# Morse inequalities at infinity for a resonant mean field equation

Mohameden Ahmedou $\&$ Mohamed Ben Ayed

_In fond memory of Abbas Bahri_

Abstract

In this paper we study the following mean field type equation

$(MF)\qquad-\Delta_{g}u\,=\varrho(\frac{Ke^{u}}{\int_{\Sigma}Ke^{u}dV_{g}}\,-\,1)\,\mbox{
in }\Sigma,$

where $(\Sigma,g)$ is a closed oriented surface of unit volume
$Vol_{g}(\Sigma)$ = 1, $K$ positive smooth function and $\varrho=8\pi m$,
$m\in\mathbb{N}$. Building on the critical points at infinity approach
initiated in [1] we develop, under generic condition on the function $K$ and
the metric $g$, a full Morse theory by proving Morse inequalities relating the
Morse indices of the critical points, the indices of the critical points at
infinity, and the Betti numbers of the space of formal barycenters
$B_{m}(\Sigma)$.
We derive from these _Morse inequalities at infinity_ various new existence as
well as multiplicity results of the mean field equation in the resonant case,
i.e. $\varrho\in 8\pi\mathbb{N}$.

Key Words: Critical points at infinity, Morse theory, Space of formal
barycenters.

AMS subject classification: 35C60, 58J60, 35J91.

##  1 Introduction and statement of the results

Let $(\Sigma,g)$ be a closed and oriented surface of unit volume
$Vol_{g}(\Sigma)$ = 1. For $\varrho\in\mathbb{R}^{+}$ real number and $K$
positive smooth function, we consider the following Mean Field equation

(1)
$(MF)\qquad-\Delta_{g}u\,=\varrho(\frac{Ke^{u}}{\int_{\Sigma}Ke^{u}dV_{g}}\,-\,1)\,\mbox{
in }\Sigma.$

Problem $(MF)$ arises as limiting equation in some mean field approximation in
the study of limit point vortices of Euler flows, in Onsager vortex theory,
and also in the description of self dual condensates in some Chern-Simons-
Higgs models. See for example the papers [2, 8, 9, 12, 13, 14, 21, 22, 33,
29], and the monographs of Tarantello [30] and of Yang [32] as well as the
references therein. Furthermore its study is also motivated by the prescribed
Gauss curvature problem in differential geometry. Indeed, if $\Sigma$ is the
standard $2$-sphere $\mathbb{S}^{2}$ endowed with its standard metric
$g_{stand}$ and $\varrho=8\pi$, then Problem $(MF)$ is the so called
_Nirenberg’s problem_ in conformal geometry, which consists of finding a
metric $g$ conformally equivalent to $g_{stand}$ and whose Gauss curvature is
given by the function $K$. See [10, 19, 11] and the references therein.
Equation $(MF)$ has a variational structure and the associated Euler Lagrange
functional $J_{\varrho}$ is defined by

$J_{\varrho}(u)\,:=\,\frac{1}{2}||u||^{2}\,-\varrho\ln(\int_{\Sigma}Ke^{u}dV_{g})\quad\mbox{
for }u\in\mathring{H^{1}}(\Sigma)$

where

$||u||:=||\nabla u||_{L^{2}}$

is the norm used to equip

(2) $\mathring{H}^{1}(\Sigma):=\Big{\\{}u\in
H^{1}(\Sigma):\int_{\Sigma}udV_{g}\,=0\Big{\\}}.$

It turns out that the analytical aspect of this variational problem depends on
the range of values taken by the parameter $\varrho$. Indeed

  * •

If $\varrho\notin 8\pi\mathbb{N}$ then the associated variational problem is
compact in the sense that changes of the topology of the sublevel sets of
$J_{\varrho}$ are induced only by critical points. We call it the _non-
resonant case._

  * •

If $\varrho\in 8\pi\mathbb{N}$ then the associated variational problem is not
compact, i.e. changes in the difference of topology between its sublevel sets
might be induced by critical points at infinity, these are non compact orbits
of the gradient flow whose level sets are bounded (see [3] for a related
notion for Yamabe type problem). We call it the _resonant case._

In the non-resonant case, Yanyan Li [22] proved that the Leray-Schauder degree
of the solutions of $(MF)$ is well defined and is a topological invariant. He
also advised a method to compute it by analyzing the _degree jump_ across the
critical values $8m\pi$. C.C. Chen and C.S. Lin [12, 13], proved a priori
estimates for the solutions, and used the strategy advised by Yanyan Li to
compute the Leray-Schauder degree. Then they derived that Problem $(MF)$ is
solvable provided that the surface has a positive genus. Later Z. Djadli [15],
using the topological argument of [16], proved the existence of a solution of
$(MF)$ in this case for surfaces of every genus. Furthermore A. Malchiodi [26]
developed a Morse theory in this case and F. De Marchis [17] proved full Morse
inequalities as well as multiplicity results for generic data $(K,g)$.
For the _resonant case_ the main difficulty lies in the fact that to find
critical points of $J_{\varrho}$ one has to understand the contributions of
the critical points at infinity to the topology of its sublevel sets. This
amounts to proving a Morse lemma at infinity in a highly degenerate setting
where gradient flow lines may converge only in the sense of measures to a sum
of Dirac masses. Based on ideas going back to Lions [24] and Struwe [28] one
can often perform a ”Lyapunov-Schmidt reduction at infinity”, and finds that
the Dirac deltas are located at critical points of a finite-dimensional
reduced function. Such an approach has been developed by the authors in
collaboration with M. Lucia in [1]. There, a Morse reduction in the
neighborhood of the critical points has been performed and the induced
difference of topology between the levels of $J_{\varrho}$ has been computed.
In this paper the authors want to develop further this approach by proving
_Morse inequalities_ between _critical points at Infinity_ and use them to
deduce new existence and multiplicity results.
To state our main results we introduce the following notation and assumptions:
For $m\in\mathbb{N}$, we let

$\mathbb{F}_{m}(\Sigma):=\left\\{(a_{1},\cdots,a_{m})\in\Sigma^{m};a_{i}\neq
a_{j}\mbox{ if }i\neq j\right\\}$

denote the space of configurations of $\Sigma^{m}$ and we define the following
reduced energy functional

$\mathcal{F}^{K}_{m}:\,\mathbb{F}_{m}(\Sigma)\,\,\longrightarrow\,\mathbb{R},\quad\mbox{
by }$ (3) $\displaystyle\mathcal{F}^{K}_{1}(a)\,:=\,\ln K(a)\,+4\pi
H(a,a)\quad(\mbox{for }m=1;\,a\in\Sigma),$ (4)
$\displaystyle\mathcal{F}^{K}_{m}(A)\,:=\,\sum_{i=1}^{m}\bigg{(}\ln
K(a_{i})\,+\,4\pi H(a_{i},a_{i})\,+4\pi\sum_{j\neq
i}G(a_{i},a_{j})\bigg{)}\,(\mbox{for }m\geq 2;\,A=(a_{1},\cdot,a_{m})),$

where $G$ is the Green’s function of $-\Delta_{g}$ and $H$ its regular part.
See (12) and (14) for the precise definition. We notice that the function
$\mathcal{F}^{K}_{m}$ achieves its infimum for each $m\geq 1$ and for $m=1$ it
also achieves its maximum.

Next for $A=(a_{1},\cdots,a_{m})\in\mathbb{F}_{m}$ a critical point of
$\mathcal{F}^{K}_{m}$ we define a vector
$(\mathcal{F}^{A}_{1},\cdots,\mathcal{F}^{A}_{m})$, where

$\mathcal{F}^{A}_{i}:\,\Sigma\longrightarrow\mathbb{R},$

are real valued functions defined as follows:

(5) $\displaystyle\mathcal{F}_{1}^{a}(x)\,:=\,K(x)\exp(\,8\pi
H(a,x)),\quad\mbox{ if }m=1,$ (6)
$\displaystyle\mathcal{F}^{A}_{i}(x)\,:=\,K(x)\exp\Big{(}\,8\pi
H(a_{i},x)\,+\,8\pi\sum_{j\neq i}G(x,a_{j})\Big{)}\quad\mbox{ if }m\geq
2\,\mbox{ for }\,i=1,\cdots,m.$

Moreover we associate to every critical point $A=(a_{1},\cdots,a_{m})$ of
$\mathcal{F}^{K}_{m}$ the following quantity

(7)
$\mathcal{L}(A)\,:=\,\sum_{i=1}^{m}\left(\Delta_{g}\mathcal{F}^{A}_{i}(a_{i})-2K_{g}(a_{i})\,\mathcal{F}^{A}_{i}(a_{i})\right),$

where $K_{g}$ denotes the Gauss curvature of $(\Sigma,g)$. Next we define the
following set

(8)
$\mathcal{K}^{-}_{m}\,:=\,\left\\{A\in\mathbb{F}_{m}(\Sigma):\,\nabla\mathcal{F}^{K}_{m}(A)\,=\,0,\,\mathcal{L}(A)<0\right\\}.$

Our first type of results are based on the construction of solutions of sub-
and sup-approximation with fixed morse indices, which we show that, under
appropriate assumptions of critical points of the related reduced energy, they
give rise to solutions of Equation $(MF)$.

###### Theorem 1.1

Let $m\geq 1$ and $\varrho=8\pi m$ and assume that the reduced energy
$\mathcal{F}^{K}_{m}$ has only non degenerate critical points. Then

  1. 1)

if at every local minimum $A\in\mathbb{F}_{m}(\Sigma)$ of
$\mathcal{F}^{K}_{m}$ we have that $\mathcal{L}(A)<0.$ Then Equation $(MF)$
has at least one solution whose generalized Morse index is $3m$,

  2. 2)

for $m\geq 2$, if at every critical point $A\in\mathbb{F}_{m}(\Sigma)$ of
$\mathcal{F}^{K}_{m}$ with morse index $2$ we have that $\mathcal{L}(A)>0$.
Then Equation $(MF)$ has at least one solution whose generalized Morse index
is $3m-3$,

where the _generalized Morse index_ of a solution $\omega$ of $(MF)$ is the
sum of its Morse index and the dimension of the Kernel of the linearized
operator

$\mathcal{T}_{\omega}(\varphi):=-\Delta_{g}\varphi\,-\,\varrho\frac{Ke^{\omega}\varphi}{\int_{\Sigma}Ke^{\omega}}.$

We observe that, for $m=1$, the functional $J_{8\pi}$ is bounded below and
therefore if there exists a local maximum point $A$ of $\mathcal{F}^{K}_{1}$
such that $\mathcal{L}(A)>0$ then $J_{8\pi}$ achieves its minimum (see [1],
Theorem 1.1). We point out that there is no such points for the Nirenberg’s
problem on $\mathbb{S}^{2}$ and more generally there are no such points if the
surface has a positive Gauss curvature.

Our next and main result of this paper is to prove _Morse inequalities_. To
that aim we introduce the following non degeneracy assumption:

_We say that the pair $(g,K)$ satisfies the condition $(\mathcal{N}_{m})$ if
the following conditions are satisfied:_

  1. (i)

_The critical points of_ $\mathcal{F}^{K}_{m}$ are non degenerate and for
every critical point $A$, we have $\mathcal{L}(A)\neq 0,$

  2. (ii)

_All the critical points of_ $J_{8\pi m}$ are non degenerate.

###### Remark 1.1

The non degeneracy condition $(\mathcal{N}_{m})$ is generic. Indeed, denoting
by $\mathcal{M}$ the set of riemannian metrics on $\Sigma$ and
$C^{0,1}_{+}(\Sigma)$ the space of positive Lipschitz functions on $\Sigma$.
It follows from transversality arguments, there is an open and dense subset
$\mathcal{D}\in\mathcal{M}\times C^{0,1}_{+}$ such that for
$(g,h)\in\mathcal{D}$, the functional $J_{\varrho}$ is a Morse function. See
for example [17], pp. 2179-80, for an argument based on some generic
properties for nonlinear boundary value problems proved by Saut and Temam
[27]. Moreover the condition $(i)$ is also a generic one.

Furthermore, under the non degeneracy condition $(i)$ of $(\mathcal{N}_{m})$,
we associate to every $A\in\mathcal{K}^{-}_{m}$ an index

(9) $\iota_{\infty}:\mathcal{K}^{-}_{m}\to\mathbb{N}\quad\mbox{ defined by
}\quad\iota_{\infty}(A)\,:=\,3m-1-Morse(\mathcal{F}^{K}_{m},A),$

where $Morse(\mathcal{F}^{K}_{m},A)$ stands for the Morse index of
$\mathcal{F}^{K}_{m}$ at its critical point $A$.

Next for $m\in\mathbb{N}$, let

$B_{m}(\Sigma):=\left\\{\sum_{i=1}^{m}\gamma_{i}\delta_{a_{i}},\,a_{i}\in\Sigma,\,\gamma_{i}\geq
0;\,\sum_{i=1}^{m}\gamma_{i}\,=\,m\right\\}$

denote the set of formal barycenters of order $m$. For $i\geq 0$, we set

$\beta^{m}_{i}:=\,rank\,\,H_{i}(B_{m}(\Sigma),\mathbb{Z}_{2}).$

Then $B_{m}(\Sigma)$ is a stratified set of top dimension $3m-1$ and therefore
$\beta_{i}^{m}=0$ for $i\geq 3m$.

In the next result we provide Morse inequalities relating the Morse indices of
the solutions of $(MF)$, the $\iota_{\infty}$-indices of the critical points
at infinity and the Betti numbers $\beta_{i}^{m}$ of the set of formal
barycenters $B_{m}(\Sigma)$. Before stating these inequalities, we recall that
it follows from the blow up analysis of solutions of $(MF)$, see [23, 12, 13]
that their Dirichlet energy is uniformly bounded. See please statement $(ii)$
in Theorem 3.1, p. 1686 in [13]. Hence it follows from Moser-Trudinger
inequality and elliptic theory that solutions of $(MF)$ with zero mean value
integral are uniformly bounded and hence their Morse indices are uniformly
bounded.
We are now ready to state our Morse inequalities in the case $\varrho=8\pi
m;m\geq 2$:

###### Theorem 1.2

Let $\varrho=8\pi m$, $m\geq 2$ and assume that the function $K$ satisfies the
non degeneracy condition $(\mathcal{N}_{m})$ and let $\overline{m}$ be the
highest Morse index of the critical points of $J_{\varrho}$. Then the
following Morse inequalities hold

  1. (a)

For every $2\leq k\leq\overline{N}:=\max(3m-1,\overline{m})$ there holds

(10) $\nu_{k}\,+\nu^{\infty}_{k}\,\geq\,\beta_{k-1}^{m-1},$

where $\nu_{i}$ denotes the number of critical points of $J_{\varrho}$ with
Morse index $i$ and

(11) $\nu^{\infty}_{q}:=\\#\\{A\in\mathcal{K}^{-}_{m};\iota_{\infty}(A)=q\\}.$

  2. (b)

Let $\chi(\Sigma)$ be the Euler Characteristic of $\Sigma$, it holds

$\sum_{i=0}^{\overline{m}}(-1)^{i}\nu_{i}\,+\sum_{A\in\mathcal{K}^{-}_{m}}(-1)^{\iota_{\infty}(A)}\,=\,\binom{m-1-\chi(\Sigma)}{m-1}.$

In particular the number of critical points of $J_{\varrho}$ is lower bounded
by

$\left|\binom{m-1-\chi(\Sigma)}{m-1}-\sum_{A\in\mathcal{K}^{-}_{m}}(-1)^{\iota_{\infty}(A)}\right|.$

As corollary of the above _Morse inequalities at infinity_ we derive the
following existence result:

###### Theorem 1.3

Let $m\geq 2$ and $\varrho=8\pi m$ and assume that the function $K$ satisfies
the non degeneracy condition $(\mathcal{N}_{m})$. Suppose there exists $2\leq
q_{0}\leq 3m-3$ such that

  * •

there is no element in $\mathcal{K}^{-}_{m}$ of index $q_{0}$

  * •

$\beta_{q_{0}-1}^{m-1}\neq 0$.

Then Equation $(MF)$ has at least one solution whose Morse index is $q_{0}$.
In particular, since $\beta_{3m-4}^{m-1}=H_{3m-4}(B_{m-1}(\Sigma))\neq 0$, if
there is no element in $\mathcal{K}^{-}_{m}$ of index $3m-3$, then Equation
$(MF)$ has at least one solution whose Morse index is $3m-3$. Furthermore, the
number of solutions is lower bounded by $\beta_{3m-4}^{m-1}$.

