_{k}}\to L_{i}\geq
1,\quad\frac{\hat{d}_{i,k}}{r_{k}}\to\hat{L}_{i}\geq
1,\quad\frac{\hat{d}_{i,k}}{\delta_{k}^{\prime}}\to\rho_{i}\in(0,2),\quad\text{as
}k\to\infty,\quad i\in\mathcal{D}.$

Up to a rotation $(\Sigma_{2\delta_{k}^{\prime}})_{k}$ converges to
$H\cap\hat{H}$, where

$H=\\{(x,y)\in\mathbb{R}^{2}:y<L_{1}\\},\quad\hat{H}=\\{(x,y)\in\mathbb{R}^{2}:y>-\hat{L}_{1}\\}.$

Considering now that

$L_{i}+o(1)=\frac{d_{i,k}}{r_{k}}=d(z_{i,k},\partial\Omega_{k})=d(z_{i,\infty},\partial
H)+o(1),\quad\text{as }k\to\infty,\quad\text{for }i\in\mathcal{D},$

we compute

$\begin{split}\nabla_{z}\sigma\left(\frac{r_{k}d(z_{i,k},\partial\Omega_{k})}{\delta^{\prime}_{k}}\right)&=\frac{r_{k}}{\delta^{\prime}_{k}}\sigma^{\prime}\left(\frac{d_{i,k}}{\delta^{\prime}_{k}}\right)\nabla_{z}d(z_{i,k},\partial\Omega_{k})\\\
&\to\gamma\sigma^{\prime}(\gamma L_{i})\nabla_{z}d(z_{i,\infty},\partial H)\\\
&=\nabla_{z}\sigma(\gamma d(z_{i,\infty},\partial H)),\quad\text{as
}k\to\infty,\quad\text{for }i\in\mathcal{D}.\end{split}$

Then $Z^{\prime}_{\infty}$ is a critical point of

$\hat{\Xi}_{\mathcal{D},\hat{H}}(Z^{\prime})=8\pi\sum_{\scriptstyle
i,j\in\mathcal{D}\atop\scriptstyle i\neq
j}\mathcal{G}_{H}(z_{i},z_{j})+4\pi\sum_{i\in\mathcal{D}}\mathcal{H}_{H}(z_{i})+\sum_{i\in\mathcal{D}}\sigma(\gamma
d(z_{i},H)).$

This contradicts Lemma A.6.

These contradictions, arising from (3.36), complete the proof of the lemma. ∎

######  Lemma 3.8.

Given $1\leq J\leq N$ and $\delta^{\prime}$ as in Lemma 3.7 (or smaller),
there exists $\tilde{\delta}\in(0,\tfrac{\delta^{\prime}}{2})$ sufficiently
small such that $\hat{\Phi}_{J}$ (defined as in (3.32), hence depending on
$\tilde{\delta}$) has no critical points in
$((\Sigma_{2\delta^{\prime}})^{J}\setminus(\Sigma_{\delta^{\prime}})^{J})\setminus
D_{J}.$

###### Proof.

This follows as in the proof of Lemma 3.2 by a contradiction argument, using
Lemma 3.7 instead of Lemma 3.1. Here $2\delta^{\prime}$ plays the role of
$\delta$ in Lemma 3.7. ∎

###  3.4. Proof of Proposition 2.4 (completed)

By Corollary 2.3 we can assume $h\equiv 1$ and work with $\Phi_{N}$ instead of
$\Phi_{N,h}$. Using Lemma 3.1, considering that $\Phi_{N}$ coincides with
$\tilde{\Phi}_{N}$ on $(\Omega_{\delta})^{N}\setminus D_{N}$, using formula
(3.26) and Lemmata 3.5 and 3.6 we now get

