# Variational theory for the resonant $T$-curvature equation

Cheikh Birahim NDIAYE

Department of Mathematics Howard University
Annex 3, Graduate School of Arts and Sciences, # 217
DC 20059 Washington, USA

11footnotetext: E-mail addresses<EMAIL_ADDRESS>

The author was partially supported by NSF grant DMS–2000164.

Abstract

We study the resonant prescribed $T$-curvature problem on a compact
$4$-dimensional Riemannian manifold with boundary. We derive sharp energy and
gradient estimates of the associated Euler-Lagrange functional to characterize
the critical points at infinity of the associated variational problem under a
non-degeneracy on a naturally associated Hamiltonian function. Using this, we
derive a Morse type lemma around the critical points at infinity. Using the
Morse lemma at infinity, we prove new existence results of Morse theoretical
type. Combining the Morse lemma at infinity and the Liouville version of the
Barycenter technique of Bahri-Coron[13] developed in [46], we prove new
existence results under a topological hypothesis on the boundary of the
underlying manifold, and the entry and exit sets at infinity.

Key Words: $T$-curvature, $Q$-curvature, Morse Theory, Critical points at
infinity, Barycenter Technique.

AMS subject classification: 53C21, 35C60, 58J60, 55N10.

##  1 Introduction and statement of the results

On a four-dimensional compact Riemannian manifolds with boundary
$(\overline{M},g)$, there exists a fourth-order operator $P_{g}$ called
Paneitz operator discovered by Paneitz[49] and an associated curvature
quantity $Q_{g}$ called $Q$-curvature introduced by Branson-Oersted[14]. The
Paneitz operator $P_{g}$ and the $Q$-curvature $Q_{g}$ are defined in terms of
the Ricci tensor $Ric_{g}$ and the scalar curvature $R_{g}$ of
$(\overline{M},g)$ by

$P^{4}_{g}=\Delta_{g}^{2}-div_{g}\left((\frac{2}{3}R_{g}g-2Ric_{g})\nabla_{g}\right);\;\;\;\;\;\;\;Q_{g}=-\frac{1}{12}(\Delta_{g}R_{g}-R_{g}^{2}+3|Ric_{g}|^{2}),$

where $div_{g}$ is the divergence and $\nabla_{g}$ is the covariant derivative
of with respect to $g$.

On the other hand, Chang-Qing[23] have discovered an operator $P^{3}_{g}$
which is associated to the boundary $\partial M$ of $\overline{M}$ and a
curvature quantity $T_{g}$ naturally associated to $P^{3}_{g}$. They are
defined by the formulas

$P^{3}_{g}=\frac{1}{2}\frac{\partial{\Delta_{g}}}{\partial
n_{g}}+\Delta_{\hat{g}}\frac{\partial}{\partial
n_{g}}-2H_{g}\Delta_{\hat{g}}+L_{g}(\nabla_{\hat{g}},\nabla_{\hat{g}})+\nabla_{\hat{g}}H_{g}.\nabla_{\hat{g}}+(F_{g}-\frac{R_{g}}{3})\frac{\partial}{\partial
n_{g}}.$ $T_{g}=-\frac{1}{12}\frac{\partial R_{g}}{\partial
n_{g}}+\frac{1}{2}R_{g}H_{g}-<G_{g},L_{g}>+3H_{g}^{3}-\frac{1}{3}tr_{g}(L_{g}^{3})-\Delta_{\hat{g}}H_{g},$

where $\hat{g}$ is the metric induced by $g$ on $\partial M$,
$\frac{\partial}{\partial n_{g}}$ is the inward Neuman operator on $\partial
M$ with respect to $g$, $L_{g}$ is the second fundamental form of $\partial M$
with respect to $g$, $H_{g}$ is the mean curvature of $\partial M$ with
respect to $g$, $R^{k}_{g,ijl}$ is the Riemann curvature tensor of
$(\overline{M},g)$, $R_{g,ijkl}=g_{mi}R^{m}_{g,jkl}$ ($g_{ij}$ are the entries
of the metric $g$), $F_{g}=R^{a}_{g,nan}$ (with $n$ denoting the index
corresponding to the normal direction in local coordinates) and
$<G_{g},L_{g}>=\hat{g}^{ac}\hat{g}^{bd}R_{g,anbn}L_{g,cd}$. Moreover, the
notation $L_{g}(\nabla_{\hat{g}},\nabla_{\hat{g}})$, means
$L_{g}(\nabla_{\hat{g}},\nabla_{\hat{g}})(u)=\nabla_{\hat{g}}^{a}(L_{g,ab}\nabla_{\hat{g}}^{b}u)$.
We point out that in all those notations above $i,j,k,l=1,\cdots 4$ and
$a,b,c,d=1,\cdots 3$, and Einstein summation convention is used for repeated
indices.