As a complement of the statement made in the above theorem, we prove the
following result:

###### Theorem 1.4

Let $m\geq 2$ and $\varrho=8\pi m$ and assume that the function $K$ satisfies
the non degeneracy condition $(\mathcal{N}_{m})$. Suppose there exists $q_{0}$
such that

  1. 1.

there exists $Q_{0}\in\mathcal{K}^{-}_{m}$ with $\iota_{\infty}(Q_{0})=q_{0}$,

  2. 2.

for each $Q\in\mathcal{K}^{-}_{m}$, we have
$\iota_{\infty}(Q)\notin\\{q_{0}-1,q_{0}+1\\}$,

  3. 3.

$\displaystyle{\beta_{q_{0}-1}^{m-1}\,=\,0}$.

Then Equation $(MF)$ has at least one solution.

###### Remark 1.2

We notice that the indices of the critical points at infinity are lower
bounded by $m-1$. Hence for $m=4$ suppose

  1. 1.

there exists $Q_{0}\in\mathcal{K}^{-}_{m}$ such that $\iota_{\infty}(Q_{0})=3$
(i.e. $\mathcal{F}_{m}^{K}(Q_{0})$ is a local maximum of
$\mathcal{F}_{m}^{K}$),

  2. 2.

for all $Q\in\mathcal{K}^{-}_{m}$, we have $\iota_{\infty}(Q)\neq 4$,

then it follows from Theorem 1.4 that Equation $(MF)$ has at least one
solution.

More generally any violation of the above _Morse inequalities at infinity_ of
Theorem 1.2 induces the existence of a solution to Equation $(MF)$. In
particular we have the following existence result:

###### Theorem 1.5

Let $m\geq 2$ and $\varrho=8\pi m$ and assume that the function $K$ satisfies
the non degeneracy condition $(\mathcal{N}_{m})$ . If one of the following
inequalities is not satisfied

  * •

$\nu^{\infty}_{3m-2}\geq\nu^{\infty}_{3m-1}$.

  * •

$\nu^{\infty}_{3m-3}-\beta_{3m-4}^{m-1}\geq\nu^{\infty}_{3m-2}-\nu^{\infty}_{3m-1}$.

Then Equation $(MF)$ has at least one solution (where $\nu_{q}^{\infty}$ is
defined in (11)).

The remainder of this paper is organized as follows: we set up some notation
and definitions in Section 2, introduce an $\varepsilon-$neighborhood of
potential critical points at infinity in Section 3 and provide in Section 4
useful expansions of the gradient of the Euler-Lagrange functional in _the
neighborhood at infinity_. Section 5 is devoted to the characterization of
such critical points at infinity and we provide the proof of our main results
in Section 6. Finally we provide useful estimates of the projected bubble (see
(17) for the definition) in the appendix.

##  2 Notation and definitions

To state our results we need to fix some notation.
For $\xi\in\Sigma$, let $y_{\xi}$ be a local chart defined in a neighborhood
of $\xi$ onto $B(0,2\eta_{0})$ such that $y_{\xi}(\xi)=0$. In the sequel, we
will denote $B_{\xi}(\eta):=y_{\xi}^{-1}(B(0,\eta))$. Furthermore, thanks to
the isothermal coordinates, we have that for every $a\in\Sigma$, there exists
a function $u_{a}\in C^{\infty}(\Sigma)$, satisfying $u_{a}(a)=0$ and $\nabla
u_{a}(a)=0$, such that for the conformal metric $g_{a}:=e^{u_{a}}g$, there
hold

$g_{a}\,=\,\delta_{ij}\mbox{ in
}B_{a}(2\eta_{0}),\,\,\Delta_{g}=e^{u_{a}}\Delta_{g_{a}},\,\,dV_{g}=e^{-u_{a}}dV_{g_{a}},\,\mbox{
and }\,\Delta_{g_{a}}u_{a}=2K_{g}(.)e^{-u_{a}}\mbox{ in }B_{a}(2\eta_{0}),$

where $0<\eta_{0}<\eta_{a}<\frac{1}{4}\,inj_{g_{a}}(\Sigma)$ where $inj$
stands for the injectivity radius.
Next for $a\in\Sigma$, let $G(a,.)$ be the Green’s function of $\Delta_{g}$
defined by

(12) $\begin{cases}-\Delta_{g}\,G(a,x)\,+\,1&=\,\delta_{a}\,\mbox{in
}\Sigma,\\\ \int_{\Sigma}G(a,x)dVg(x)&=\,0.\end{cases}$

For $0<\eta<\frac{1}{4}\eta_{0}$ and $a\in\Sigma$ we define the smooth cut-off
function

(13) $\psi_{a}(x):=\psi(|y_{a}(x)|)\quad\mbox{ where
}\quad\psi(t):=\begin{cases}t,&\mbox{if }t\in[0,{\eta}],\\\ 2\eta,&\mbox{if
}t\geq{2\eta},\\\ \psi(t)\in[\eta,2\eta],\psi\in
C^{\infty}&\mbox{otherwise}.\end{cases}$

Next for $\xi\in\Sigma$, as usual we decompose $G(\xi,.)$ as follows

(14) $G(\xi,x)=-\frac{1}{2\pi}\ln(\psi_{\xi}(x))+H(\xi,x).$

From the definition of $G(\xi,.)$ we derive that $H(\xi,.)$ has to satisfy

(15)
$\Delta_{g}H(\xi,.)\,-1\,=\Delta_{g}G(\xi,.)-1+\frac{1}{2\pi}\Delta_{g}\ln(\psi_{\xi}(.))=\begin{cases}0\mbox{
in }(\Sigma\setminus B_{\xi}(2\eta))\cup B_{\xi}(\eta),\\\
\frac{e^{u_{\xi}}}{2\pi}\Delta_{g_{\xi}}\ln(\psi_{\xi}(.))\mbox{ in
}B_{\xi}(2\eta)\setminus B_{\xi}(\eta).\end{cases}$

In the next section, we will describe the lack of compactness occurring in the
variational problem associated to Equation $(MF)$. To do so we need to
introduce some highly concentrated functions, the so called _bubbles_ and a
neighborhood of such bubbles.
First, we recall that the following functions

$\widetilde{\delta}_{a,\lambda}(x):=\ln(\frac{8\lambda^{2}}{(1+\lambda^{2}|a-x|^{2})^{2}}),\mbox{
for }a\in\mathbb{R}^{2},\mbox{ and }\lambda>0,$

are the solutions of

$-\Delta u=e^{u}\quad\mbox{in }\mathbb{R}^{2}\quad\mbox{ with
}\quad\int_{\mathbb{R}^{2}}e^{u}<\infty.$

Next for $a\in\Sigma$ and $\lambda>0$ we define the standard bubble as

(16)
$\delta_{a,\lambda}(x):=\ln(\frac{8\lambda^{2}}{(1+\lambda^{2}\psi_{a}(x)^{2})^{2}}),$

where $\psi_{a}$ is defined in (13). In order to construct a suitable
_approximated solution_ of $(MF)$ we follow the strategy introduced by A.
Bahri and J.M. Coron in their study of the Yamabe equation [4] and we
introduce the _projected bubble_ $\varphi_{a,\lambda}$ to be the unique
solution of

(17)
$\begin{cases}-\Delta_{g}\varphi_{a,\lambda}\,=\,e^{\delta_{a,\lambda}+u_{a}}\,-\,\int_{\Sigma}e^{\delta_{a,\lambda}+u_{a}}dV_{g}\,\mbox{
in }\Sigma,\\\ \int_{\Sigma}\varphi_{a,\lambda}\,dV_{g}\,=\,0.\end{cases}$

We observe that for $A:=(a_{1},\cdots,a_{m})\in\Sigma^{m}$ and
$\Lambda:=(\lambda_{1},\cdots,\lambda_{m})\in(\mathbb{R}^{+})^{m}$ the
sequence of functions $(U_{A,\Lambda})_{\Lambda}$ defined by

(18) $U_{A,\Lambda}:=\sum_{i=1}^{m}\varphi_{a_{i},\lambda_{i}}$

is a Palais-Smale sequence for Equation $(MF)$ for $\varrho=8\pi m$ if $A$ and
$\Lambda$ satisfy the following balancing condition

(19) $\forall i\neq
j\quad\lambda_{i}^{2}\mathcal{F}^{A}_{i}(a_{i})\,=\,\lambda_{j}^{2}\mathcal{F}^{A}_{j}(a_{j})(1\,+\,o_{\lambda}(1)),$

where $o_{\lambda}(1)\to 0\mbox{ if }\lambda\to+\infty$. See please Lemma 7.5
for the equation satisfied by $U_{A,\Lambda}$. Furthermore we collected in the
appendix some useful estimates on $\varphi_{a,\lambda}$. Such estimates are
used in the expansion of the Euler Lagrange functional and its gradient in the
neighborhood at infinity $V(m,\varepsilon)$. See please (20) for the
definition of this set.

##  3  Lack of compactness and a neighborhood at infinity

Our approach is variational. Therefore, in order to detect critical points for
the functional $J_{\varrho}$ with $\varrho=8\pi m$, we have to find all
possible obstructions in deforming sublevel sets
$J_{\varrho}^{a}:=\\{u:J_{\varrho}(u)\leq a\\}$. To this aim we make use of a
pseudogradient for $J_{\varrho}$, constructed in Horak-Lucia [20] (see also
[25, 26]), whose flow lines have the following property:
For any fixed initial data $u_{0}$, the flow line $\eta(t,u_{0})$ generated by
this pseudogradient satisfies the following property:

  * •

either $J_{\varrho}(\eta(t,u_{0}))\to-\infty$ as $t\to+\infty$,

  * •

or $\eta(t,u_{0})$ enters any arbitrary small neighborhood of the set of
solutions $u_{\beta}$ of $(\mathcal{P}_{8m\pi-\beta})$ with
$\beta\in[0,\overline{\varepsilon})$ for some $\overline{\varepsilon}>0$.

Furthermore, for a solution $u_{\beta}$ with $\beta>0$ we have the following
alternative: $(i)$ either $u_{\beta}$ converges to a solution $\overline{u}$
of $(\mathcal{P}_{8m\pi})$ as $\beta\to 0$ $(ii)$ or it has to blow up when
$\beta\to 0$. In the latter case, it follows from Proposition 7.6 (see also
[12, 13]), that the solutions $(u_{\beta})$ have to belong to some subset
${V}(m,\varepsilon)$, called in the sequel neighborhood at Infinity, which is
defined as:

(20) $\displaystyle{V}(m,\varepsilon)$
$\displaystyle:=\Big{\\{}u\in\mathring{H}^{1}(\Sigma)\,:\,\|\nabla
J_{\varrho}(u)\|<\varepsilon\,;\,\,\exists\,\lambda_{1},\cdots,\lambda_{m}>{\varepsilon^{-1}}\mbox{
with }{\lambda_{i}}<C_{1}{\lambda_{j}}\,\,\forall i\neq j;$
$\displaystyle\quad\exists\,a_{1},\cdots,a_{m}\mbox{ with
}d_{g}(a_{i},a_{j})\geq 2\eta\,\,\forall i\neq j\quad\mbox{such that
}\|u-\sum_{i=1}^{m}\varphi_{a_{i},\lambda_{i}}\|<\varepsilon\Big{\\}},$

where the space $\mathring{H}^{1}(\Sigma)$ is defined in (2), $\varepsilon$ is
a small positive constant and $C_{1}$, $\eta$ are fixed positive constants.
Hence, we are led to study the obstructions to decrease the functional
$J_{\varrho}$ in the set $V(m,\varepsilon)$. A first step consists in finding
an appropriate parametrization of this set. To that aim, following the ideas
of A. Bahri and J.M. Coron we consider the following minimization problem

(21)
$\min_{\alpha_{i}>0;a_{i}\in\Sigma;\lambda_{i}>0}\big{\|}u-\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\big{\|}\,.$

We have the following Lemma whose proof is identical to the proof of
Proposition 7 in [5] (see also Chen and Lin [13, Lemma 3.2]). Namely we have

###### Lemma 3.1

For $\varepsilon$ small, Problem (21) has for any $u\in{V}(m,\varepsilon)$
only one solution (up to permutations on the indices). The variables
$\alpha_{i}$’s satisfy $|\alpha_{i}-1|=O(\varepsilon)$.

Hence every $u\in{V}(m,\varepsilon)$ can be written as

(22) $u\,=\,\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\,+\,w,$

where $\alpha_{i}$, $w$ satisfy

(23) $\begin{cases}&|\alpha_{i}-1|\leq c\varepsilon\quad\forall
i,\qquad\|w\|\leq c\varepsilon,\\\
&\big{<}w,\varphi_{a_{i},\lambda_{i}}\big{>}_{g}\,=\,\big{<}w,{\partial\varphi_{a_{i},\lambda_{i}}}/{\partial\lambda_{i}}\big{>}_{g}\,=\,0,\,\quad\,\big{<}w,{\partial\varphi_{a_{i},\lambda_{i}}}/{\partial
a_{i}}\big{>}_{g}\,=0\quad\forall\,\,i.\end{cases}$

In the following, for ${A}=(a_{1},...,a_{m})$ and
$\Lambda=(\lambda_{1},...,\lambda_{m})$, we denote

(24) $E_{{A},\Lambda}^{m}:=\\{w\in\mathring{H}^{1}(\Sigma):w\mbox{ satisfies
\eqref{f:orthog}}\\}.$

To keep the notation short we will write $\varphi_{i}$ instead of
$\varphi_{a_{i},\lambda_{i}}$. The next Proposition shows how to deal with the
infinite dimensional part $w$ in the above representation:

###### Proposition 3.2

[1] Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}\in V(m,\varepsilon)$. Then
there exists a unique $\overline{w}:=\overline{w}(u)\in E_{{A},\Lambda}^{m}$
such that:

$J_{\varrho}(u+\overline{w})\,=\,\min\\{J_{\varrho}(u+w)\,:\,w\in
E_{{A},\Lambda}^{m}\\}.$

Furthermore, there exists a constant $C$ such that

(25)
$\|\overline{w}\|\,\leq\,C\sum_{i=1}^{m}\Big{(}|\alpha_{i}-1|+\frac{1}{\lambda_{i}}\Big{)}.$

##  4  The expansion of the gradient in the neighborhood at infinity

In this section we provide an asymptotic expansion of the gradient of the
Euler Lagrange functional $J_{\varrho}$ in the neighborhood at infinity
$V(m,\varepsilon)$. For the sake of simplicity of the notation and since the
variables $\lambda_{i}$’s are of the same order, we will write in this section
and in the sequel $O(1/\lambda^{\gamma})$ instead of $\sum
O(1/\lambda_{k}^{\gamma})$.
In the first proposition we expand the gradient with respect to the
concentration rates. Namely we have

###### Proposition 4.1

Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}+w\in V(m,\varepsilon)$ with $w\in
E^{m}_{{A},\Lambda}$ and $\varrho:=8m\pi(1+\mu)$ with $\mu$ a small real
number. It holds

$\Big{\langle}\nabla
J_{\varrho}(u),\lambda_{i}\frac{\partial\varphi_{i}}{\partial\lambda_{i}}\Big{\rangle}_{g}=16\pi\alpha_{i}(\tau_{i}-\mu+\mu\tau_{i})-64\pi^{2}\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}+O\left(\sum|\alpha_{k}-1|^{2}+\|w\|^{2}\right)+o\left(\frac{\ln\lambda}{\lambda^{2}}\right)$

with

(26)
$\tau_{i}\,:=\,1\,-\,\frac{m\pi}{2\alpha_{i}-1}\,\frac{\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}^{A}_{i}(a_{i})g^{A}_{i}(a_{i})}{\int_{\Sigma}Ke^{u}dV_{g}},$

where $\mathcal{F}^{A}_{i}$ and $g^{A}_{i}$ are defined in (49). Furthermore,
we have the estimate

(27) $|\tau_{i}|=O(\varepsilon+|\mu|),\quad\forall i\in\\{1,\cdots,m\\}.$

Proof. Recall that

(28) $\Big{\langle}\nabla
J_{\varrho}(u),h\Big{\rangle}_{g}=\Big{\langle}u,h\Big{\rangle}_{g}-\frac{\varrho}{\int_{\Sigma}Ke^{u}}\int_{\Sigma}Ke^{u}h.$

Using Lemma 7.2 we get that

$\Big{\langle}u,\lambda_{i}\frac{\partial\varphi_{i}}{\partial\lambda_{i}}\Big{\rangle}_{g}=-64\pi^{2}\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\sum_{j=1}^{m}\alpha_{j}+16\pi\alpha_{i}+O(\frac{1}{\lambda^{2}})=-64\pi^{2}m\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}+16\pi\alpha_{i}+o(\frac{\ln\lambda}{\lambda^{2}}).$