$\begin{split}\deg(\nabla\Phi_{N},\Omega^{N}\setminus
D_{N},0)&=\deg(\nabla\Phi_{N},(\Omega_{\delta})^{N}\setminus D_{N},0)\\\
&=\deg(\nabla\tilde{\Phi}_{N},(\Omega_{\delta})^{N}\setminus D_{N},0)\\\
&=\deg(\nabla\tilde{\Phi}_{N},\Omega^{N}\setminus D_{N},0)-\sum_{0\leq I\leq
N-1}\deg(\nabla\tilde{\Phi}_{N},\Theta_{\delta,\delta^{\prime},I},0)\\\
&=\deg(\nabla\tilde{\Phi}_{N},\Omega^{N}\setminus D_{N},0).\end{split}$

By Lemma 3.4 we finally obtain

$\deg(\nabla\Phi_{N},\Omega^{N}\setminus
D_{N},0)=\chi(\Omega)(\chi(\Omega)-1)...(\chi(\Omega)-N+1).$

$\square$

##  Appendix A Properties of the Green’s function

For some fixed $L,\hat{L}>0$, let $H$ and $\hat{H}$ denote the half spaces

$H=\\{(x,y)\in\mathbb{R}^{2}:y<L\\},\quad\hat{H}=\\{(x,y)\in\mathbb{R}^{2}:y>-\hat{L}\\}\,.$
(A.1)

On the half-space $H$, the Green’s function and its regular part are given
explicitly by

$\mathcal{G}_{H}(z,\eta)=\frac{1}{2\pi}\ln\frac{|z-\eta^{*}|}{|z-\eta|},\quad\mathcal{H}_{H}(z,\eta)=\frac{1}{2\pi}\ln|z-\eta^{*}|,$
(A.2)

where $\eta^{*}$ is the reflection of $\eta$ across $\partial H$.

Consider now $\Omega\subset\mathbb{R}^{2}$ smoothly bounded. For
$x_{k}\in\Omega$, $d_{k}:=d(x_{k},\partial\Omega)$, $r_{k}>0$, let
$\mathcal{G}_{k}(z,\eta)$ and $\mathcal{H}_{k}(z,\eta)$ denote the Green’s
function of $\Delta$ in $\Omega_{k}:=r_{k}^{-1}(\Omega-x_{k})$ and its regular
part.

######  Lemma A.1.

With the above notation, assume that, up to a subsequence, $r_{k}=o(d_{k})$,
so that $\Omega_{k}$ invades $\mathbb{R}^{2}$ as $k\to\infty$. Then, for every
compact set $K\subset\mathbb{R}^{2}$, there exists $C_{K}$ such that

$\left|\nabla_{z}\left(\mathcal{G}_{k}(z,\eta)-\frac{1}{2\pi}\ln\frac{1}{|z-\eta|}\right)\right|\leq
C_{K}\frac{r_{k}}{d_{k}}$

and

$|\nabla_{z}\mathcal{H}_{k}(z,z)|\leq C_{K}\frac{r_{k}}{d_{k}},$

uniformly for $z\in K$, $\eta\in\Omega_{k}$.

###### Proof.

This follows from Appendix B in [18]. ∎

######  Lemma A.2.

With the above notation, assume that, up to a subsequence and a rotation,
$r_{k}\to 0$, $d_{k}/r_{k}\to L\in(0,\infty)$, and $\Omega_{k}$ converges to
the half-space $H$ as $k\to\infty$. Then for every compact set $K\subset H$
and for $k$ large enough

$\|\nabla(\mathcal{H}_{k}-\mathcal{H}_{H})\|_{L^{\infty}(K\times\Omega_{k,\rho})}\leq
C_{K}r_{k}\max\\{1,\rho^{-1}\\},$

and

$\|\nabla(\mathcal{G}_{k}-\mathcal{G}_{H})\|_{L^{\infty}(K\times\Omega_{k,\rho})}\leq
C_{K}r_{k}\max\\{1,\rho^{-1}\\},$

where $\Omega_{k,\rho}:=\\{z\in\Omega_{k}:d(z,\partial\Omega_{k})>\rho\\}.$

###### Proof.

We have

$\left\\{\begin{array}[]{ll}\Delta_{z}\mathcal{H}_{k}(z,\eta)=0&\text{in
}\Omega_{k}\times\Omega_{k}\\\
\mathcal{H}_{k}(z,\eta)=\frac{1}{2\pi}\ln|z-\eta|&\text{for
}z\in\partial\Omega_{k},\;y\in\Omega_{k}.\end{array}\right.$ (A.3)