As the Laplace-Beltrami operator and the Neumann operator are conformally
covariant, we have that $P^{4}_{g}$ is conformally covariant of bidegree
$(0,4)$ and $P^{3}_{g}$ of bidegree $(0,3)$. Furthermore, as they govern the
transformation laws of the Gauss curvature and the geodesic curvature on
compact surfaces with boundary, the couple $(P^{4}_{g},P^{3}_{g})$ does the
same for $(Q_{g},T_{g})$ on a compact four-dimensional Riemannian manifold
with boundary $(\overline{M},g)$. In fact, under a conformal change of metric
$g_{u}=e^{2u}g$, we have

(1) $\left\\{\begin{split}P^{4}_{g_{u}}=e^{-4u}P^{4}_{g},\\\
P^{3}_{g_{u}}=e^{-3u}P^{3}_{g},\end{split}\right.\qquad\mbox{and}\qquad\left\\{\begin{split}P^{4}_{g}u+2Q_{g}=2Q_{g_{u}}e^{4u}\;\;\text{in
}\;\;M,\\\ P^{3}_{g}u+T_{g}=T_{g_{u}}e^{3u}\;\;\text{on}\;\;\partial
M.\end{split}\right.$

Apart from this analogy, we have also an extension of the Gauss-Bonnet
identity $\eqref{eq:gbc}$ which is known as the Gauss-Bonnet-Chern formula

(2) $\int_{M}(Q_{g}+\frac{|W_{g}|^{2}}{8})dV_{g}+\oint_{\partial
M}(T_{g}+Z_{g})dS_{g}=4\pi^{2}\chi(\overline{M})$

where $W_{g}$ denote the Weyl tensor of $(\overline{M},g)$ and $Z_{g}$ is
given by the following formula

$Z_{g}=R_{g}H_{g}-3H_{g}Ric_{g,nn}+\hat{g}^{ac}\hat{g}^{bd}R_{g,anbn}L_{g,cd}-\hat{g}^{ac}\hat{g}^{bd}R_{g,acbc}L_{g,cd}+6H_{g}^{3}-3H_{g}|L_{g}|^{2}+tr_{g}(L_{g}^{3}),$

with $tr_{g}$ denoting the trace with respect to the metric induced on
$\partial M$ by $g$ (namely $\hat{g}$) and $\chi(\overline{M})$ the Euler-
Poincaré characteristic of $\overline{M}$. Concerning the quantity $Z_{g}$, we
have that it vanishes when the boundary is totally geodesic and
$\oint_{\partial M}Z_{g}dV_{g}$ is always conformally invariant, see [23].
Thus, setting

(3)
$\kappa_{(P^{4},P^{3})}:=\kappa_{(P^{4},P^{3})}[g]:=\int_{M}Q_{g}dV_{g}+\oint_{\partial
M}T_{g}dS_{g}$

we have that thanks to $\eqref{eq:gbc}$, and to the fact that
$|W_{g}|^{2}dV_{g}$ is pointwise conformally invariant,
$\kappa_{(P^{4},P^{3})}$ is a conformal invariant (which justifies the
notation used above). We remark that $4\pi^{2}$ is the the total integral of
the $(Q,T)$-curvature of the standard four-dimensional Euclidean unit ball
$\mathbb{B}^{4}.$

As was asked in [2], a natural question is whether every compact four-
dimensional Riemannian manifold with boundary $(\overline{M},g)$ carries a
conformal metric $g_{u}$ for which the corresponding $Q$-curvature $Q_{g_{u}}$
is zero, the corresponding $T$-curvature $T_{g_{u}}$ is a prescribed function
and such that $(\overline{M},g_{u})$ has minimal boundary. Thanks to
$\eqref{eq:law}$, this problem is equivalent to finding a smooth solution to
the following BVP:

$\left\\{\begin{split}P^{4}_{g}u+2Q_{g}&=0\;\;&\text{in}\;\;M,\\\
P^{3}_{g}u+T_{g}&=Ke^{3u}\;\;&\text{on}\;\;\partial M,\\\ -\frac{\partial
u}{\partial n_{g}}+H_{g}u&=0\;\;&\text{on}\;\;\partial M.\end{split}\right.$

where $K:\partial M\longrightarrow\mathbb{R}_{+}$ is a positive smooth
function on $\partial M$.