For the other term of (28), using Lemma 7.1, we have

$\displaystyle\int_{\Sigma}Ke^{u}\lambda_{i}\frac{\partial\varphi_{i}}{\partial\lambda_{i}}dV_{g}$
$\displaystyle=\int_{\Sigma}Ke^{u}\left(\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}-8\pi\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}+O(\frac{1}{\lambda^{2}})\right)dV_{g}$
$\displaystyle=-8\pi\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\int_{\Sigma}Ke^{u}dV_{g}+\int_{B_{i}}Ke^{u}\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}dV_{g}+O(\frac{1}{\lambda^{2}})\int_{\Sigma}Ke^{u}dV_{g}.$

Now we need to estimate the second integral. Letting $\overline{u}:=u-w$ and
using Lemma A.4 in [1], we have

$\displaystyle\int_{B_{i}}Ke^{u}\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}$
$\displaystyle=\int_{B_{i}}Ke^{\overline{u}}\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}+\int_{B_{i}}Ke^{\overline{u}}w\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}+\int_{B_{i}}Ke^{\overline{u}}(e^{w}-1-w)\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}$
$\displaystyle=\int_{B_{i}}Ke^{\overline{u}}\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}dV_{g}+O\bigg{(}(\sum\lambda_{k}^{4\alpha_{k}-2})\|w\|\Big{(}\|w\|+\frac{1}{\lambda}+|\alpha_{i}-1|\Big{)}\bigg{)}.$

Concerning the last integral, using Lemma 7.3 we get

$\displaystyle\int_{B_{i}}Ke^{\overline{u}}\frac{4}{1+\lambda_{i}^{2}\psi_{i}^{2}}dV_{g}$
$\displaystyle=\Big{(}1+4\pi\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}\int_{B_{i}}\frac{4\lambda_{i}^{4\alpha_{i}}\mathcal{F}_{i}^{A}g_{i}^{A}e^{-u_{a_{i}}}}{(1+\lambda_{i}^{2}|y_{a_{i}}(.)|^{2})^{2\alpha_{i}+1}}dV_{g_{a_{i}}}+o\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}\int_{B_{i}}Ke^{\overline{u}}$
$\displaystyle=\Big{(}1+4\pi\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}\frac{2\pi}{\alpha_{i}}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}g_{i}^{A}(a_{i})+o\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}\sum\lambda_{k}^{4\alpha_{k}-2},$

by using

$\int_{0}^{\lambda\eta}\frac{4\,r}{(1+r^{2})^{2\alpha+1}}dr=\frac{1}{\alpha}+O\Big{(}\frac{1}{\lambda^{4\alpha}}\Big{)}\quad\mbox{
and
}\quad\int_{0}^{\lambda\eta}\frac{r^{3}}{(1+r^{2})^{2\alpha+1}}dr=O\big{(}1\big{)}.$

Thus we get

$\displaystyle\frac{\varrho}{\int_{\Sigma}Ke^{u}}\int_{\Sigma}Ke^{u}\lambda_{i}\frac{\partial\varphi_{i}}{\partial\lambda_{i}}dV_{g}=$
$\displaystyle\Big{(}1+4\pi\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}16\pi(1+\mu)\frac{2\alpha_{i}-1}{\alpha_{i}}(1-\tau_{i})$
$\displaystyle+o\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}+O\Big{(}\|w\|^{2}+|\alpha_{i}-1|^{2}\Big{)}.$

The result follows by summing the previous estimates.

In the next proposition we expand the gradient $\nabla J_{\varrho}$ with
respect to the gluing parameters $\alpha_{i}$’s.

###### Proposition 4.2

Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}+w\in V(m,\varepsilon)$ with $w\in
E^{m}_{{A},\Lambda}$ and $\varrho:=8m\pi(1+\mu)$ with $\mu$ a small real
number. It holds

$\displaystyle\Big{\langle}\nabla
J_{\varrho}(u),\varphi_{i}\Big{\rangle}_{g}=$ $\displaystyle
32\pi\ln\lambda_{i}\big{(}(\alpha_{i}-1)+(\tau_{i}-\mu+\mu\tau_{i})\big{)}+O\left(|\alpha_{i}-1|+\|w\|^{2}\ln\lambda\right)$
$\displaystyle+O\left(\|w\|(1+|\alpha_{i}-1|\ln\lambda)+\sum|\tau_{k}-\mu+\mu\tau_{k}|+\frac{\ln^{2}\lambda}{\lambda^{2}}+\frac{\ln\lambda_{i}}{\lambda_{i}^{4\alpha_{i}-2}}\right).$

Proof. Using Lemma 7.2 we get that

$\displaystyle\Big{\langle}u,\varphi_{i}\Big{\rangle}_{g}$
$\displaystyle=\alpha_{i}\Big{(}32\pi\ln\lambda_{i}+64\pi^{2}H(a_{i},a_{i})-16\pi\Big{)}+64\pi^{2}\sum_{j\neq
i}\alpha_{j}G(a_{i},a_{j})+O\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)},$
$\displaystyle\int_{\Sigma}Ke^{u}\varphi_{i}dV_{g}$
$\displaystyle=\int_{\Sigma}Ke^{\overline{u}}\varphi_{i}dV_{g}+\int_{\Sigma}Ke^{\overline{u}}w\varphi_{i}dV_{g}+\int_{\Sigma}Ke^{\overline{u}}(e^{w}-1-w)\varphi_{i}dV_{g}$
$\displaystyle=\int_{\Sigma}Ke^{\overline{u}}\varphi_{i}dV_{g}+O\Big{(}\|w\|\Big{(}1+|\alpha_{i}-1|\ln\lambda_{i}+\|w\|\ln\lambda_{i}\Big{)}\sum\lambda_{k}^{4\alpha_{k}-2}\Big{)}.$

For the last integral we notice that it follows from Lemma 7.1 that
$\varphi_{i}$ and $e^{\overline{u}}$ are bounded outside the balls $B_{k}$.
Moreover inside each ball $B_{k}$, for $k\neq i$, using Lemmas 7.1 and 7.3, we
derive

$\displaystyle\int_{B_{k}}Ke^{\overline{u}}\varphi_{i}dV_{g}$
$\displaystyle=8\pi\int_{B_{k}}\frac{\lambda_{k}^{4\alpha_{k}}\mathcal{F}_{k}^{A}g_{k}^{A}e^{-u_{a_{k}}}}{(1+\lambda_{k}^{2}|y_{a_{k}}(.)|^{2})^{2\alpha_{k}}}G(a_{i},.)dV_{g_{a_{k}}}+O\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}\int
Ke^{\overline{u}}$
$\displaystyle=8\pi\lambda_{k}^{4\alpha_{k}-2}\mathcal{F}_{k}^{A}g_{k}^{A}(a_{k})G(a_{i},a_{k})\frac{\pi}{2\alpha_{k}-1}+O\Big{(}\Big{(}\frac{\ln\lambda}{\lambda^{2}}+\sum\frac{|\alpha_{j}-1|}{\lambda^{3/2}}\Big{)}\sum\lambda_{j}^{4\alpha_{j}-2}\Big{)}$

by using the fact that (using (55))

$\displaystyle\int_{0}^{\lambda\eta}\frac{r^{3}dr}{(1+r^{2})^{2\alpha}}=\int_{0}^{\lambda\eta}\frac{r^{3}\xi(r)dr}{(1+r^{2})^{2}}$
$\displaystyle=\int_{0}^{\lambda\eta}\frac{r^{3}dr}{(1+r^{2})^{2}}+O\Big{(}|\alpha-1|\int_{0}^{\lambda\eta}\frac{r^{3}\sqrt{r}dr}{(1+r^{2})^{2}}\Big{)}$
$\displaystyle=O\Big{(}\ln\lambda+|\alpha-1|\sqrt{\lambda}\Big{)}.$

In the ball $B_{i}$, it holds

$\displaystyle\int_{B_{i}}K$ $\displaystyle e^{\overline{u}}\varphi_{i}dV_{g}$
$\displaystyle=\int_{B_{i}}\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}_{i}^{A}g_{i}^{A}e^{-u_{a_{i}}}}{(1+\lambda_{i}^{2}|y_{a_{i}}|^{2})^{2\alpha_{i}}}\Big{(}4\ln\lambda_{i}-2\ln(1+\lambda_{i}^{2}|y_{a_{i}}|^{2})+8\pi
H(a_{i},.)\Big{)}dV_{g_{a_{i}}}+O\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}\int
Ke^{\overline{u}}$ $\displaystyle=\Big{(}4\ln\lambda_{i}+8\pi
H(a_{i},a_{i})\Big{)}\Big{(}\frac{\pi}{2\alpha_{i}-1}+O\Big{(}\frac{1}{\lambda_{i}^{4\alpha_{i}-2}}+\frac{\ln\lambda}{\lambda^{2}}+\frac{|\alpha_{i}-1|}{\lambda^{3/2}}\Big{)}\Big{)}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}g_{i}^{A}(a_{i})$
$\displaystyle-2\int_{B_{i}}\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}_{i}^{A}g_{i}^{A}e^{-u_{a_{i}}}}{(1+\lambda_{i}^{2}|y_{a_{i}}|^{2})^{2\alpha_{i}}}\ln(1+\lambda_{i}^{2}|y_{a_{i}}|^{2})dV_{g_{a_{i}}}+O\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}\int
Ke^{\overline{u}}.$

Observe that

$\displaystyle\int_{0}^{\lambda\eta}\frac{2r}{(1+r^{2})^{2\alpha}}\ln(1+r^{2})dr=\frac{1}{(2\alpha-1)^{2}}+O\Big{(}\frac{\ln\lambda}{\lambda^{4\alpha-2}}\Big{)},$
$\displaystyle\int_{0}^{\lambda\eta}\frac{r^{3}}{(1+r^{2})^{2\alpha}}\ln(1+r^{2})dr\leq
c\,\ln\lambda\int_{0}^{\lambda\eta}\frac{r^{3}}{(1+r^{2})^{2\alpha}}dr\leq
c\,\ln\lambda\Big{(}\ln\lambda+|\alpha-1|\sqrt{\lambda}\Big{)}.$

Thus we obtain

$\displaystyle\int_{B_{i}}Ke^{\overline{u}}\varphi_{i}dV_{g}=$
$\displaystyle\Big{(}4\ln\lambda_{i}+8\pi
H(a_{i},a_{i})\Big{)}\frac{\pi}{2\alpha_{i}-1}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}g_{i}^{A}(a_{i})-\frac{2\pi}{(2\alpha_{i}-1)^{2}}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}g_{i}^{A}(a_{i})$
$\displaystyle+O\Big{(}\ln\lambda+\Big{(}\frac{\ln^{2}\lambda}{\lambda^{2}}+|\alpha_{i}-1|\frac{\ln\lambda}{\lambda^{3/2}}\Big{)}\sum\lambda_{j}^{4\alpha_{j}-2}\Big{)}.$

As in the proof of Proposition 4.1, summing the previous estimates, we derive
the result.

Combining Propositions 4.1 and 4.2, we obtain

###### Corollary 4.3

Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}+w\in V(m,\varepsilon)$ with $w\in
E^{m}_{{A},\Lambda}$ and $\varrho:=8m\pi(1+\mu)$ with $\mu$ a small real
number. It holds

$\displaystyle\Big{\langle}$ $\displaystyle\nabla
J_{\varrho}(u),\frac{\varphi_{i}}{\ln\lambda_{i}}-\frac{2}{\alpha_{i}}\lambda_{i}\frac{\partial\varphi_{i}}{\partial\lambda_{i}}\Big{\rangle}_{g}$
$\displaystyle=32\pi(\alpha_{i}-1)+O\left(\|w\|^{2}+\frac{\|w\|}{\ln\lambda}+\frac{1}{\ln\lambda}\sum(|\tau_{k}-\mu+\mu\tau_{k}|+|\alpha_{k}-1|)+\frac{\ln\lambda}{\lambda^{2}}+\frac{1}{\lambda^{4\alpha_{i}-2}}\right).$

###### Lemma 4.4

Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}+w\in V(m,\varepsilon)$ with $w\in
E^{m}_{{A},\Lambda}$ and $\varrho:=8m\pi(1+\mu)$ with $\mu$ a small real
number. Let $\tau_{i}$ be defined in (26) and assume that
$\sum|\alpha_{i}-1|\ln\lambda_{i}$ is small. Then, it holds

$\sum_{i=1}^{m}\tau_{i}=\frac{\pi}{2}\frac{m\ln\lambda_{1}}{\int_{\Sigma}Ke^{u}dV_{g}}\mathcal{L}(A)+4\pi
m\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}+o\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}+\sum
O\Big{(}|\alpha_{k}-1|^{2}\Big{)}.$

Proof. In this case, since we assumed that $\sum|\alpha_{i}-1|\ln\lambda_{i}$
is small, we derive that
$\lambda_{i}^{4(\alpha_{i}-1)}=1+O(|\alpha_{i}-1|\ln\lambda_{i})$ for each
$i$. Moreover we have

(29)
$\int_{\Sigma}Ke^{\overline{u}}wdV_{g}+\int_{\Sigma}Ke^{\overline{u}}(e^{w}-1-w)dV_{g}=O\Big{(}\Big{(}\|w\|^{2}+\sum|\alpha_{k}-1|^{2}+\frac{1}{\lambda^{2}}\Big{)}\sum\lambda_{k}^{4\alpha_{k}-2}\Big{)}.$

Hence the result follows from the previous estimate and (51).

Lastly arguing as above, we derive the following asymptotic expansion of the
gradient with respect to the concentration points $(a_{1},\cdots,a_{m})$.
Namely we prove that

###### Proposition 4.5

Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}+w\in V(m,\varepsilon)$ with $w\in
E^{m}_{{A},\Lambda}$ and $\varrho:=8m\pi(1+\mu)$ with $\mu$ a small real
number. It holds

$\displaystyle\Big{\langle}\nabla J_{\varrho}(u),$
$\displaystyle\frac{1}{\lambda_{i}}\frac{\partial\varphi_{i}}{\partial
a_{i}}\Big{\rangle}_{g}\,=\,-8\pi(1+\mu)\frac{\nabla\mathcal{F}^{A}_{i}(a_{i})}{\lambda_{i}}+O\left(\frac{|\mu|}{\lambda}+\frac{|\tau_{i}|}{\lambda}+\frac{\ln\lambda}{\lambda^{2}}+\sum_{k=1}^{m}|\alpha_{k}-1|^{2}+\|w\|^{2}\right).$

##  5  Critical points at infinity and their indices

Critical points at infinity of the functional $J_{\varrho}$ are accumulation
points of some orbits of the negative gradient flow $-\nabla J_{\varrho}$
which enter some $V(m,\varepsilon)$ and remain there indefinitely. In this
section we recall for the sake of completeness the full characterization of
these critical points proved in [1] and restate their contribution to the
difference of topology between the level sets of the functional $J_{\varrho}$.
We start with an expansion of $J_{\varrho}$ in the neighborhood at infinity
$V(m,\varepsilon)$.