We apply the maximum principle to the function

$\mathcal{H}_{k}(z,\cdot)-\mathcal{H}_{H}(z,\cdot):\Omega_{k}\to\mathbb{R},$

where $z\in K$ and $K\subset H$ is a fixed compact set. We want to prove that

$\sup_{z\in
K,\eta\in\partial\Omega_{k}}|\mathcal{H}_{k}(z,\eta)-\mathcal{H}_{H}(z,\eta)|\leq
C(K)r_{k},$ (A.4)

for $k$ sufficiently large. By definition we have

$\mathcal{H}_{k}(z,\eta)=\frac{1}{2\pi}\ln|z-\eta|,\quad\mathcal{H}_{H}(z,\eta)=\frac{1}{2\pi}\ln|z-\eta^{*}|\quad\text{for
}\eta\in\partial\Omega_{k}.$

Let $\delta>0$ be so small that $\partial\Omega\cap B_{\delta}(p(x_{k}))$ can
be written as the graph over $[-\delta,\delta]$ of a function $f$, where
$p(x_{k})$ is the nearest point projection of $x_{k}$ onto $\partial\Omega$.
Let $\mathcal{L}\subset\mathbb{R}^{2}$ be the tangent line to $\partial\Omega$
at $p(x_{k})$ and $f:\mathcal{L}\cap B_{\delta}(x_{k})\to\mathbb{R}$. Up to a
rotation we can assume that $f^{\prime}(p(x_{k}))=0$ and by a Taylor expansion
we obtain

$|f(t)|\leq\frac{\sup_{[-\delta,\delta]}f^{\prime\prime}}{2}t^{2},\quad
t\in[-\delta,\delta].$

In particular for each $x\in\partial\Omega\cap B_{\delta}(x_{k})$ we have
$d(x,L)\leq C_{\Omega,\delta}|x-x_{k}|^{2}$. Scaling to $\Omega_{k}$ we then
obtain

$\eta\in\partial\Omega_{k},\quad|\eta|\leq\frac{\delta}{r_{k}}\quad\Rightarrow\quad
d(\eta,\partial H)=O(r_{k}).$

Then, for such $\eta$ and for $z\in K$, we have
$|z-\eta|\leq|z-\eta^{*}|+O(r_{k})$ and

$|\mathcal{H}_{k}-\mathcal{H}_{H}|=\left|\ln\frac{|z-\eta|}{|z-\eta^{*}|}\right|\leq\ln\left(1+\frac{O(r_{k})}{|z-\eta^{*}|}\right)=C(K)O(r_{k}),$

since $\frac{1}{|z-\eta^{*}|}\geq\varepsilon(K)>0$ uniformly for $z\in K$. On
the other hand

$\eta\in\partial\Omega_{k},\quad|\eta|\geq\frac{\delta}{r_{k}}\quad\Rightarrow\quad|z-\eta^{*}|-|z-\eta|\leq
C(K).$

Then

$|\mathcal{H}_{k}-\mathcal{H}_{H}|=\left|\ln\frac{|z-\eta|}{|z-\eta^{*}|}\right|\leq\ln\left(1+\frac{r_{k}C(K)}{\delta}\right)=C(K)O(r_{k}),$

so that (A.4) is proven. By the maximum principle we obtain

$\sup_{z\in
K,\eta\in\Omega_{k}}|\mathcal{H}_{k}(z,\eta)-\mathcal{H}_{H}(z,\eta)|=C(K)O(r_{k}),$
(A.5)

and the bound on $\nabla(\mathcal{H}_{k}-\mathcal{H}_{H})$ follows by elliptic
estimates, since both $\mathcal{H}_{H}(z,\cdot)$ and
$\mathcal{H}_{k}(z,\cdot)$ are harmonic. The estimate of
$\nabla(\mathcal{G}_{k}-\mathcal{G}_{H})$ follows at once since
$\mathcal{G}_{k}-\mathcal{G}_{H}=\mathcal{H}_{k}-\mathcal{H}_{H}$. ∎