Since we are interested to find a metric in the conformal class of $g$, then
we can assume that $H_{g}=0$, since this can be always obtained through a
conformal transformation of the background metric. Thus, we are lead to solve
the following BVP with Neumann homogeneous boundary condition:

(4) $\left\\{\begin{split}P^{4}_{g}u+2Q_{g}&=0\;\;&\text{in}\;\;M,\\\
P^{3}_{g}u+T_{g}&=Ke^{3u}\;\;&\text{on}\;\;\partial M,\\\ \frac{\partial
u}{\partial n_{g}}&=0\;\;&\text{on}\;\;\partial M.\end{split}\right.$

Defining $\mathcal{H}_{\frac{\partial}{\partial n}}$ as

$\mathcal{H}_{\frac{\partial}{\partial n}}=\Big{\\{}u\in
W^{2,2}(M):\;\;\;\frac{\partial u}{\partial
n_{g}}=0\;\;\;\text{on}\;\;\partial M\Big{\\}},$

where $W^{2,2}(M)$ denotes the space of functions on $M$ which are square
integrable together with their first and second derivatives, and

$\mathbb{P}^{4,3}_{g}(u,v)=\left<P^{4}u,u\right>_{L^{2}(M)}+2\left<P^{3}_{g}u,u\right>_{L^{2}(\partial
M)},\;\;\;\;u,v\in\mathcal{H}_{\frac{\partial}{\partial n}},$

we have integration by part implies

(5)
$\begin{split}\mathbb{P}^{4,3}_{g}(u,v)=\int_{M}\left(\Delta_{g}u\Delta_{g}v+\frac{2}{3}R_{g}\nabla_{g}u\cdot\nabla_{g}v\right)dV_{g}-2\int_{M}Ric_{g}(\nabla_{g}u,\nabla_{g}v)dV_{g}\\\
-2\oint_{\partial
M}L_{g}(\nabla_{\hat{g}}u,\nabla_{\hat{g}}v)dS_{g},\;\;\;u,v\in\mathcal{H}_{\frac{\partial}{\partial
n}}\end{split}$

and is clearly a bilinear form on $\mathcal{H}_{\frac{\partial}{\partial n}}$.
We set

(6) $\ker\mathbb{P}^{4,3}_{g}:=\\{u\in\mathcal{H}_{\frac{\partial}{\partial
n}}:\;\;\;\mathbb{P}^{4,3}_{g}(u,v)=0,\;\;\forall
v\in\mathcal{H}_{\frac{\partial}{\partial n}}\\}$

On the other hand, standard regularity theory implies that smooth solutions to
(4) can be found by looking at critical points of the geometric functional

$\begin{split}\mathcal{E}_{g}(u)=\mathbb{P}^{4,3}_{g}(u,u)+4\int_{M}Q_{g}udV_{g}+4\oint_{\partial
M}T_{g}udS_{g}-\frac{4}{3}\kappa_{(P^{4},P^{3})}\log\oint_{\partial
M}Ke^{3u}dS_{g},\;\;\;u\in\mathcal{H}_{\frac{\partial}{\partial
n}}.\end{split}$

As a Liouville type problem, the analytic features of equation (4) and of the
associated Euler-Lagrange functional $\mathcal{E}_{g}$ depend strongly on the
conformal invariant $\kappa_{(P^{4},P^{3})}$ . Indeed, depending on whether
$\kappa_{(P^{4},P^{3})}$ is a positive integer multiple of $4\pi^{2}$ or not,
the noncompactness of equation (4) and the way of finding critical points of
$\mathcal{E}_{g}$ changes drastically. As far as existence questions are
concerned, we have that problem (4) has been solved in a work of Chang-
Qing[24] under the assumption that
$\ker\;\mathbb{P}^{4,3}_{g}\simeq\mathbb{R}$ , $\mathbb{P}^{4,3}_{g}$ is non-
negative and $\kappa_{(P^{4},\;P^{3})}<4\pi^{2}$. In [44], we show existence
of solutions for (4) under the assumption
$\ker\mathbb{P}^{4,3}_{g}\simeq\mathbb{R}$ and $\kappa_{(P^{4},\;P^{3})}\notin
4\pi^{2}\mathbb{N}^{*}$.