###### Proposition 5.1

Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}+w\in V(m,\varepsilon)$ with $w\in
E^{m}_{{A},\Lambda}$. Assume that $|\alpha_{i}-1|\ln\lambda_{i}$ is small for
each $i$ and (25) holds. Then

$\displaystyle J_{8m\pi}$ $\displaystyle(u)=-8\pi
m(1+\ln(m\pi))-8\pi\mathcal{F}_{m}^{K}(a_{1},...,a_{m})-4\pi\sum_{i=1}^{m}|\tau^{\prime}_{i}|^{2}+16\pi\sum_{i=1}^{m}(\alpha_{i}-1)^{2}\ln\lambda_{i}$
$\displaystyle-4\pi\frac{\ln\lambda_{1}}{\lambda_{1}^{2}\mathcal{F}^{A}_{1}(a_{1})}\sum_{i=1}^{m}\Big{(}\Delta\mathcal{F}^{A}_{i}(a_{i})-2K_{g}(a_{i})\mathcal{F}^{A}_{i}(a_{i})\Big{)}+\sum_{k=1}^{m}\Big{\\{}O\Big{(}(\alpha_{k}-1)^{2}+\frac{1}{\lambda_{k}^{2}}\Big{)}+o(|{\tau}^{\prime}_{k}|^{2})\Big{\\}},$

where $\mathcal{F}^{A}_{i}$ and $g^{A}_{i}$ are defined in (49) and

${\tau}^{\prime}_{i}=1-\frac{{m}\,\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}^{A}_{i}(a_{i})}{\sum_{k=1}^{m}\lambda_{k}^{4\alpha_{k}-2}\mathcal{F}^{A}_{k}(a_{k})}.$

Proof. The proof follows from Lemmas 7.2 and 7.4 and the following
computations. Let us denote by

$\Gamma:={\sum_{i=1}^{m}\frac{\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}^{A}_{i}(a_{i})g_{i}^{A}(a_{i})}{2\alpha_{i}-1}}\qquad\mbox{
and
}\qquad\tilde{\tau}_{i}=1-\frac{\frac{m}{2\alpha_{i}-1}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}^{A}_{i}(a_{i})g_{i}^{A}(a_{i})}{\sum\frac{1}{2\alpha_{k}-1}\lambda_{k}^{4\alpha_{k}-2}\mathcal{F}^{A}_{k}(a_{k})g_{k}^{A}(a_{k})}.$

Since $w\in E^{m}_{{A},\Lambda}$, then (29) holds and using Lemma 7.4, we have

$\displaystyle\ln\Big{(}\int_{\Sigma}Ke^{u}\Big{)}$
$\displaystyle=\ln(\pi\Gamma)+\ln\Big{\\{}1+\frac{1}{2\Gamma}\sum_{i=1}^{m}\Big{(}\Delta\mathcal{F}^{A}_{i}(a_{i})-2K_{g}(a_{i})\mathcal{F}^{A}_{i}(a_{i})\Big{)}\ln\lambda_{i}$
$\displaystyle+\frac{4\pi}{\Gamma}\Big{(}\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}\sum_{i=1}^{m}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}(a_{i})+O\big{(}\sum_{k=1}^{m}\big{\\{}|\alpha_{k}-1|^{2}+\frac{1}{\lambda^{2}_{k}}\big{\\}}\big{)}\Big{\\}}$
$\displaystyle=\ln\pi-\frac{1}{m}\sum_{i=1}^{m}\ln(1-\tilde{\tau}_{i})+\frac{1}{2\Gamma}\sum_{i=1}^{m}\Big{(}\Delta\mathcal{F}^{A}_{i}(a_{i})-2K_{g}(a_{i})\mathcal{F}^{A}_{i}(a_{i})\Big{)}\ln\lambda_{i}$
$\displaystyle+\frac{1}{m}\sum_{i=1}^{m}\ln\Big{(}\frac{m\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}^{A}_{i}(a_{i})g_{i}^{A}(a_{i})}{2\alpha_{i}-1}\Big{)}+\frac{4\pi}{\Gamma}\Big{(}\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}\sum_{i=1}^{m}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}(a_{i})$
$\displaystyle+O\Big{(}\sum_{k=1}^{m}\big{\\{}|\alpha_{k}-1|^{2}+\frac{1}{\lambda^{2}_{k}}\big{\\}}\Big{)}$
$\displaystyle=\ln\pi+\frac{1}{m}\sum_{i=1}^{m}\Big{\\{}\tilde{\tau}_{i}+\frac{\tilde{\tau}_{i}^{2}}{2}\Big{\\}}+\frac{1}{2\Gamma}\sum_{i=1}^{m}\Big{(}\Delta\mathcal{F}^{A}_{i}(a_{i})-2K_{g}(a_{i})\mathcal{F}^{A}_{i}(a_{i})\Big{)}\ln\lambda_{i}$
$\displaystyle+\ln
m+\frac{1}{m}\sum_{i=1}^{m}\\{-\ln(2\alpha_{i}-1)+(4\alpha_{i}-2)\ln\lambda_{i}+\ln(\mathcal{F}^{A}_{i}(a_{i}))+\ln(g_{i}^{A}(a_{i}))\\}$
$\displaystyle+\frac{4\pi}{\Gamma}\Big{(}\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}\sum_{i=1}^{m}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}(a_{i})+O\Big{(}\sum_{k=1}^{m}\Big{\\{}|\alpha_{k}-1|^{2}+|\tilde{\tau}_{k}|^{3}+\frac{1}{\lambda_{k}^{2}}\Big{\\}}\Big{)}.$

We remark that $\sum_{i=1}^{m}\tilde{\tau}_{i}=0$ and for each $i$, we have
$\tilde{\tau}_{i}=O(\varepsilon)$ which implies that
$\lambda_{i}^{2}\mathcal{F}^{A}_{i}(a_{i})=\lambda_{j}^{2}\mathcal{F}^{A}_{j}(a_{j})(1+O(\sum|\alpha_{k}-1|\ln\lambda_{k}))$
for each $i,j$ and $\ln\lambda_{i}=\ln\lambda_{1}+O(1)$. Furthermore, we have

$\displaystyle\ln\big{(}2\alpha_{i}-1\big{)}=\ln\big{(}1+2(\alpha_{i}-1)\big{)}=2(\alpha_{i}-1)+O(|\alpha_{i}-1|^{2}),$
$\displaystyle
g_{i}^{A}(a_{i})=1+O\Big{(}\sum_{k=1}^{m}|\alpha_{k}-1|\Big{)}\quad;\quad\Gamma=\sum_{i=1}^{m}\lambda_{i}^{2}\mathcal{F}^{A}_{i}(a_{i})+O\Big{(}\sum_{k=1}^{m}|\alpha_{k}-1|\lambda_{k}^{2}\ln\lambda_{k}\Big{)}$
$\displaystyle\mbox{and}\quad|\tilde{\tau}_{i}|^{2}=|\tau^{\prime}_{i}|^{2}+o(|\tau^{\prime}_{i}|^{2})+\sum
O(|\alpha_{k}-1|^{2}).$

Hence, the result follows.

In [1] we constructed the following decreasing pseudogradient of the Euler
Lagrange functional $J_{\varrho}$ in the neighborhood at Infinity
$V(m,\varepsilon)$:

###### Proposition 5.2

[1] Let $\varrho=8\pi m$ with $m\geq 1$ and assume that the function $K$
satisfies the condition $(i)$ of $(\mathcal{N}_{m})$. Then there exists a
pseudogradient $W$ defined in $V(m,\varepsilon)$ and satisfying the following
properties: There exists a constant $C$ independent of
$u=\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\bar{w}$ such that

  1. (1)

$\displaystyle{\langle-\nabla
J_{\varrho}(u),W\rangle\,\geq\,C\sum_{i=1}^{m}\Big{(}|\alpha_{i}-1|\,+\,|\tau_{i}|\,+\,\frac{|\nabla\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}\,+\,\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\Big{)}}$,

  2. (2)

$\displaystyle{\langle-\nabla
J_{\varrho}(u),W+\frac{\partial\overline{w}(W)}{\partial(\alpha,\lambda,a)}\rangle\,\geq\,C\sum_{i=1}^{m}\Big{(}|\alpha_{i}-1|\,+\,|\tau_{i}|\,+\,\frac{|\nabla\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}\,+\,\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\Big{)}}$,

  3. (3)

$|W|$ is bounded and the only region where the variables $\lambda_{i}$’s
increase along the flow lines of $W$ is the region where
$(a_{1},\cdots,a_{m})$ is very close to a critical point
$q:=(q_{1},\cdots,q_{m})$ of $\mathcal{F}^{K}_{m}$ with
$q\in\mathcal{K}^{-}_{m}$.

Following the program developed in [1], we deduce from Proposition 5.2 the
following characterization of the critical points at infinity:

###### Proposition 5.3

[1] Let $\varrho=8\pi m$, $m\geq 1$. The critical points at Infinity of
$J_{\varrho}$ are in one to one correspondence with critical points
$Q:=(q_{1},\cdots,q_{m})$ of $\mathcal{F}_{m}^{K}$ satisfying
$Q\in\mathcal{K}^{-}_{m},$ that will be denoted by $(Q)_{\infty}$. Furthermore
the energy level of such a critical point at Infinity $(Q)_{\infty}$ denoted
$C_{\infty}(Q)_{\infty}$ is given by:

$C_{\infty}(Q)_{\infty}\,=\,-8\pi
m(1+\ln(m\pi))\,-8\pi\mathcal{F}^{K}_{m}(Q).$

Moreover the Morse index of such a critical point at Infinity $(Q)_{\infty}$
is given by:

$\iota_{\infty}(Q):=\,3m-1-Morse(\mathcal{F}^{K}_{m},Q).$

We point out that around _a critical point at infinity_ there is a Morse type
reduction. Indeed denoting by

$\displaystyle
V(m,Q,\varepsilon)\,:=\\{u=\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\,+w\,\in
V(m,\varepsilon):\,\forall i=1,\cdots,m,\,|a_{i}-q_{i}|<\varepsilon;\,$ (30)
$\displaystyle|\alpha_{i}-1|\ln\lambda_{i}<\varepsilon,\mbox{ and }\forall
i\neq
j\quad|\frac{\lambda_{i}^{2}\mathcal{F}^{A}_{i}(a_{i})}{\lambda_{j}^{2}\mathcal{F}^{A}_{j}(a_{j})}\,-1|<\varepsilon\\},$

where $Q:=(q_{1},\cdots,q_{m})$ is a critical point of $\mathcal{F}^{K}_{m}$
with $\mathcal{L}(Q)<0$, we have the following Morse Lemma at Infinity,whose
proof is inspired by the proof of a similar statement for Yamabe type flows
written down by A. Bahri in [6] (pages 415-417). We state our result as
follows:

###### Lemma 5.4

Let $u=\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\,+w\,\in
V(m,Q,\varepsilon)$, then there exists a change of variables

$(a_{1},\cdots,a_{m},\alpha_{1},\cdots,\alpha_{m},\lambda_{1},\cdots,\lambda_{m},w)\mapsto(\overline{a}_{1},\cdots,\overline{a}_{m},\beta_{1},\cdots,\beta_{m},\Lambda_{1},y_{2},\cdots,y_{m},V)$

(where $(\overline{a}_{1},\cdots,\overline{a}_{m})$ is close to $Q$, the
$\beta_{i}$’s and the $y_{i}$’s are small and $\Lambda_{1}$ is very large)
such that

$J_{\varrho}(u)=-8\pi
m(1+\ln(m\pi))-8\pi\mathcal{F}_{m}^{K}(\overline{a}_{1},...,\overline{a}_{m})+\sum_{i=1}^{m}\beta_{i}^{2}-\sum_{i=2}^{m}y_{i}^{2}+(4\pi-\sigma)\frac{(-\mathcal{L}(Q))}{\mathcal{F}^{Q}_{1}(q_{1})}\frac{\ln{\Lambda}_{1}}{\Lambda_{1}^{2}}+\|V\|^{2}$

where $\sigma$ is a small positive constant.

Proof. We recall that for $\sum\alpha_{i}\varphi_{i}\in V(m,\varepsilon)$, we
have by Proposition 3.2 that there exists a unique $\overline{w}$ which
minimizes $J_{\varrho}(\sum\alpha_{i}\varphi_{i}+w)$ in the space
$E_{A,\Lambda}^{m}$. Hence, by the classical Morse Lemma, there exists a
change of variable $w-\overline{w}\to V$ so that

(31) $J_{\varrho}(u)=J_{\varrho}(\overline{u})+\|V\|^{2}\quad\mbox{ where
}{u}:=\sum\alpha_{i}\varphi_{i}+{w}\quad\mbox{ and
}\quad\overline{u}:=\sum\alpha_{i}\varphi_{i}+\overline{w}.$

Furthermore for $\varepsilon^{\prime}>0$ small (with
$\varepsilon/\varepsilon^{\prime}$ small) and $W$ the pseudogradient
constructed in Proposition 5.2, we have that for
$\overline{u}:=\sum\alpha_{i}\varphi_{i}+\overline{w}\in
V(m,\varepsilon^{\prime})$ there holds:

(32) $\langle\nabla
J_{\varrho}(\overline{u}),W\rangle\leq-c\sum\Big{(}|\alpha_{i}-1|+|\tau_{i}|+\frac{|\nabla\mathcal{F}_{i}^{A}(a_{i})|}{\lambda_{i}}+\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\Big{)}.$

Next let $\sigma>0$ be a small constant and set

(33) $\displaystyle I(\overline{u}):=$ $\displaystyle-8\pi
m(1+\ln(m\pi))-8\pi\mathcal{F}_{m}^{K}(a_{1},...,a_{m})-(4\pi-\sigma)\sum_{i=1}^{m}|{\tau}^{\prime}_{i}|^{2}$
$\displaystyle+(16\pi+\sigma)\sum_{i=1}^{m}(\alpha_{i}-1)^{2}\ln\lambda_{i}^{2\alpha_{i}-1}-(4\pi-\sigma)\frac{\ln\lambda_{1}^{2\alpha_{1}-1}}{\lambda_{1}^{4\alpha_{1}-2}\mathcal{F}^{Q}_{1}(q_{1})}\mathcal{L}(Q).$

where $Q:=(q_{1},\cdots,q_{m})$. Since we assumed that
$|\alpha_{i}-1|\ln\lambda_{i}$ is small for each $i$, it is easy to see that

(34) $0<I(\overline{u})-J_{\varrho}(\overline{u})\leq
2\sigma\sum\Big{(}(\alpha_{i}-1)^{2}\ln\lambda_{i}+\frac{\ln\lambda_{1}}{\lambda_{1}^{2}}+|{\tau}^{\prime}_{i}|^{2}\Big{)}\quad\mbox{
for each }\overline{u}.$

Furthermore it follows from Proposition 5.2 that

(35) $\langle\nabla
I(\overline{u}),W\rangle\leq-c\sum\Big{(}|\alpha_{i}-1|+|\tau_{i}|+\frac{|\nabla\mathcal{F}_{i}^{A}(a_{i})|}{\lambda_{i}}+\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\Big{)}.$

Next for $\overline{u}_{0}\in V(m,\varepsilon)$ we consider the following
differential equation

(36) $\frac{\partial u}{\partial s}=W(u)\quad;\quad u(0)=\overline{u}_{0},$

whose solution is $h_{s}(\overline{u}_{0})$ where $h_{s}$ is the 1-parameter
group generated by $W$.

Note that, for
$\overline{u}:=\sum\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\overline{w}$ as far
as $h_{s}(\overline{u})\in V(m,Q,\varepsilon)$ we have that

$h_{s}(\overline{u})=\sum\alpha_{i}(s)\varphi_{a_{i}(s),\lambda_{i}(s)}+\overline{w(s)},$

that is $\overline{w(s)}$ satisfies conclusions of Proposition 3.2.
Next we claim:
CLAIM 1: There exists $\overline{s}>0$ such that
$I(h_{\overline{s}}(\overline{u}_{0}))=J_{\varrho}(\overline{u}_{0})$.
Observe that $I(h_{s}(\overline{u}_{0}))$ is a decreasing function with
respect to $s$. Hence there exists at most a unique solution to the equation
$I(h_{s}(\overline{u}_{0}))=J_{\varrho}(\overline{u}_{0})$. The only cases
where there could be no solution are

  * •

either $h_{s}(\overline{u}_{0})$ exits $V(m,\varepsilon^{\prime})$ (outside
this set, we loose (32) since $W$ is defined only in
$V(m,\varepsilon^{\prime})$) before reaching this level.

  * •

or $h_{s}(\overline{u}_{0})$ will build a critical point at infinity before
reaching the level $J_{\varrho}(\overline{u}_{0})$.