We state Lemmata A.3, A.4 and A.5 below with general positive coefficients
$\alpha_{i,j}$ and $\beta_{i}$ for future reference, and also to remark that
the coefficients $8\pi$ and $4\pi$ appearing in the definitions of
$\Phi_{N,h}$, $\Phi_{N}$, etc…, can be replaced by any other positive
constants.

######  Lemma A.3.

Consider for some $J\geq 2$ the functional
$\Xi_{J,\mathbb{R}^{2}}:(\mathbb{R}^{2})^{J}\setminus D_{J}\to\mathbb{R}$

$\Xi_{J,\mathbb{R}^{2}}(z_{1},\dots,z_{J}):=\sum_{\scriptstyle 1\leq i,j\leq
J\atop\scriptstyle i\neq j}\alpha_{i,j}\ln\frac{1}{|z_{i}-z_{j}|},$

with $\alpha_{i,j}>0$ for $1\leq i,j\leq J$, $i\neq j$. Then
$\Xi_{J,\mathbb{R}^{2}}$ has no critical points.

###### Proof.

Consider $1\leq i\leq J$ such that $z_{i}$ lies on the boundary of the convex
hull of $\\{z_{1},\dots,z_{J}\\}$, and observe that the derivative with
respect to $z_{i}$ in a direction pointing outwards of the convex hull is
negative, as $|z_{i}-z_{j}|$ is increasing for every $1\leq i\neq j\leq J$. ∎

######  Lemma A.4.

Consider for some $J\geq 1$ the functional
$\hat{\Xi}_{J,\hat{H}}:\hat{H}^{J}\setminus D_{J}\to\mathbb{R}$

$\hat{\Xi}_{J,\hat{H}}(z_{1},\dots,z_{J}):=\sum_{\scriptstyle 1\leq i,j\leq
J\atop\scriptstyle i\neq j}\alpha_{i,j}\ln\frac{1}{|z_{i}-z_{j}|}+\sum_{1\leq
i\leq J}\gamma_{i}\ln\frac{1}{d(z_{i},\partial\hat{H})},$

with $\alpha_{i,j}>0$ for $1\leq i<j\leq J$ and $\gamma_{i}>0$ for $1\leq
i\leq J$. Then $\hat{\Xi}_{J,\hat{H}}$ has no critical point.

###### Proof.

Given $(z_{1},\dots,z_{J})$, write $z_{i}=(a_{i},b_{i})$ and up to reordering,
assume that $b_{1}\leq\dots\leq b_{J}$, i.e. $z_{J}$ is one of the points
farthest from $\partial\hat{H}$. Then

$\frac{\partial}{\partial b_{J}}\ln\frac{1}{|z_{i}-z_{J}|}\leq
0,\quad\text{for }1\leq i<J,\quad\frac{\partial}{\partial
b_{J}}\ln\frac{1}{d(z_{J},\partial\hat{H})}<0,$

hence

$\frac{\partial\hat{\Xi}_{J,\hat{H}}(z_{1},\dots,z_{J})}{\partial b_{J}}<0.$

∎

######  Lemma A.5.

Consider for some $J\geq 1$ the functionals $\Xi_{J,H}^{\pm}:H^{J}\setminus
D_{J}\to\mathbb{R}$ given as

$\Xi_{J,H}^{\pm}(z_{1},\dots,z_{J}):=\sum_{\scriptstyle 1\leq i,j\leq
J\atop\scriptstyle i\neq
j}\alpha_{i,j}\mathcal{G}_{H}(z_{i},z_{j})\pm\sum_{1\leq i\leq
J}\beta_{j}\mathcal{H}_{H}(z_{i},z_{i}),$

with $\alpha_{i,j}>0$ for $1\leq i,j\leq J$, $i\neq j$, $\beta_{j}>0$ for
$1\leq i\leq J$. Then $\Xi_{J,H}^{\pm}$ has no critical points.