As a Liouviile type problem, the assumption
$\ker\mathbb{P}^{4,3}_{g}\simeq\mathbb{R}$ and $\kappa_{(P^{4},P^{3})}\notin
4\pi^{2}\mathbb{N}^{*}$ will be referred to as nonresonant case. This
terminology is motivated by the fact that in that situation the set of
solutions to some perturbations of equation (7) (including it) is compact.
Naturally, we call resonant case when
$\ker\mathbb{P}^{4,3}_{g}\simeq\mathbb{R}$ and $\kappa_{(P^{4},\;P^{3})}\in
4\pi^{2}\mathbb{N}^{*}$. With these terminologies, we have that the works of
Chang-Qing[24] and our work in [44] answer affirmatively the question raised
above in the nonresonant case. However, for the resonant case, there are no
known existence results to the best of our knowledge.

In this work, beside existence results for (4), we are interested in a
complete variational theory for the boundary value problem (4) in the resonant
case, namely when $\ker\mathbb{P}^{4,3}_{g}\simeq\mathbb{R}$ and
$\kappa_{(P^{4},P^{3})}=4\pi^{2}k$ for some $k\in\mathbb{N}^{*}$. To present
the main results of the paper, we need to set first some notation and make
some definitions. We define the Hamiltonian function (at infinity)
$\mathcal{F}_{K}:(\partial M)^{k}\setminus F_{k}(\partial
M)\longrightarrow\mathbb{R}$ by

$\mathcal{F}_{K}((a_{1},\cdots,a_{k})):=\sum_{i=1}^{k}\left(H(a_{i},a_{i})+\sum_{j=1,\;j\neq
i}^{k}G(a_{i},\;a_{j})+\frac{2}{3}\log(K(a_{i}))\right)$

where where $F_{k}(\partial M)$ denotes the fat Diagonal of $(\partial
M)^{k}$, namely

$F_{k}(\partial M):=\\{A:=(a_{1},\cdots,a_{k})\in(\partial
M)^{k}:\;\;\text{there exists}\;\;i\neq j\;\,\text{with}\;\,a_{i}=a_{j}\\},$

$G$ is the Green’s function defined by (45) , and $H$ is its regular defined
as in (47). Furthermore, we define

(7) $Crit(\mathcal{F}_{K}):=\\{A\in(\partial M)^{k}\setminus F_{k}(\partial
M),\;\;A\;\;\;\text{critical point of}\;\;\mathcal{F}_{K}\\}.$

Moreover, for $A=(a_{1},\cdots,a_{k})\in(\partial M)^{k}\setminus
F_{k}(\partial M)$, and $i=1\cdots k$, we set

(8) $\mathcal{F}^{A}_{i}(x):=e^{3(H(a_{i},x)+\sum_{j+1,\;j\neq
i}^{k}G(a_{j},x))+\frac{1}{3}\log(K(x))},$

and define

(9)
$\mathcal{L}_{K}(A):=-\sum_{i=1}^{k}(\mathcal{F}^{A}_{i})^{\frac{1}{2}}L_{\hat{g}}((\mathcal{F}^{A}_{i})^{\frac{1}{6}})(a_{i}),$

where

$L_{\hat{g}}:=-\Delta_{\hat{g}}+\frac{1}{8}R_{\hat{g}}$

is the conformal Laplacian associated to $\hat{g}$. We also set

(10) $\mathcal{F}_{\infty}:=\\{A\in
Crit(\mathcal{F}_{K}):\>\;\mathcal{L}_{K}(A)<0\\},$ (11)
$i_{\infty}(A):=4k-1-Morse(A,\;\mathcal{F}_{K}),$

and define

(12) $m_{i}^{k}:=card\\{A\in
Crit(\mathcal{F}_{K}):\;i_{\infty}(A)=i\\},\;\;i=0,\cdots,4k-1,$

where $Morse(\mathcal{F}_{K},A)$ denotes the Morse index of $\mathcal{F}_{K}$
at $A$. We point out that for $k\geq 2$, $m_{i}^{k}=0$ for $0\leq i\leq k-2$.
For $k\geq 2$, we use the notation $B_{k-1}(\partial M)$ to denote the set of
formal barycenters of order $k-1$ of $\partial M$, namely