We will prove that these two cases cannot occur. In fact, for the first one,
since $\overline{u}_{0}\in V(m,\varepsilon)$ then, to exit
$V(m,\varepsilon^{\prime})$, the flow line has to travel from
$V(m,\varepsilon^{\prime}/2)$ to the boundary of $V(m,\varepsilon^{\prime})$.
Note that, using (35), we have that: $\partial
I(h_{{s}}(\overline{u}_{0}))/\partial s\leq-c(\varepsilon^{\prime})$ along
this path, independent of $\varepsilon$, but depending on
$\varepsilon^{\prime}$. Also, by (35), the time to travel from
$V(m,\varepsilon^{\prime}/2)$ to the boundary of $V(m,\varepsilon^{\prime})$
is lower-bounded by a constant $c^{\prime}(\varepsilon^{\prime})$, because
$|W|$ is bounded and the distance to travel is lower-bounded by a constant
$c>0$. Therefore, $I(h_{{s}}(\overline{u}_{0}))$ decreases at least by
$c(\varepsilon^{\prime})c^{\prime}(\varepsilon^{\prime})$ during this trip.
However, using (34), since $\overline{u}_{0}\in V(m,\varepsilon)$, it follows
that $J_{\varrho}(\overline{u}_{0})<I(\overline{u}_{0})\leq
J_{\varrho}(\overline{u}_{0})+c(\varepsilon)$. Hence we have to choose
$\varepsilon$ small with respect to $\varepsilon^{\prime}$ so that
$I(h_{{s}}(\overline{u}_{0}))$ reaches the level
$J_{\varrho}(\overline{u}_{0})$ before leaving the set
$V(m,\varepsilon^{\prime})$.
Concerning the second case, the flow line $h_{{s}}(\overline{u}_{0})$ will
enter $V(m,\varepsilon_{1})$ for each $\varepsilon_{1}>0$. Observe that

$J_{\varrho}(h_{s}(\overline{u}_{0}))<J_{\varrho}(\overline{u}_{0})\quad;\quad
0<I(h_{{s}}(\overline{u}_{0}))-J_{\varrho}(h_{{s}}(\overline{u}_{0}))\to
0\quad\mbox{ as }s\to\infty\,\,(\mbox{for }\varepsilon_{1}\to 0).$

Hence $I(h_{{s}}(\overline{u}_{0}))$ has to reach the level
$J_{\varrho}(\overline{u}_{0})$ and therefore this case cannot occur.
Our claim is thereby proved.
Conversely, taking $\varepsilon^{\prime\prime}>0$ small with respect to
$\varepsilon$ and given $\overline{u}^{\prime}_{0}\in
V(m,\varepsilon^{\prime\prime})$, arguing by the same way (and using $-W$ as
an increasing pseudogradient) : there exists $\overline{s}^{\prime}>0$ such
that
$J_{\varrho}(h_{-\overline{s}^{\prime}}(\overline{u}^{\prime}_{0}))=I(\overline{u}^{\prime}_{0})$.
Hence, $h_{s}$ is the required isomorphism.
It follows from (33) and the previous claim that

$\displaystyle
J_{\varrho}(\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\,+\,\overline{w}):=$
$\displaystyle-8\pi
m(1+\ln(m\pi))-8\pi\mathcal{F}_{m}^{K}(\overline{a}_{1},...,\overline{a}_{m})-(4\pi-\sigma)\sum_{i=1}^{m}|{\overline{\tau}}^{\prime}_{i}|^{2}$
$\displaystyle+(16\pi+\sigma)\sum_{i=1}^{m}(\overline{\alpha}_{i}-1)^{2}\ln(\overline{\lambda}_{i})^{2\overline{\alpha}_{i}-1}-(4\pi-\sigma)\frac{\ln(\overline{\lambda}_{1})^{2\overline{\alpha}_{1}-1}}{(\overline{\lambda}_{1})^{4\overline{\alpha}_{1}-2}\mathcal{F}^{Q}_{1}(q_{1})}\mathcal{L}(Q),$

where $\overline{\tau}^{\prime}_{i}$ has the same definition as
$\tau^{\prime}_{i}$ (see Proposition 5.1) using the new variables.
In order to achieve the split of variables as claimed in the Lemma we need to
perform some changes of variables. We first consider the following change of
variables

$\displaystyle\psi_{1}:Y:=$
$\displaystyle(\overline{a}_{1},\cdots,\overline{a}_{m},\overline{\alpha}_{1},\cdots,\overline{\alpha}_{m},\overline{\lambda}_{1},\cdots,\overline{\lambda}_{m})\mapsto(\overline{a}_{1},\cdots,\overline{a}_{m},\overline{\alpha}_{1},\cdots,\overline{\alpha}_{m},\Lambda_{1},\cdots,\Lambda_{m})$
$\displaystyle\mbox{ with
}\Lambda_{i}:=\overline{\lambda}_{i}^{2\alpha_{i}-1}$

By computing the determinant $\det(D\psi_{1}(Y))$, it is easy to show that
$\psi_{1}$ is a diffeomorphism. With this change of variables, the functional
reads as follows:

$\displaystyle
J_{\varrho}(\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\,+\,\overline{w}):=$
$\displaystyle-8\pi
m(1+\ln(m\pi))-8\pi\mathcal{F}_{m}^{K}(\overline{a}_{1},...,\overline{a}_{m})-(4\pi-\sigma)\sum_{i=1}^{m}|{{\tau}}^{\prime\prime}_{i}|^{2}$
$\displaystyle+(16\pi+\sigma)\sum_{i=1}^{m}(\overline{\alpha}_{i}-1)^{2}\ln{\Lambda}_{i}-(4\pi-\sigma)\frac{\ln{\Lambda}_{1}}{{\Lambda}_{1}^{2}\mathcal{F}^{Q}_{1}(q_{1})}\mathcal{L}(Q),$

where

${\tau}^{\prime\prime}_{i}=1-\frac{{m}\,\Lambda_{i}^{2}\mathcal{F}^{\overline{A}}_{i}(\overline{a}_{i})}{\sum_{k=1}^{m}\Lambda_{k}^{2}\mathcal{F}^{\overline{A}}_{k}(\overline{a}_{k})}.$

Furthermore we perform the following change of variables:

$\psi_{2}:Y:=(\overline{a}_{1},\cdots,\overline{a}_{m},\overline{\alpha}_{1},\cdots,\overline{\alpha}_{m},{\Lambda}_{1},\cdots,{\Lambda}_{m})\mapsto(\overline{a}_{1},\cdots,\overline{a}_{m},{\beta}_{1},\cdots,{\beta}_{m},x_{1},\cdots,x_{m})$

with

$\beta_{i}:=\sqrt{16\pi+\sigma}(\overline{\alpha}_{i}-1)\sqrt{\ln\Lambda_{i}}\,\,\forall\,\,1\leq
i\leq m\quad;\quad x_{1}:={\Lambda}_{1}\quad;\quad
x_{i}:=\sqrt{4\pi-\sigma}\tau^{\prime\prime}_{i}\,\,\forall\,\,2\leq i\leq m.$

Next we claim:
CLAIM 2: $\psi_{2}$ is a diffeomorphism on the set $(\alpha,a,\lambda)$
satisfying the conditions in the definition of $V(m,\varepsilon,Q)$, see (5).
Indeed let
$H:=(b_{1},\cdots,b_{m},\gamma_{1},\cdots,\gamma_{m},\xi_{1},\cdots,\xi_{m})$.
We need to solve $D\psi_{2}(Y)(H)=0$ and to prove that the unique solution is
$H=0$. Let $\psi_{2}^{j}$ be the $j$-th component of $\psi_{2}$. It holds

$D\psi_{2}(Y)(H)=0\Longleftrightarrow D\psi_{2}^{j}(H)=0\,\,\forall\,\,1\leq
j\leq 3m\Longleftrightarrow\begin{cases}b_{j}=0\,\,\forall\,\,j=1,\cdots,m,\\\
\sqrt{\ln\Lambda_{i}}\gamma_{i}+\frac{(\overline{\alpha}_{i}-1)}{2\Lambda_{i}\sqrt{\ln\Lambda_{i}}}\xi_{i}=0\,\,\forall\,\,i\leq
m,\\\ \xi_{1}=0\\\ \sum_{j=2}^{m}\frac{\partial
x_{i}}{\partial\Lambda_{j}}\xi_{j}=0\,\,\forall\,\,i=2,\cdots,m.\end{cases}$

We will focus on the last equation. Let
$\mathcal{F}_{k}:=\mathcal{F}_{k}^{\overline{A}}(\overline{a}_{k})$ and
$D:=\sum_{k=1}^{m}\Lambda_{k}^{2}\mathcal{F}_{k}(a_{k})$. Observe that

$\Lambda_{i}\frac{\partial
x_{i}}{\partial\Lambda_{i}}=-\frac{2m}{D^{2}}\Lambda_{i}^{2}\mathcal{F}_{i}\Big{(}\sum_{k\neq
i}\Lambda_{k}^{2}\mathcal{F}_{k}\Big{)}\quad;\quad\Lambda_{j}\frac{\partial
x_{i}}{\partial\Lambda_{j}}=\frac{2m}{D^{2}}\Lambda_{i}^{2}\mathcal{F}_{i}\Lambda_{j}^{2}\mathcal{F}_{j}\,\,\forall\,j\neq
i.$

Hence the last equation of the system is equivalent to

$-\Big{(}\sum_{k\neq
i}\Lambda_{k}^{2}\mathcal{F}_{k}\Big{)}\frac{\xi_{i}}{\Lambda_{i}}+\sum_{j\geq
2;j\neq
i}\Lambda_{j}^{2}\mathcal{F}_{j}\frac{\xi_{j}}{\Lambda_{j}}=0\,\,\forall\,\,i\geq
2\Longleftrightarrow\sum_{j\geq
2}\Lambda_{j}^{2}\mathcal{F}_{j}\frac{\xi_{j}}{\Lambda_{j}}=\Big{(}\sum_{k=1}^{m}\Lambda_{k}^{2}\mathcal{F}_{k}\Big{)}\frac{\xi_{i}}{\Lambda_{i}}\,\,\forall\,\,i\geq
2.$

Thus we derive that $\frac{\xi_{i}}{\Lambda_{i}}=\frac{\xi_{2}}{\Lambda_{2}}$
for each $i\geq 2$. Putting this information in the last equation implies

$\Lambda_{1}^{2}\mathcal{F}_{1}\frac{\xi_{2}}{\Lambda_{2}}=0$

which implies that $\xi_{2}=0$ and therefore $H=0$. This implies that
$D\psi_{2}(Y)$ is invertible and the claim follows from the inverse function
theorem.

Next we observe that, using the coordinates provided by the diffeomorphism
$\psi_{2}(Y)$, the functional $J_{8m\pi}$ reads as follows

$\displaystyle
J_{\varrho}(\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\,+\,\overline{w})=$
$\displaystyle-8\pi
m(1+\ln(m\pi))-8\pi\mathcal{F}_{m}^{K}(\overline{a}_{1},...,\overline{a}_{m})+\sum_{i=1}^{m}\beta_{i}^{2}$
$\displaystyle-\sum_{i=2}^{m}x_{i}^{2}-\Big{(}\sum_{i=2}^{m}x_{i}\Big{)}^{2}+(4\pi-\sigma)\frac{(-\mathcal{L}(Q))}{\mathcal{F}^{Q}_{1}(q_{1})}\frac{\ln{\Lambda}_{1}}{\Lambda_{1}^{2}}.$

Finally, it follows from elementary linear algebra, that there exists a change
of variables: $(x_{2},\cdots,x_{m})\mapsto(y_{2},\cdots,y_{m})$ such that
$J_{8m\pi}$ reads as

$J_{\varrho}(\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}\,+\,\overline{w})=-8\pi
m(1+\ln(m\pi))-8\pi\mathcal{F}_{m}^{K}(\overline{A})+\sum_{i=1}^{m}\beta_{i}^{2}-\sum_{i=2}^{m}y_{i}^{2}+(4\pi-\sigma)\frac{(-\mathcal{L}(Q))}{\mathcal{F}^{Q}_{1}(q_{1})}\frac{\ln{\Lambda}_{1}}{\Lambda_{1}^{2}}$

where $\overline{A}:=(\overline{a}_{1},...,\overline{a}_{m})$. This ends the
proof.

Next as consequence of Proposition 5.3 of and the Morse reduction in Lemma 5.4
one derives the topological contribution of the _critical points at infinity_
to the difference of topology between the level sets of the functional
$J_{\varrho}$. Namely we have the following corollary

###### Corollary 5.5

Let $(Q)_{\infty}$ be a critical point at infinity corresponding to the
critical point $Q$ of $\mathcal{F}^{K}_{m}$ at the level
$C_{\infty}(Q)_{\infty}$ with index $\iota_{\infty}(Q)$. Assume that at the
level $C_{\infty}(Q)_{\infty}$ there is no other critical point/critical point
at infinity. Then for $\theta$ a small positive number and a field
$\mathbb{F}$, we have that

(37) $\displaystyle
H_{l}(J_{\varrho}^{C_{\infty}(Q)_{\infty}+\theta},J_{\varrho}^{C_{\infty}(Q)_{\infty}-\theta};\mathbb{F})=\begin{cases}\mathbb{F}&\mbox{
if }\quad l=\iota_{\infty}(Q),\\\ 0,&\mbox{otherwise}.\end{cases}$

where $H_{l}$ denotes the $l-$dimensional homology group with coefficient in
the field $\mathbb{F}$.

##  6  Proof of the main results

Proof of Theorem 1.1. We start by proving the statement $1)$. To this aim we
consider the following sup-approximation of $(MF)$

(38)
$(MF)_{\mu}\quad-\Delta_{g}u\,=\,8m\pi(1+\mu)(\frac{Ke^{u}}{\int_{\Sigma}Ke^{u}dV_{g}}\,-\,1)\quad\mbox{
in }\Sigma,$

where $\mu>0$ is a small positive real number.
Regarding Problem $(MF)_{\mu}$ we prove the following :
Claim 1: For a sequence of $\mu_{k}\to 0$ , Problem $(MF)_{\mu_{k}}$ admits a
solution $u_{\mu_{k}}$ whose generalized Morse index $Morse(u_{\mu_{k}})$ is
$3m$.

Indeed let $J_{8\pi m(1+\mu)}$ be the Euler Lagrange functional associated to
$(MF)_{\mu}$ then it follows from [15, 26, 16] that, for large $L$ the
sublevel set $J_{8\pi m(1+{\mu})}^{-L}$ has the same homotopy type as the set
of formal barycenters $B_{m}(\Sigma)$ of order $m$ . Note that (see [18, 26]),
we have

$H_{3m-1}(J_{8\pi
m(1+\mu)}^{-L},\mathbb{Z}_{2})=H_{3m-1}(B_{m}(\Sigma),\mathbb{Z}_{2})\,\neq\,0.$

Let $\varrho_{\mu}:=8\pi m(1+\mu)$, using the fact that
$J^{L}_{\varrho_{\mu}}$ is a contractible set and the exact sequence of the
pair $(J^{L}_{\varrho_{\mu}},J^{-L}_{\varrho_{\mu}})$ we derive that

$\begin{CD}0=H_{3m}(J_{\varrho_{\mu}}^{L})\to
H_{3m}(J_{\varrho_{\mu}}^{L},J_{\varrho_{\mu}}^{-L})\to
H_{3m-1}(J_{\varrho_{\mu}}^{-L})\to
H_{3m-1}(J_{\varrho_{\mu}}^{L})=0.\end{CD}$

Hence it follows that

$H_{3m}(J^{L}_{8\pi m(1+\mu)},J^{-L}_{8\pi m(1+\mu)})\,\neq 0.$

Therefore $J_{8\pi m(1+\mu)}$ has a critical point whose Morse index is $3m$.

To conclude the proof of the theorem we prove the following claim:
Claim 2: $\displaystyle{u_{\mu_{k}}\to u_{\infty}\mbox{ in
}C^{2,\alpha}(\Sigma),\,}$ where $u_{\infty}$ is a solution of Equation
$(MF)$.
To prove the claim it is enough to rule out the blow up of $u_{\mu_{k}}$.
Arguing by contradiction it follows from Proposition 7.6 that $u_{\mu_{k}}\in
V(m,\varepsilon)$ for $k$ large . Hence this function has to be written as

$u_{\mu_{k}}=\sum_{i=1}^{m}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+w,$

where the function $w\in E^{m}_{{A},\Lambda}$ and satisfies

(39) $\|w\|\leq c\Big{(}\sum|\alpha_{k}-1|+\frac{1}{\lambda}\Big{)}.$

Now, using the fact that $\nabla J_{8\pi m(1+\mu)}(u_{\mu_{k}})=0$ and
Corollary 4.3, we get

(40) $\sum|\alpha_{i}-1|\leq
c\Big{(}\frac{1}{\lambda\ln\lambda}+\frac{1}{\ln\lambda}\sum|\tau_{j}-\mu+\mu\tau_{j}|\Big{)}.$

Now, using Proposition 4.1 we derive that

(41)
$16\pi\alpha_{i}(\tau_{i}-\mu+\mu\tau_{i})-64\pi^{2}\sum\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}=O\Big{(}\sum|\alpha_{j}-1|^{2}\Big{)}+o\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}.$

Using (40) and (41), we get that

(42)
$\sum|\tau_{i}-\mu+\mu\tau_{i}|=O\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}.$

Hence, using (40), (42) and summing (41) for $i=1,\cdots,m$, we derive that

$\sum_{i=1}^{m}\tau_{i}-m\mu+\mu\sum_{i=1}^{m}\tau_{i}=4\pi
m\sum\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}+o\Big{(}\frac{\ln\lambda}{\lambda^{2}}\Big{)}.$

Now using Lemma 4.4 we obtain

(43)
$\mu=\frac{\pi}{2}\,\mathcal{L}({A})\frac{\ln\lambda_{1}}{\int_{\Sigma}Ke^{u}}(1+o(1)),$

which implies that $\mathcal{L}({A})$ has to be positive. Furthermore it
follows from Proposition 4.5, (40), (42) and (39) that
${A}:=(a_{1},\cdots,a_{m})$ converges to a critical point of
$\mathcal{F}_{m}^{K}$.
Next expanding the functional $J_{8m\pi(1+\mu_{k})}$ in ${V}(m,\varepsilon)$
(following the proof of Proposition 5.1), we obtain