###### Proof.

Given $(z_{1},\dots,z_{J})$, write in coordinates $z_{i}=(a_{i},b_{i})\in H$
for $1\leq i\leq J$ and, up to reordering, assume first that $J>1$, $a_{1}\leq
a_{2}\leq\dots\leq a_{J}$ and $a_{1}<a_{J}$. Then, it follows from (A.2)

$\frac{\partial\mathcal{G}_{H}(z_{1},z_{J})}{\partial
a_{1}}>0,\quad\frac{\partial\mathcal{G}_{H}(z_{1},z_{j})}{\partial a_{1}}\geq
0,\text{ for }2\leq j\leq
J-1,\quad\frac{\partial\mathcal{H}_{H}(z_{1},z_{1})}{\partial a_{1}}=0,$

hence $\nabla_{z_{1}}\Xi_{J,H}^{\pm}(z_{1},\dots,z_{J})\neq 0$.

If $J=1$ or all the $z_{i}^{\prime}s$ are vertically, i.e.
$a_{1}=\dots=a_{J}$, then, up to reordering, assume that $b_{1}<\dots<b_{J}$.
Then, again from (A.2), we get

$\frac{\partial\mathcal{G}_{H}(z_{1},z_{j})}{\partial b_{1}}<0,\quad\text{for
}2\leq j\leq J,\quad\frac{\partial\mathcal{H}_{H}(z_{1},z_{1})}{\partial
b_{1}}<0,$ (A.6)

hence $\nabla_{z_{1}}\Xi_{J,H}^{+}(z_{1},\dots,z_{J})\neq 0$, and

$\frac{\partial\mathcal{G}_{H}(z_{J},z_{j})}{\partial b_{J}}>0,\quad\text{for
}2\leq j\leq J,\quad\frac{\partial\mathcal{H}_{H}(z_{J},z_{J})}{\partial
b_{J}}<0,$

hence $\nabla_{z_{N}}\Xi_{J,H}^{-}(z_{1},\dots,z_{J})\neq 0$. ∎

######  Lemma A.6.

Consider for some $J\geq 1$ the functional
$\Xi_{J,H,\hat{H}}^{\varphi}:(H\cap\hat{H})^{J}\setminus D_{J}\to\mathbb{R}$
given as

$\Xi_{J,H,\hat{H}}^{\varphi}(z_{1},\dots,z_{J}):=\Xi^{+}_{J,H}(z_{1},\dots,z_{J})+\sum_{1\leq
i\leq J}\gamma_{i}\varphi(z_{i}),$

where $H$ and $\hat{H}$ are as in (A.1) for some $L,\hat{L}>0$,
$\Xi^{+}_{J,H}$ is as in Lemma A.5, $\gamma_{i}\geq 0$, for $1\leq i\leq J$,
$\varphi\in C^{1}(H\cap\hat{H})$ and $\nabla\varphi(z)\cdot(0,1)\leq 0$ for
every $z\in H\cap\hat{H}$. Then $\Xi_{J,H,\hat{H}}^{\varphi}$ has no critical
points.

###### Proof.

The proof is identical to the proof of Lemma A.5, using (A.6) together with

$\frac{\partial\varphi(z_{1})}{\partial
b_{1}}=\nabla\varphi(z_{1})\cdot(0,1)\leq 0.$

∎

##  Appendix B The Euler characteristic of configuration spaces

We will need the following general fact about configuration spaces.