(13) $B_{k-1}(\partial
M):=\\{\sum_{i=1}^{k-1}\alpha_{i}\delta_{a_{i}},\;\;a_{i}\in\partial
M,\;\;\alpha_{i}\geq 0,i=1,\cdots,k-1,\;\,\sum_{i=1}^{k-1}\alpha_{i}=1\\}.$

Furthermore, we define

(14) $c^{k-1}_{p}=dim\;H_{p}(B_{k-1}(\partial M)),\;\;p=1,\cdots 4k-5,$

where $H_{p}(B_{k-1}(\partial M)$ denotes the $p$-th homology group of
$B_{k-1}(\partial M)$ with $\mathbb{Z}_{2}$ coefficients. Finally, we say

(15) $(ND)\;\;\;\text{holds if }\;\;\mathcal{F}_{K}\;\text{is a Morse function
and for every}\;\;A\in Crit(\mathcal{F}_{K}),\;\;\mathcal{L}_{K}(A)\neq 0.$

Now, we are ready to state our existence results of Morse theoretical type
starting with the critical case, namely when $k=1$.

######  Theorem 1.1.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that $H_{g}=0$,
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$ and
$\kappa_{(P^{4},\;P^{3})}=4\pi^{2}$. Assuming that $K$ is a smooth positive
function on $\partial M$ such that $(ND)$ holds and the system

(16) $\begin{cases}m^{1}_{0}=1+x_{0},\\\
m^{1}_{i}=x_{i}+x_{i-1},\;&i=1,\cdots,3,\\\ 0=x_{3}\\\ x_{i}\geq
0,\;\;&i=0,\cdots,3\end{cases}$

has no solutions, then $K$ is the $T$-curvature of a Riemannian metric on
$\overline{M}$ conformally related to $g$ with zero $Q$-curvature in $M$ and
zero mean curvature on $\partial M$.

The system (16) not having a solution traduces the violation of a strong Morse
type inequalities (SMTI) for the critical points at infinity of
$\mathcal{E}_{g}$. Since (SMTI) imply Poincare-Hopf type formulas, then we
have Theorem 1.1 implies the following Poincare-Hopf index type result.

######  Corollary 1.2.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that $H_{g}=0$,
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$ and
$\kappa_{(P^{4},\;P^{3})}=4\pi^{2}$. Assuming that $K$ is a smooth positive
function on $\partial M$ such that $(ND)$ holds and

(17) $\sum_{A\in\mathcal{F}_{\infty}}(-1)^{i_{\infty}(A)}\neq 1,$

then $K$ is the $T$-curvature of a Riemannian metric on $\overline{M}$
conformally related to $g$ with zero $Q$-curvature in $M$ and zero mean
curvature on $\partial M$

The formula (17) says that the Euler number of the space of variations is
different from the total contribution of the the true critical points at
infinity and is of global character. Localizing the arguments of Corollary 1.2
in the case of the presence of a jump in the Morse index of the critical
points of the Hamiltonian function $\mathcal{F}_{K}$, we have the following
extension of Corollary 1.2.

######  Theorem 1.3.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that $H_{g}=0$,
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$ and
$\kappa_{(P^{4},\;P^{3})}=4\pi^{2}$ and $K$ be a smooth positive function on
$\partial M$ satisfying the non degeneracy condition $(ND)$. Assuming that
there exists a positive integer $1\leq l\leq 3$ such that

$\begin{split}\sum_{A\in\mathcal{F}_{\infty},\;i_{\infty}(A)\leq
l-1}&(-1)^{i_{\infty}(A)}\neq 1\\\ &\text{and}\\\ \forall
A\in\mathcal{F}_{\infty},\;\;&\quad i_{\infty}(A)\neq l,\end{split}$

then $K$ is the $T$-curvature of a Riemannian metric on $\overline{M}$
conformally related to $g$ with zero $Q$-curvature in $M$ and zero mean
curvature on $\partial M$.

In the supercritical case, i.e $k\geq 2$, the Euler-Lagrange functional
$\mathcal{E}_{g}$ is not bounded from below, and taking into account the
topological contribution of very large negative sublevels of
$\mathcal{E}_{g}$, we have the following analogue of Theorem 1.1.