$\begin{split}J_{8m\pi(1+\mu_{k})}(u_{\mu_{k}})\,&=\,C(m,\mu_{k})\,-\,8\pi(1+\mu_{k})\mathcal{F}^{K}_{m}(A)\,\,+16\pi\sum_{i=1}^{m}(\alpha_{i}-1)^{2}\ln\lambda_{i}\\\
&-\,4\pi\,\sum_{i=1}^{m}\widetilde{\tau}_{i}^{2}\,-\,4\pi\frac{\mathcal{L}(A)}{\mathcal{F}^{A}_{1}(a_{1})}\frac{\ln\lambda_{1}}{\lambda_{1}^{2}}\,+\,o(\frac{\ln\lambda}{\lambda^{2}}).\end{split}$

Hence we derive from this expansion, arguing as in Corollary 5.3 that

$Morse(u_{\mu_{k}})\,=3m-Morse(\mathcal{F}^{K}_{m},A).$

Since $Morse(u_{\varepsilon})=3m$, we have that $A$ is a local minimum of
$\mathcal{F}_{m}^{K}$ satisfying that $\mathcal{L}(A)>0.$ Hence we reach a
contradiction to the assumption of Theorem 1.1.
The proof of statement $2)$ follows the same argument as above. The only
difference is that we use a sub-approximation:

(44)
$(MF)_{\mu}\quad-\Delta_{g}u\,=\,8m\pi(1-\mu)\Big{(}\frac{Ke^{u}}{\int_{\Sigma}Ke^{u}dV_{g}}\,-\,1\Big{)}\quad\mbox{
in }\Sigma,$

where $\mu>0$ is a small positive real number. Using the fact that
$H_{3m-4}(B_{m-1}(\Sigma),\mathbb{Z}_{2})\neq 0$ (see [18], Lemma 8.7) we have
that

$H_{3m-4}(J_{8\pi
m(1-\mu)}^{-L},\mathbb{Z}_{2})=H_{3m-4}(B_{m-1}(\Sigma),\mathbb{Z}_{2})\,\neq\,0.$

Hence we have that

$H_{3m-3}(J^{L}_{8\pi m(1-\mu)},J^{-L}_{8\pi m(1-\mu)})\,\neq 0.$

Therefore $J_{8\pi m(1-\mu)}$ has a critical point whose Morse index is
$3m-3$. Next we claim
Claim 3: $\displaystyle{u_{\mu_{k}}\to u_{\infty}\mbox{ in
}C^{2,\alpha}(\Sigma),\,}$ where $u_{\infty}$ is a solution of Equation
$(MF)$.
By elliptic regularity, it is enough to rule out the blow up of $u_{\mu_{k}}$.
Arguing by contradiction we have by Proposition 7.6 that for $k$ large
$u_{\mu_{k}}\in V(m,\varepsilon)$. It follows then from Proposition 5.3 that
$u_{\mu_{k}}\in V(m,Q,\varepsilon)$, where $Q$ is a critical point of
$\mathcal{F}^{K}_{m}$ with $\mathcal{L}(Q)<0.$ Moreover we have that

$Morse(u_{\mu_{k}})\,=\,3m-1-Morse(\mathcal{F}^{K}_{m},Q).$

That is $Q$ is a critical point of $\mathcal{F}^{K}_{m}$ whose Morse index is
2 and $\mathcal{L}(Q)<0.$ A contradiction to the assumption $2)$ of the
theorem. Hence the proof of statement 2) is complete.

Proof  of Theorem 1.2 We first observe that since the function $K$ satisfies
the non degeneracy condition $(\mathcal{N}_{m})$, $J_{\varrho}$ is a Morse
function. Moreover the Morse indices of its critical points are uniformly
bounded, say by $\bar{m}$ and it follows from Corollary 5.3 that the Morse
indices of the critical points at Infinity of $J_{8\pi m}$ are bounded by
$3m-1$.
Without loss of generality, we may assume that $J_{\varrho}$ separates its
critical as well as its critical points at Infinity. That is at any critical
value there is only one critical point or one critical point at infinity. This
can be arranged by perturbing $J_{\varrho}$ slightly in disjoint neighborhoods
of its critical points (resp. its critical points at Infinity). Next we choose
$L$ such that all critical values, respectively, critical values at infinity
are contained in the open interval $(-L,L)$ and we order these critical values
as

$-L<C_{1}<\cdots<C_{p}<L.$

Now choose regular values $a_{0}<\cdots<a_{p}$ such that

$a_{0}=-L,a_{p}=L\mbox{ and }\,\,a_{i-1}<C_{i}<a_{i},\,\forall i=1,\cdots,p.$

Moreover we denote by $N_{i}$ the Morse index of the critical point $u_{i}$
such that $J_{\varrho}(u_{i})=C_{i}$, resp. by $N^{\infty}_{i}$ the index
$\iota_{\infty}$ of the critical point at infinity $u^{\infty}_{i}$ such that
$C_{\infty}(u^{\infty}_{i})=C_{i}$ and set
$M_{i}:=J_{\varrho}^{a_{i}}:=\\{u:J_{\varrho}(u)<a_{i}\\}$. We notice that it
follows from the standard Morse Lemma in the case of usual critical points and
from Lemma 5.4 in the case of critical points at infinity, that $M_{i}$ is
obtained from $M_{i-1}$ by attaching a $N(i)-$cell (resp.
$N^{\infty}(i)-$cell). We set

$\theta(i,j):=dim\,H_{i}(M_{j},M_{j-1})\quad;\quad\mu(i,j):=dim\,H_{i}(M_{j},J_{\varrho}^{-L})$

and observe that $\theta(i,j)=\delta_{i,N_{j}}$ (resp.
$\theta(i,j)=\delta_{i,N^{\infty}_{j}}$) and $\mu(i,j)=0$ if
$i>\overline{N}:=\max(\bar{m},3m-1).$ Next, considering the triple
$(M_{j},M_{j-1},J_{\varrho}^{-L})$ we have the exact sequence

$\begin{CD}0@>{}>{}>H_{*}(M_{j-1},J_{\varrho}^{-L})@>{}>{}>H_{*}(M_{j},J_{\varrho}^{-L}),@>{}>{}>H_{*}(M_{j},M_{j-1})\\\
@>{}>{}>H_{*-1}(M_{j-1},J_{\varrho}^{-L})@>{}>{}>\cdots
@>{}>{}>H_{0}(M_{j},M_{j-1})\end{CD}$

We recall that exactness implies the vanishing of corresponding alternating
sum of dimensions of the vector spaces in the sequence.

Denoting by $\mathcal{N}_{q,j}$ the kernel of
$H_{q}(M_{j-1},J_{\varrho}^{-L})\to H_{q}(M_{j},J_{\varrho}^{-L})$ and
$\nu_{q,j}:=dim\mathcal{N}_{q,j}$ and using the exactness of the triple
$(M_{j};M_{j-1},J_{\varrho}^{-L})$ we derive that (by grouping 3 terms at a
time):

$\nu_{q,j}\,=\,\sum_{i=0}^{q}(-1)^{i+q}\Big{(}\mu(i,j-1)\,-\mu(i,j)\,+\,\theta(i,j)\Big{)}.$

Summing over $j=1,\cdots,p$ yields

$\sum_{j=1}^{p}\nu_{q,j}\,=\,\sum_{i=0}^{q}(-1)^{i+q}\Big{(}\mu(i,0)\,-\mu(i,p)\,+\,\sum_{j=1}^{p}\theta(i,j)\Big{)}.$

Note that $\mu(i,0)=0$ and
$\mu(i,p)=\mbox{dim}(H_{i}(J_{\varrho}^{L},J_{\varrho}^{-L}))$. Using the
exact sequence of the pair $(J_{\varrho}^{L},J_{\varrho}^{-L})$ we derive that

(45)
$\begin{cases}&H_{0}(J_{\varrho}^{L},J_{\varrho}^{-L})\,\simeq\,H_{1}(J_{\varrho}^{L},J_{\varrho}^{-L})\simeq
0\quad\mbox{ and }\\\
&H_{i}(J_{\varrho}^{L},J_{\varrho}^{-L})\,\simeq\,H_{i-1}(J_{\varrho}^{-L})\simeq
H_{i-1}(B_{m-1}(\Sigma)):=\beta_{i-1}^{m-1}\quad\forall i\geq 2.\end{cases}$

Moreover, denoting by $\nu_{i}$ the number of critical points of Morse index
$i$ resp. by $\nu^{\infty}_{i}$ the number of critical points at Infinity of
index $i$, it follows that

$\sum_{j=1}^{p}\theta(i,j)\,=\nu_{i}+\nu^{\infty}_{i}.$

Hence we get

(46)
$\sum_{j=1}^{p}\nu_{q,j}+\sum_{i=2}^{q}(-1)^{q+i}\beta_{i-1}^{m-1}\,=\,\sum_{(A\in\mathcal{K}^{-}_{m};\iota_{\infty}(A)\leq
q)}(-1)^{\iota_{\infty}(A)+q}\,+\,\sum_{i=0}^{q}(-1)^{i+q}\nu_{i}.$

Now, for $2\leq k\leq 3m-1$, summing (46) for $q=k$ and $q=k-1$, we obtain

$\nu_{k}\,+\,\nu^{\infty}_{k}\,-\,\beta^{m-1}_{k-1}\,=\sum_{j=1}^{\overline{N}}\nu_{k,j}+\sum_{j=1}^{\overline{N}}\nu_{k-1,j}\geq\,0,$

and the statement $(a)$ is proved.
The second claim follows by taking $q=\overline{N}$ in (46) and using
$\nu_{\overline{N},j}=0$ for each $j$ and

$\sum(-1)^{i}\beta_{i}^{m-1}=\chi(B_{m-1}(\Sigma))=1-\binom{m-1-\chi(\Sigma)}{m-1}.$

The proof of Theorem 1.2 is complete.

Proof  of Theorem 1.3 Since there are no critical points at infinity of index
$q_{0}$, it follows from (10) that:

$\nu_{q_{0}}\,\geq\,\beta^{m-1}_{q_{0}-1}.$

Since by assumption $\beta_{q_{0}-1}^{m-1}\neq 0$, we deduce that Equation
$(MF)$ has at least $\beta_{q_{0}-1}^{m-1}$ solutions. In particular since
$\beta^{m-1}_{3m-4}=H_{3m-4}(B_{m-1}(\Sigma),\mathbb{Z}_{2})\,\neq 0$, if
there are no critical points at infinity of index $3m-3$, we obtain a solution
of $(MF)$ whose Morse index is $3m-3$.

Proof  of Theorem 1.5 First observe that if there is no solution and there is
no critical point at infinity of index $q_{0}$, then there holds:

(47) $\nu_{q_{0},j}=0\quad\mbox{ and }\quad\nu_{q_{0}-1,j}=0\quad\forall\,j.$

Indeed, recall that $\nu_{q,j}:=dim\,Ker(i_{q,j})$ where $i_{q,j}$ denotes the
map

$i_{q,j}:\,H_{q}(M_{j-1},J_{\varrho}^{-L})\to H_{q}(M_{j},J_{\varrho}^{-L}).$

The claim is immediate for $q_{0}$ since there is no critical point or
critical point at infinity of index $q_{0}$ which implies that
$H_{q_{0}}(M_{j-1},J_{\varrho}^{-L})=H_{q_{0}}(M_{j},J_{\varrho}^{-L})=0$. For
$q_{0}-1$, using the exact sequence of the triple
$(M_{j},M_{j-1},J_{\varrho}^{-L})$, we get (since we assumed that there is no
critical point/critical point at infinity of index $q_{0}$)

$0=H_{q_{0}}(M_{j},M_{j-1})\to H_{q_{0}-1}(M_{j-1},J_{\varrho}^{-L})\to
H_{q_{0}-1}(M_{j},J_{\varrho}^{-L}),\quad\mbox{ for each }\,j,$

which gives the claim for $q_{0}-1$.
Next to prove Theorem 1.5, we assume that there is no solution and observe
that $\beta_{i}^{m-1}=0$ for each $i>3m-4$. Furthermore, since there is no
critical point at infinity of index $3m$, we get from the above claim (47)
that $\nu_{3m-1,j}=0$ for each $j$. Hence, summing (46) for $q=3m-1$ and
$q=3m-3$, we get

$0\leq\sum_{j=1}^{p}\nu_{3m-3,j}=-(-1)^{3m-3}\sum_{A\in\mathcal{K}_{m}^{-};\iota_{\infty}(A)=3m-1\,et\,\iota_{\infty}(A)=3m-2}(-1)^{\iota_{\infty}(A)}=\nu_{3m-2}-\nu_{3m-1}.$

In the same way, summing (46) for $q=3m-1$ and $q=3m-4$, we get

$\sum_{j=1}^{p}\nu_{3m-4,j}+\beta_{3m-4}^{m-1}=-(-1)^{3m-4}\sum_{A\in\mathcal{K}_{m}^{-};3m-3\leq\iota_{\infty}(A)\leq
3m-1}(-1)^{\iota_{\infty}(A)}=\nu_{3m-3}-\nu_{3m-2}+\nu_{3m-1}.$

Hence the result follows.

Proof  of Theorem 1.4 By contradiction, assume that $(MF)$ does not have
solutions and observe that, as in (47) it follows from assumption $2)$ that

$\forall
j,\quad\nu_{q_{0}+1,j}\,=\,\nu_{q_{0},j}\,=\,\nu_{q_{0}-1,j}\,=\,\nu_{q_{0}-2,j}=0.$

Hence summing (46) for $k=q_{0}$ and $k=q_{0}-1$ we obtain
$\nu^{\infty}_{q_{0}}\,=\,\beta^{m-1}_{q_{0}-1}.$ A contradiction to the
assumptions $1)$ and $3)$.

##  7  Appendix

In this section we collect some technical Lemmas used in this paper.

###### Lemma 7.1

Let $\varphi_{a,\lambda}$ and $\delta_{a,\lambda}$ be defined in (17) and
(16). The following expansions hold pointwise

$\displaystyle\varphi_{a,\lambda}=\delta_{a,\lambda}\,+\,\ln\big{(}\frac{\lambda^{2}}{8}\big{)}\,+\,8\pi\,H(a,.)\,+\,4\pi\frac{\ln\lambda}{\lambda^{2}}\,+\,O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad\mbox{
in }\Sigma,$
$\displaystyle\lambda\frac{\partial\varphi_{a,\lambda}}{\partial\lambda}(x)=\frac{4}{1+\lambda^{2}\psi_{a}^{2}(x)}-8\pi\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad\mbox{
for each }x\in\Sigma,$
$\displaystyle\frac{1}{\lambda}\frac{\partial\varphi_{a,\lambda}}{\partial
a}(x)=\frac{1}{\lambda}\frac{\partial\delta_{a,\lambda}}{\partial
a}(x)+8\pi\frac{1}{\lambda}\frac{\partial H(a,x)}{\partial
a}(x)+O\big{(}\frac{\ln\lambda}{\lambda^{2}}\big{)}\quad\mbox{ for each
}x\in\Sigma.$

In particular, we have

$\displaystyle\varphi_{a,\lambda}=8\pi\,G(a,.)\,+\,4\pi\frac{\ln\lambda}{\lambda^{2}}\,+\,O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad\mbox{
in }\Sigma\setminus B_{a}(\eta),$
$\displaystyle\lambda\frac{\partial\varphi_{a,\lambda}}{\partial\lambda}=-8\pi\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad\mbox{
in }\Sigma\setminus B_{a}(\eta),$
$\displaystyle\frac{1}{\lambda}\frac{\partial\varphi_{a,\lambda}}{\partial
a}=8\pi\frac{1}{\lambda}\frac{\partial G(a,.)}{\partial
a}+O\big{(}\frac{\ln\lambda}{\lambda^{2}}\big{)}\quad\mbox{ in
}\Sigma\setminus B_{a}(\eta).$

Proof. We remark that the second part of this lemma follows immediately from
the first one. Now, we start by proving the first claim. First, observe that

(48)
$\int_{\Sigma}e^{\delta_{a,\lambda}+u_{a}}dV_{g}=\int_{B_{a}(\eta)}e^{\delta_{a,\lambda}}dV_{g_{a}}+\int_{\Sigma\setminus
B_{a}(\eta)}O\big{(}\frac{1}{\lambda^{2}}\big{)}=8\pi+O\big{(}\frac{1}{\lambda^{2}}\big{)}.$

Now, let us define the following function
$k_{a,\lambda}:=\varphi_{a,\lambda}-\delta_{a,\lambda}\,-\,\ln\big{(}\frac{\lambda^{2}}{8}\big{)}\,-\,8\pi\,H(a,.)$.
We recall that $\delta_{a,\lambda}$ is a constant function in $\Sigma\setminus
B_{a}(2\eta)$. Thus, using (15), we obtain