######  Proposition B.1.

Let $\Omega$ be a smoothly bounded domain of $\mathbb{R}^{2}$ and call for any
$N\in\mathbb{N}^{\star}$

$F(\Omega,N):=\Omega^{N}\setminus
D_{N}=\left\\{(x_{1},\dots,x_{N})\in\Omega^{N}\;:\;x_{i}\neq x_{j}\hbox{ for
}i\neq j\right\\}.$

Then

$\chi(F(\Omega,N))=\chi(\Omega)(\chi(\Omega)-1)...(\chi(\Omega)-N+1).$

###### Proof.

This result follows from the fact that $F(\Omega,N)$ fibers on $F(\Omega,N-1)$
with fiber $\Omega_{N-1}$ (projecting on the last of the $N$ points), where we
denote by $\Omega_{j}$ the set $\Omega$ with $j$ points removed, see [35,
Chapter 2].

Claim The above fibration is orientable, in the sense that
$\pi_{1}(F(\Omega,N-1))$ acts trivially on the homology
$H_{*}(\Omega_{N-1},\mathbb{R})$ of the fiber.

###### Proof of the claim..

Since $\Omega_{N-1}$ is homotopically equivalent to a (connected) finite union
of circles, the only relevant homology group of $\Omega_{N-1}$ is
$H_{1}(\Omega_{N-1},\mathbb{R})$. Consider now a closed path $\gamma\in
C^{0}([0,1],F(\Omega,N-1))$ representing a homotopy class
$[\gamma]\in\pi_{1}(F(\Omega,N-1))$. Set

$(x_{1}(t),\dots,x_{N-1}(t)):=\gamma(t).$

For the point $\gamma(0)=:(x_{1},\dots,x_{N-1})\in F(\Omega,N-1)$ consider the
fiber

$\Omega\setminus\\{x_{1},\dots,x_{N-1}\\}\simeq\Omega_{N-1}$

and choose generators of its first homology group as follows. Since $\Omega$
is a two-dimensional domain, it is homotopic to a disk with $r$ points
removed, for $r=0,1,2,\dots$. Given the $i$-th hole of $\Omega$,
$i=1,\dots,r$, choose an element $\sigma_{i}$ of $H_{1}(\Omega,\mathbb{R})$ in
the form of a smooth Jordan curve that does not contain any of the points
$x_{1}(t),\dots,x_{N-1}(t)$ with $t\in[0,1]$. Then it is clear that the action
of $\gamma$ on $\sigma_{1},\dots,\sigma_{r}$ is trivial.

Now, for each $x_{j}$, $j=1,\dots,k-1$, choose a small and smooth Jordan curve
$\eta_{j}$ around $x_{j}$ not intersecting $\sigma_{1},\dots,\sigma_{r}$, for
instance the oriented boundary of $B_{\varepsilon}(x_{j})$. By compactness, if
we choose $\varepsilon$ small enough we get

$\overline{B_{\varepsilon}(x_{j}(t))}\cap\overline{B_{\varepsilon}(x_{i}(t))}=\emptyset,\quad\text{for
}1\leq i<j\leq N-1,\;t\in[0,1]$

and

$\overline{B_{\varepsilon}(x_{j}(t))}\cap\sigma_{i}=\emptyset,\quad\text{for
}1\leq j\leq N-1,\;1\leq i\leq r,\;t\in[0,1].$

Again it follows immediately that the action of $[\gamma]$ of $\eta_{j}$ is
trivial. Since

$H_{1}(\Omega_{N-1},\mathbb{R})=\text{span}\\{\sigma_{i},\eta_{j},1\leq i\leq
r,1\leq j\leq N-1\\}$

the claim is proven. ∎

By the product formula for orientable fibrations, see Spanier [35, p.481], we
infer

$\chi(F(\Omega,N))=\chi(F(\Omega,N-1))\chi(\Omega_{N-1})=\chi(F(\Omega,N-1))(\chi(\Omega)-N+1).$

Now the proposition follows at once by induction in $N$. ∎

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