$-\Delta_{g}k_{a,\lambda}=\begin{cases}&e^{\delta_{a,\lambda}+u_{a}}-\,\int_{\Sigma}e^{\delta_{a,\lambda}+u_{a}}dV_{g}+8\pi=O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad(\mbox{in
}\Sigma\setminus B_{a}(2\eta)).\\\
&e^{\delta_{a,\lambda}+u_{a}}-\,\int_{\Sigma}e^{\delta_{a,\lambda}+u_{a}}dV_{g}+e^{u_{a}}\Delta_{g_{a}}\delta_{a,\lambda}+8\pi=O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad(\mbox{in
}B_{a}(\eta)).\end{cases}$

It remains the case of $x\in B_{a}(2\eta)\setminus B_{a}(\eta)$. Using again
(15) we get

$\displaystyle-\Delta_{g}k_{a,\lambda}$
$\displaystyle=O\big{(}\frac{1}{\lambda^{2}}\big{)}-8\pi-2e^{u_{a}}\Delta_{g_{a}}\ln(1+\lambda^{2}\psi_{a}^{2})+(8\pi+2e^{u_{a}}\Delta_{g_{a}}\ln(\psi_{a}^{2})$
$\displaystyle=-2e^{u_{a}}\Delta_{g_{a}}\ln\big{(}\lambda^{2}+\frac{1}{\psi_{a}^{2}}\big{)}+O\big{(}\frac{1}{\lambda^{2}}\big{)}=O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad(\mbox{in
}B_{a}(2\eta)\setminus B_{a}(\eta)).$

Hence we obtain that $\Delta_{g}k_{a,\lambda}=O(1/\lambda^{2})$ in $\Sigma$
and therefore we get that

$k_{a,\lambda}(x)-\int_{\Sigma}k_{a,\lambda}(y)dV_{g}=O\big{(}\frac{1}{\lambda^{2}}\big{)}\quad\mbox{in
}\Sigma.$

It remains to estimate the previous integral. Using (12), (14) and (17), we
get

$\displaystyle\int_{\Sigma}k_{a,\lambda}dV_{g}$
$\displaystyle=-\int_{\Sigma}\delta_{a,\lambda}\,+\,\ln\big{(}\frac{\lambda^{2}}{8}\big{)}\,+\,8\pi\,H(a,.)dV_{g}\,=\,2\int_{\Sigma}\ln\big{(}1+\frac{1}{\lambda^{2}\psi_{a}^{2}}\big{)}dV_{g}$
$\displaystyle=2\int_{B_{a}(\eta)}\ln\big{(}1+\frac{1}{\lambda^{2}\psi_{a}^{2}}\big{)}dV_{g_{a}}+O\big{(}\int_{B_{a}(\eta)}\ln\big{(}1+\frac{1}{\lambda^{2}\psi_{a}^{2}}\big{)}|e^{-u_{a}}-1|dV_{g_{a}}+\frac{1}{\lambda^{2}}\big{)}$
$\displaystyle=\frac{4\pi}{\lambda^{2}}\int_{0}^{\lambda\eta}\ln
\big{(}1+\frac{1}{r^{2}}\big{)}rdr+O\big{(}\frac{1}{\lambda^{2}}\big{)}=4\pi\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)}.$

Hence the proof of the first claim follows.
The other claims can be proved in the same way.

###### Lemma 7.2

Let $\varphi_{a,\lambda}$ be defined in (17). The following expansions hold:

$\|\varphi_{a,\lambda}\|^{2}=32\pi\ln\lambda+64\pi^{2}H(a,a)-16\pi+64\pi^{2}\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)},$
$\langle\varphi_{a,\lambda},\lambda\frac{\partial\varphi_{a,\lambda}}{\partial\lambda}\rangle_{g}=16\pi-64\pi^{2}\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)},$
$\langle\varphi_{a_{j},\lambda_{j}},\varphi_{a_{i},\lambda_{i}}\rangle_{g}=64\pi^{2}\,G(a_{j},a_{i})\,+\,32\pi^{2}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\,+\,32\pi^{2}\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\,+\,O\big{(}\frac{1}{\lambda_{j}^{2}}+\frac{1}{\lambda_{i}^{2}}\big{)},$
$\langle\varphi_{a_{j},\lambda_{j}},\lambda_{i}\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial\lambda_{i}}\rangle_{g}=-64\pi^{2}\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}+O\big{(}\frac{1}{\lambda_{i}^{2}}+\frac{1}{\lambda_{j}^{2}}\big{)}.$

Proof. Since $\int_{\Sigma}\varphi_{a,\lambda}dV_{g}=0$, using Lemma 7.1 and
(48), we get

$\displaystyle\|\varphi_{a,\lambda}\|^{2}$
$\displaystyle=\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}\big{(}\delta_{a,\lambda}\,+\,\ln\big{(}\frac{\lambda^{2}}{8}\big{)}\,+\,8\pi\,H(a,.)\,+\,4\pi\frac{\ln\lambda}{\lambda^{2}}\,+\,O\big{(}\frac{1}{\lambda^{2}}\big{)}\big{)}dV_{g}$
$\displaystyle=32\pi^{2}\frac{\ln\lambda}{\lambda^{2}}\,+\,O\big{(}\frac{1}{\lambda^{2}}\big{)}+2\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}\ln\big{(}\frac{\lambda^{2}\psi_{a}^{2}}{1+\lambda^{2}\psi_{a}^{2}}\big{)}dV_{g}+8\pi\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}G(a,.)dV_{g}$

Note that (using the fact that $\int_{\Sigma}G(a,.)dV_{g}=0$)

$8\pi\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}G(a,.)dV_{g}=8\pi\varphi_{a,\lambda}(a)=32\pi\ln\lambda+64\pi^{2}H(a,a)+32\pi^{2}\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)}.$

Furthermore, we have

$\displaystyle\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}\ln\big{(}\frac{\lambda^{2}\psi_{a}^{2}}{1+\lambda^{2}\psi_{a}^{2}}\big{)}dV_{g}$
$\displaystyle=\int_{B_{a}(\eta)}e^{\delta_{a,\lambda}}\ln\big{(}\frac{\lambda^{2}|x-a|^{2}}{1+\lambda^{2}|x-a|^{2}}\big{)}dV_{g_{a}}+O\big{(}\frac{1}{\lambda^{4}}\big{)}$
$\displaystyle=\int_{B(0,\eta)}\frac{8\lambda^{2}}{(1+\lambda^{2}|x|^{2})^{2}}\ln\big{(}\frac{\lambda^{2}|x|^{2}}{1+\lambda^{2}|x|^{2}}\big{)}dx+O\big{(}\frac{1}{\lambda^{4}}\big{)}$
$\displaystyle=-8\pi+O\big{(}\frac{1}{\lambda^{2}}\big{)}.$

Hence the proof of the first claim follows.
Concerning the second one, using the fact that
$\int_{\Sigma}\lambda\frac{\partial\varphi_{a,\lambda}}{\partial\lambda}dV_{g}=0$,
we have

$\displaystyle\langle\varphi_{a,\lambda},\lambda\frac{\partial\varphi_{a,\lambda}}{\partial\lambda}\rangle_{g}$
$\displaystyle=\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}\bigg{(}\frac{4}{1+\lambda^{2}\psi_{a}^{2}(x)}-8\pi\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)}\bigg{)}dV_{g}$
$\displaystyle=-64\pi^{2}\frac{\ln\lambda}{\lambda^{2}}+O\big{(}\frac{1}{\lambda^{2}}\big{)}+\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}\frac{4}{1+\lambda^{2}\psi_{a}^{2}(x)}dV_{g}.$

Observe that, we have

$\int_{\Sigma}e^{u_{a}}e^{\delta_{a,\lambda}}\frac{4}{1+\lambda^{2}\psi_{a}^{2}(x)}dV_{g}=\int_{B(0,\eta)}\frac{32\lambda^{2}}{(1+\lambda^{2}|x|^{2})^{3}}dx+O\big{(}\frac{1}{\lambda^{4}}\big{)}=16\pi+O\big{(}\frac{1}{\lambda^{4}}\big{)}.$

Thus the proof of the second claim follows. Now, we will focus on the third
claim. Using $\int_{\Sigma}|\varphi_{a,\lambda}|=O(1)$ and Lemma 7.1, it holds

$\displaystyle\langle\varphi_{a_{j},\lambda_{j}},\varphi_{a_{i},\lambda_{i}}\rangle_{g}$
$\displaystyle=\int_{B_{a_{j}}(\eta)}e^{u_{a_{j}}}e^{\delta_{a_{j},\lambda_{j}}}\bigg{(}8\pi\,G(a_{i},.)\,+\,4\pi\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\,+\,O\big{(}\frac{1}{\lambda_{i}^{2}}\big{)}\bigg{)}dV_{g}+O\big{(}\frac{1}{\lambda_{j}^{2}}\big{)}$
$\displaystyle=32\pi^{2}\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}+O\big{(}\frac{1}{\lambda_{j}^{2}}+\frac{1}{\lambda_{i}^{2}}\big{)}+8\pi\int_{B_{a_{j}}(\eta)}e^{u_{a_{j}}}e^{\delta_{a_{j},\lambda_{j}}}G(a_{i},.)dV_{g}.$

Observe that (using $\int_{\Sigma}G(a_{i},.)dV_{g}=0$)

$\displaystyle\int_{B_{a_{j}}(\eta)}e^{u_{a_{j}}}e^{\delta_{a_{j},\lambda_{j}}}G(a_{i},.)dV_{g}$
$\displaystyle=\int_{\Sigma}e^{u_{a_{j}}}e^{\delta_{a_{j},\lambda_{j}}}G(a_{i},.)dV_{g}+O\big{(}\frac{1}{\lambda_{j}^{2}}\big{)}$
$\displaystyle=\varphi_{a_{j},\lambda_{j}}(a_{i})+O\big{(}\frac{1}{\lambda_{j}^{2}}\big{)}=8\pi\,G(a_{j},a_{i})\,+\,4\pi\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\,+\,O\big{(}\frac{1}{\lambda_{j}^{2}}\big{)}.$

Hence the proof of this claim follows. Concerning the last one, it holds

$\displaystyle\langle\varphi_{a_{j},\lambda_{j}},\lambda_{i}\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial\lambda_{i}}\rangle_{g}$
$\displaystyle=\int_{B_{a_{j}}(\eta)}e^{u_{a_{j}}}e^{\delta_{a_{j},\lambda_{j}}}\bigg{(}-8\pi\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}+O\big{(}\frac{1}{\lambda_{i}^{2}}\big{)}\bigg{)}dV_{g}+O\big{(}\frac{1}{\lambda_{j}^{2}}\big{)}$
$\displaystyle=-64\pi^{2}\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}+O\big{(}\frac{1}{\lambda_{j}^{2}}+\frac{1}{\lambda_{i}^{2}}\big{)}.$

Thereby the proof of this lemma follows.

From Lemma 7.1, it is easy to get the following expansion

###### Lemma 7.3

Let ${u}:=\sum_{j=1}^{m}\alpha_{j}\varphi_{a_{j},\lambda_{j}}$. In
$B_{a_{i}}(\eta)$, it holds

$\displaystyle Ke^{u}$
$\displaystyle=\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}_{i}^{A}g_{i}^{A}}{(1+\lambda_{i}^{2}|y_{a_{i}}(.)|^{2})^{2\alpha_{i}}}\Big{(}1+4\pi\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}+O\big{(}\frac{1}{\lambda^{2}}+\sum|\alpha_{j}-1|\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\big{)}\Big{)}$
$\displaystyle=\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}_{i}^{A}g_{i}^{A}}{(1+\lambda_{i}^{2}|y_{a_{i}}(.)|^{2})^{2\alpha_{i}}}\Big{(}1+4\pi\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}+O\bigg{(}\Big{(}\frac{1}{\lambda^{2}}+\sum|\alpha_{j}-1|\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}Ke^{u}\bigg{)},$

where the functions $\mathcal{F}^{A}_{i}$ (with ${A}=(a_{1},\cdots,a_{m})$)
and $g_{i}^{A}$ are defined as follows

(49) $\left\\{\begin{array}[]{cc}\mathcal{F}^{A}_{i}(x):=K(x)\exp\Big{(}8\pi
H(a_{i},x)+8\pi\sum_{j\neq i}G(a_{j},x)\Big{)},\vspace{1mm}\\\
g_{i}^{A}(x):=\exp\Big{(}8\pi(\alpha_{i}-1)H(a_{i},x)+8\pi\sum_{j\neq
i}(\alpha_{j}-1)G(a_{j},x)\Big{)}.\end{array}\right.$

###### Lemma 7.4

Let $u:=\sum_{i=1}^{m}\alpha_{i}\varphi_{i}\in V(m,\varepsilon)$. Then,

(50)
$\displaystyle\int_{\Sigma}Ke^{u}dV_{g}=\pi\sum_{i=1}^{m}\frac{\lambda_{i}^{4\alpha_{i}-2}}{2\alpha_{i}-1}(\mathcal{F}_{i}^{A}g_{i}^{A})(a_{i})+O\Big{(}1+\sum_{k=1}^{m}\lambda_{k}^{4\alpha_{k}-2}\Big{(}|\alpha_{k}-1|^{2}+\frac{\ln\lambda_{k}}{\lambda_{k}^{2}}\Big{)}\Big{)}.$

If $\sum_{i=1}^{m}|\alpha_{i}-1|\ln\lambda_{i}=o_{\varepsilon}(1)$, the
expansion (50) can be improved as follows

(51) $\displaystyle\int_{\Sigma}Ke^{u}dV_{g}$ $\displaystyle=$
$\displaystyle\pi\sum_{i=1}^{m}\frac{\lambda_{i}^{4\alpha_{i}-2}}{2\alpha_{i}-1}(\mathcal{F}_{i}^{A}g_{i}^{A})(a_{i})+\frac{\pi}{2}\sum_{i=1}^{m}\Big{(}\Delta\mathcal{F}_{i}^{A}(a_{i})-2K_{g}(a_{i})\mathcal{F}_{i}^{A}(a_{i})\Big{)}\ln\lambda_{i}$
$\displaystyle+4\pi^{2}\Big{(}\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}\sum_{i=1}^{m}\lambda_{i}^{4\alpha_{i}-2}\mathcal{F}_{i}^{A}(a_{i})+O\Big{(}1+\sum_{k=1}^{m}|\alpha_{k}-1|\ln^{2}\lambda_{k}\Big{)}.$

Proof. Let $B_{i}:=B_{a_{i}}(\eta)$. Firstly, note that Lemma 7.1 shows that
$e^{u}$ is bounded on $\Sigma\setminus(\cup B_{i})$. Hence the integral in
this set is bounded. Now, using Lemma 7.3, we have

$\displaystyle\int_{B_{i}}Ke^{u}dV_{g}$
$\displaystyle=\int_{B_{i}}\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}^{A}_{i}g_{i}^{A}}{(1+\lambda_{i}^{2}|y_{a_{i}}|^{2})^{2\alpha_{i}}}\Big{(}1+4\pi\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}dV_{g}+O\bigg{(}\big{(}\frac{1}{\lambda^{2}}+\sum|\alpha_{j}-1|\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\big{)}\int_{B_{i}}Ke^{u}dV_{g}\bigg{)}$
$\displaystyle=\Big{(}1+4\pi\sum_{j=1}^{m}\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\Big{)}\int_{B_{i}}\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}^{A}_{i}g_{i}^{A}e^{-u_{a_{i}}}}{(1+\lambda_{i}^{2}|y_{a_{i}}(x)|^{2})^{2\alpha_{i}}}dV_{g_{a_{i}}}+O\bigg{(}\big{(}\frac{1}{\lambda^{2}}+\sum|\alpha_{j}-1|\frac{\ln\lambda_{j}}{\lambda_{j}^{2}}\big{)}\int_{\Sigma}Ke^{u}dV_{g}\bigg{)}.$

Finally, we have

(52)
$\displaystyle\int_{B_{i}}\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}^{A}_{i}g_{i}^{A}e^{-u_{a_{i}}}}{(1+\lambda_{i}^{2}|y_{a_{i}}(x)|^{2})^{2\alpha_{i}}}dV_{g_{a_{i}}}$
$\displaystyle=\lambda_{i}^{4\alpha_{i}-2}(\mathcal{F}_{i}^{A}g_{i}^{A}e^{-u_{a_{i}}})(a_{i})\int_{B_{i}}\frac{\lambda_{i}^{2}}{(1+\lambda_{i}^{2}|y_{a_{i}}(x)|^{2})^{2\alpha_{i}}}dV_{g_{a_{i}}}$
$\displaystyle+\lambda_{i}^{4\alpha_{i}-2}O\left(\int_{B_{i}}\frac{\lambda_{i}^{2}|y_{a_{i}}(x)|^{2}}{(1+\lambda_{i}^{2}|y_{a_{i}}(x)|^{2})^{2\alpha_{i}}}dV_{g_{a_{i}}}\right).$

Note that ${u_{a_{i}}}(a_{i})=0$. Furthermore, easy computations imply

(53)
$\int_{B(0,\eta)}\frac{\lambda_{i}^{2}}{(1+\lambda_{i}^{2}|x|^{2})^{2\alpha_{i}}}dx=\frac{\pi}{2\alpha_{i}-1}+O\Big{(}\frac{1}{\lambda_{i}^{4\alpha_{i}-2}}\Big{)}.$

To estimate the other integral, we introduce the following function

(54) $\xi(x)=\frac{1}{(1+\lambda^{2}|x|^{2})^{2(\alpha-1)}}.$

If $(\alpha-1)\ln\lambda$ is small, one obtains that $\xi=1+o(1)$ uniformly on
$B(0,\eta)$. In general, we have

(55)
$|\xi(x)-1|=|\xi(x)-\xi(0)|\,=\,\Big{|}\int_{0}^{1}\frac{4(1-\alpha)\lambda^{2}t|x|^{2}}{(1+\lambda^{2}t^{2}|x|^{2})^{2\alpha-1}}dt\Big{|}\,\leq\,c|\alpha-1|\sqrt{\lambda|x|}.$

Now, using (55), we obtain

$\displaystyle\int_{B(0,\eta)}\frac{\lambda_{i}^{2}|x|^{2}}{(1+\lambda_{i}^{2}|x|^{2})^{2\alpha_{i}}}dx$
$\displaystyle=\int_{B(0,\eta)}\frac{\lambda_{i}^{2}|x|^{2}}{(1+\lambda_{i}^{2}|x|^{2})^{2}}dx+\int_{B(0,\eta)}\frac{\lambda_{i}^{2}|x|^{2}}{(1+\lambda_{i}^{2}|x|^{2})^{2}}(\xi_{i}-1)dx$
$\displaystyle=O\Big{(}\frac{\ln\lambda_{i}}{\lambda_{i}^{2}}\Big{)}+O\Big{(}\frac{|\alpha_{i}-1|}{\lambda_{i}^{3/2}}\Big{)}.$

Hence, the proof of (50) follows. Now, we will focus on (51). In this case, we
have $|\alpha_{i}-1|\ln\lambda_{i}$ is small for each $i$ and we need to
improve the estimate of (52). Using (53), we obtain

$\displaystyle\int_{B_{i}}$
$\displaystyle\frac{\lambda_{i}^{4\alpha_{i}}\mathcal{F}^{A}_{i}g_{i}^{A}e^{-u_{a_{i}}}}{(1+\lambda_{i}^{2}|y_{a_{i}}(x)|^{2})^{2\alpha_{i}}}dV_{g_{a_{i}}}=\frac{\pi}{2\alpha_{i}-1}\lambda_{i}^{4\alpha_{i}-2}(\mathcal{F}_{i}^{A}g_{i}^{A})(a_{i})+O(1)$
$\displaystyle+\frac{1}{4}\Delta_{g_{a_{i}}}(\mathcal{F}_{i}^{A}g_{i}^{A}e^{-u_{a_{i}}})(a_{i})\int_{B(0,\eta)}\frac{\lambda_{i}^{4\alpha_{i}}|x|^{2}}{(1+\lambda_{i}^{2}|x|^{2})^{2\alpha_{i}}}dx+O\Big{(}\int_{B(0,\eta)}\frac{\lambda_{i}^{4\alpha_{i}}|x|^{3}}{(1+\lambda_{i}^{2}|x|^{2})^{2\alpha_{i}}}dx\Big{)}.$

Observe that

$\displaystyle\int_{B(0,\eta)}\frac{\lambda_{i}^{4\alpha_{i}}|x|^{3}}{(1+\lambda_{i}^{2}|x|^{2})^{2\alpha_{i}}}dx=\lambda_{i}^{4\alpha_{i}-5}2\pi\int_{0}^{\lambda_{i}\eta}.\frac{r^{4}}{(1+r^{2})^{2\alpha_{i}}}dr=O(1),$
$\displaystyle\int_{B(0,\eta)}\frac{\lambda_{i}^{4\alpha_{i}}|x|^{2}}{(1+\lambda_{i}^{2}|x|^{2})^{2\alpha_{i}}}dx=2\pi\ln\lambda_{i}+O(1+|\alpha_{i}-1|\ln^{2}\lambda_{i})\quad(\mbox{by
using }|\alpha_{i}-1|\ln\lambda_{i}\mbox{ is small}).$

To conclude, we need the following information. We know that the function
$\widetilde{u}_{a}(x):=u_{a}(y_{a}(x))$ (for $x\in
B(a,\eta)\subset\mathbb{R}^{2}$) satisfies

$-\Delta\widetilde{u}_{a}=-2K_{g}(y_{a}^{-1}(.))e^{-\widetilde{u}_{a}}\quad\mbox{
in }B(a,\eta)$

and therefore we derive that

$\Delta_{g_{a_{i}}}(\mathcal{F}_{i}^{A}g_{i}^{A}e^{-u_{a_{i}}})(a_{i})=\Delta_{g_{a_{i}}}\mathcal{F}_{i}^{A}(a_{i})-2\mathcal{F}_{i}^{A}(a_{i})K_{g}(a_{i})+O(\sum|\alpha_{j}-1|),$

by using the fact that

$g_{i}^{A}(a_{i})=1+O\Big{(}\sum|\alpha_{j}-1|\Big{)},\quad|\nabla
g_{i}^{A}(a_{i})|=O\Big{(}\sum|\alpha_{j}-1|\Big{)},\quad|\Delta
g_{i}^{A}(a_{i})|=O\Big{(}\sum|\alpha_{j}-1|\Big{)}.$

Summing the above estimates, the result follows.

###### Lemma 7.5

Let $A$ and $\Lambda$ satisfying the balancing condition (19) then the
approximate solution $U_{A,\Lambda}$ defined in (18) satisfies the following
equation

$-\Delta_{g}U_{A,\Lambda}\,+\,8\pi m\,=\,8\pi
m\frac{Ke^{U_{A,\Lambda}}}{\int_{\Sigma}Ke^{U_{A,\Lambda}}}\,+\,f_{A,\Lambda},\quad\mbox{
where }$ $f_{A.\Lambda}\rightharpoonup 0\mbox{ weakly in }H^{1}(\Sigma)\mbox{
as }\lambda_{i}\to\infty\,\forall\,i=1,\cdots,m.$

Proof. We first notice that according to Equation (17) satisfied by
$\varphi_{a,\lambda}$, using (48), the function $U_{A,\lambda}$ satisfies the
equation

(56) $-\Delta_{g}U_{A,\Lambda}\,+\,8\pi
m\,+\,O(\sum_{i=1}^{m}\frac{1}{\lambda_{i}^{2}})=\,\sum_{i=1}^{m}e^{\delta_{a_{i},\lambda_{i}}+u_{a_{i}}}.$

Moreover it follows from Lemmas 7.1 and 7.4 that in
$\Sigma\setminus\bigcup_{i=1}^{m}B_{\eta}(a_{i})$ we have that

$8\pi
m\frac{Ke^{U_{A,\Lambda}}}{\int_{\Sigma}Ke^{U_{A,\Lambda}}}\,=\,O(\frac{\ln\lambda}{\lambda^{2}}).$

Furthermore it follows from Lemmas 7.3 and 7.4 that in $B_{\eta}(a_{i})$ there
hold

$Ke^{U_{a,\lambda}}\,=\,\frac{1}{8}\lambda_{i}^{2}\mathcal{F}_{i}^{A}(a_{i})e^{\delta_{a_{i},\lambda_{i}}}(1+O(\frac{\ln\lambda}{\lambda^{2}}))\quad\mbox{
and
}\quad\int_{\Sigma}Ke^{U_{A,\lambda}}\,=\,\pi\sum_{i=1}^{m}\lambda_{i}^{2}\mathcal{F}_{i}^{A}(a_{i})(1+O(\frac{\ln\lambda}{\lambda^{2}})).$

Therefore in $B_{\eta}(a_{i})$ we have that

$8\pi
m\frac{Ke^{U_{A,\Lambda}}}{\int_{\Sigma}Ke^{U_{A,\Lambda}}}\,=\,\frac{m\lambda_{i}^{2}\mathcal{F}^{A}_{i}(a_{i})e^{\delta_{a_{i},\lambda_{i}}}}{\sum_{j=1}^{m}\lambda_{j}^{2}\mathcal{F}^{A}_{j}(a_{j})}(1+O(\frac{\ln\lambda}{\lambda^{2}})).$

Hence we derive that

$\sum_{j=1}^{m}e^{\delta_{a_{j},\lambda_{j}}+u_{a_{j}}}\,-\,8\pi
m\frac{Ke^{U_{A,\Lambda}}}{\int_{\Sigma}Ke^{U_{A,\Lambda}}}\,=e^{\delta_{a_{i},\lambda_{i}}}(\tilde{\tau}_{i}+O(|x-a_{i}|^{2}\frac{\ln\lambda}{\lambda^{2}}))\quad\mbox{
in }B_{\eta}(a_{i}),$

where ${\tau}^{\prime}_{i}$ is defined in Proposition 5.1 by taking
$\alpha_{k}=1$ for each $k$.
We notice that it follows from the balancing condition (19) that $\forall
i=1,\cdots,m$ we have that $|{\tau}^{\prime}_{i}|=o_{\lambda}(1)$. Summarizing
we have that $U_{A,\Lambda}$ satisfies the equation

$\displaystyle-\Delta_{g}U_{A,\Lambda}\,+\,8\pi m\,=\,8\pi
m\frac{Ke^{U_{A,\Lambda}}}{\int_{\Sigma}Ke^{U_{A,\Lambda}}}\,+\,f_{A,\Lambda},\quad\mbox{
where }$ $\displaystyle
f_{A,\Lambda}\,=\,\begin{cases}O({\ln\lambda}/{\lambda^{2}})\mbox{ in
}\Sigma\setminus\bigcup_{i=1}^{m}B_{\eta}(a_{i})\\\
e^{\delta_{a_{i},\lambda_{i}}}(\tilde{\tau}_{i}+O({|x-a_{i}|^{2}+\ln\lambda_{i}}/{\lambda_{i}^{2}}))\mbox{
in }B_{\eta}(a_{i}).\end{cases}$

Hence the Lemma follows.

Lastly for the sake of completeness we provide the following characterization
of approximate blowing up solutions of Equation $(MF)$. Namely we prove:

###### Proposition 7.6

Let $(\Sigma,g)$ be a closed surface of unit volume and $u_{k}$ be a blowing
up solution of

$-\Delta_{g}u_{k}\,=\,\varrho_{k}(\frac{Ke^{u_{k}}}{\int_{\Sigma}Ke^{u_{k}}}\,-1)\quad\mbox{
in }\Sigma;\qquad\int_{\Sigma}u_{k}dV_{g}=0,$

where $\varrho_{k}\to 8\pi m,\,m\in\mathbb{N}$. Then for $\varepsilon$ small
and $k$ large we have that $u_{k}\in V(m,\varepsilon)$.

Proof. We set $\tilde{u}_{k}:=u_{k}\,-\,\ln(\int_{\Sigma}Ke^{u_{k}})$ and
observe that $\tilde{u}_{k}$ satisfies the equation

$-\Delta_{g}\tilde{u}_{k}\,=\,\varrho_{k}(Ke^{\tilde{u}_{k}}-1)\,\mbox{ in
}\Sigma.$

Now it follows from the refined blow up analysis performed in [23, 12, 13]
that $\tilde{u}_{k}$ blows up at $m$-points $a_{1},\cdots,a_{m}\in\Sigma$ with
comparable blow up rates $\lambda_{i}:=e^{u_{k}(a_{i})}$ such that
$d_{g}(a_{i},a_{j})\geq 2\eta$ for some $\eta>0$. Moreover we have that

$e^{\tilde{u}_{k}}\,=\,O(\sum_{i=1}^{m}\frac{1}{\lambda_{i}^{2}})\,\mbox{ in
}\Sigma\setminus\bigcup_{i=1}^{m}B_{\eta}(a_{i})$

and inside the balls $B_{i}:=B_{\eta}(a_{i})$ there holds (See Theorem 1.4 in
[12])

$\tilde{u}_{k}\,+\,\ln(\varrho_{k}K(a_{i}))\,=\,{\delta_{a_{i},\lambda_{i}}}\,+O(d_{g}(a_{i},x)).$

Next we set

$\tilde{v}_{k}:=\tilde{u}_{k}\,-\,\sum_{i=1}^{m}\varphi_{a_{i},\lambda_{i}}$

and we notice that $\tilde{v}_{k}$ satisfies the equation

(57)
$-\Delta_{g}\tilde{v}_{k}\,+\,\varrho_{k}-8m\pi\,=\varrho_{k}Ke^{\tilde{u}_{k}}-\sum_{i=1}^{m}e^{\delta_{a_{i},\lambda_{i}}+u_{a_{i}}}+O(\frac{1}{\lambda^{2}}):=f_{k}.$

Next we claim that:
Claim: For $1<p<2$ there exists a constant $C>0$ such that

$\|f_{k}\|_{L^{p}}\,\leq\,C\sum_{i=1}^{m}\frac{1}{\lambda_{i}^{(2/p)-1}}.$

Indeed we have that

$\int_{\Sigma}|f_{k}|^{p}dV_{g}\,=\,\sum_{i=1}^{m}\int_{B_{i}}|f_{k}|^{p}dV_{g}\,+\,O(\sum_{i=1}^{m}\frac{1}{\lambda_{i}^{2p}}).$

Furthermore in $B_{i}$ we have, in geodesic local coordinates around $a_{i}$,
that

$\displaystyle f_{k}$
$\displaystyle=e^{\tilde{u}_{k}+\ln(\varrho_{k}K)}\,-\sum_{i=1}^{m}e^{\delta_{a_{i},\lambda_{i}}+u_{a_{i}}}+O(\frac{1}{\lambda^{2}})$
$\displaystyle=e^{\delta_{a_{i},\lambda_{i}}+\ln(\varrho_{k}K)-\ln(\varrho_{k}K(a_{i})+O(|x-a_{i}|))}-e^{\delta_{a_{i},\lambda_{i}}+u_{a_{i}}}+O(\frac{1}{\lambda^{2}})$
(58)
$\displaystyle=O(|x-a_{i}|e^{\delta_{a_{i},\lambda_{i}}}+\frac{1}{\lambda^{2}}).$

Hence for $1<p<2$ and a constant $C>0$ there holds

$\int_{\Sigma}|f_{k}|^{p}dV_{g}\,\leq\,C\,\sum_{i=1}^{m}\frac{1}{\lambda_{i}^{2-p}}.$

Next setting
$v_{k}:=\tilde{v}_{k}\,+\,\ln(\int_{\Sigma}Ke^{u_{k}})\,=\,u_{k}-\sum_{i=1}^{m}\varphi_{a_{i},\lambda_{i}}$
we have that $v_{k}$ satisfies the equation

$-\Delta_{g}\,v_{k}+\,\varrho_{k}-8\pi m\,=\,f_{k}\,\mbox{ in
}\Sigma;\,\int_{\Sigma}v_{k}dV_{g}\,=\,0,$

where $f_{k}\in L^{p}(\Sigma)$ for $p\in(1,2)$. Hence it follows from the
Calderon-Zygmund a priori estimate that

$\|v_{k}\|_{W^{2,p}}\,\leq\,C\,\|f_{k}\|_{L^{p}}\,\leq\,C\,\sum_{i=1}^{m}\frac{1}{\lambda_{i}^{(2/p)-1}}.$

Therefore it follows from the Sobolev embedding
$W^{2,p}(\Sigma)\hookrightarrow W^{1,2}(\Sigma)$ that

$\|v_{k}\|_{H^{1}}\,\leq\,\sum_{i=1}^{m}\frac{1}{\lambda_{i}^{(2/p)-1}}.$

Hence choosing $p=\frac{4}{3}$ we obtain that

$\|u_{k}\,-\,\sum_{i=1}^{m}\varphi_{a_{i},\lambda_{i}}\|\,\leq\,C\,\sum_{i=1}^{m}\frac{1}{\sqrt{\lambda_{i}}}.$

Therefore the proposition is fully proven.

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Mohameden Ahmedou
Mathematisches Institut der Justus-Liebig-Universität Giessen
Arndtsrasse 2, D-35392 Giessen
Germany
<EMAIL_ADDRESS>Mohamed Ben Ayed
Université de Sfax, Faculté des Sciences de Sfax
Département de Mathématiques
Route de Soukra, Sfax, Tunisia
<EMAIL_ADDRESS>