######  Theorem 1.4.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that $H_{g}=0$,
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$, and
$\kappa_{(P^{4},P^{3})}=4k\pi^{2}$ with $k\geq 2$. Assuming that $K$ is a
smooth positive function on $\partial M$ such that $(ND)$ holds and the
following system

(18) $\begin{cases}0=x_{0},\\\ m^{k}_{1}=x_{1},\\\
m_{i}^{k}=c^{k-1}_{i-1}+x_{i}+k_{i-1},\;&i=2,\cdots,4k-4,\\\
m_{i}^{k}=x_{i}+x_{i-1},\;&i=4k-3,\cdots,4k-1,\\\ 0=x_{4k-1},\\\ x_{i}\geq
0,\;\;&i=0,\cdots,4k-1,\end{cases}$

has no solutions, then $\;K$ is the $T$-curvature of a Riemannian metric on
$\overline{M}$ conformally related to $g$ with zero $Q$-curvature in $M$ and
zero mean curvature on $\partial M$.

######  Remark 1.5.

The presence of the number $c^{k-1}_{i-1}=dimH_{i-1}(B_{i-1}(\partial M))$ in
(18) account for the contribution of the topology of very negative sublevels
of $\mathcal{E}_{g}$. The relation between the topology of very negative
sublevels of the Euler-Lagrange functional of Liouville type problems and the
space of formal barycenters was first observed by Djadli-Malchiodi[29].

As in the critical case, we have that Theorem 1.4 implies the following
Poincaré-Hopf index type criterion for existence.

######  Corollary 1.6.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that $H_{g}=0$,
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$, and
$\kappa_{(P^{4},P^{3})}=4k\pi^{2}$ with $k\geq 2$. Assuming that $K$ is a
smooth positive function on $\partial M$ such that $(ND)$ holds and

(19)
$\sum_{A\in\mathcal{F}_{\infty}}(-1)^{i_{\infty}(A)}\neq\frac{1}{(k-1)!}\Pi_{i=1}^{k-1}(i-\chi(\partial
M)),$

then $K$ is the $T$-curvature of a Riemannian metric on $\overline{M}$
conformally related to $g$ with zero $Q$-curvature in $M$ and zero mean
curvature on $\partial M$.

As in the critical case, we have that a localization of the arguments of
Corollary 1.6 implies the following jumping index type result.

######  Theorem 1.7.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that $H_{g}=0$,
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$, $\kappa_{(P^{4},P^{3})}=4k\pi^{2}$
with $k\geq 2$, and let $K$ be a smooth positive function on $\partial M$
satisfying the non degeneracy condition (ND). Assuming that there exists a
positive integer $1\leq l\leq 4k-1$ and $A^{l}\in\mathcal{F}_{\infty}$ with
$i_{\infty}(A^{l})\leq l-1$ such that

$\begin{split}\sum_{A\in\mathcal{F}_{\infty},\;i_{\infty}(A)\leq
l-1}(-1)^{i_{\infty}(A)}&\neq\frac{1}{(k-1)!}\Pi_{j=1}^{k-1}(j-\chi(\partial
M))\\\ &\text{and}\\\ \forall A\in\mathcal{F}_{\infty},\;\;&\qquad
i_{\infty}(A)\neq l,\end{split}$

then $K$ is the $T$-curvature of a Riemannian metric on $\overline{M}$
conformally related to $g$ with zero $Q$-curvature in $M$ and zero mean
curvature on $\partial M$.

The Morse theoretical results stated above depend only the Morse Lemma at
infinity around true critical points at infinity (see Lemma 3.24) which
justify the condition $\mathcal{L}_{K}<0$ in the definition of
$\mathcal{F}_{\infty}$. However, our existence result of algebraic topological
type are based on the Morse lemma at infinity around all critical points at
infinity. Thus, to state our existence result of algebraic topological type,
we need first to introduce the neighborhood of potential critical points at
infinity of $\mathcal{E}_{g}$. In order to do that, we first fix $\nu$ to be a
positive and small real number, $\Lambda$ to be a large positive constant, and
$R$ to be a large positive constant too. Next, for $\epsilon$ small and
positive, and $\Theta\geq 0$, we denote by $V(k,\epsilon,\Theta)$ the
$(k,\epsilon,\Theta)$-neighborhood of potential critical points at infinity,
namely

(20)
$\begin{split}&V(k,\epsilon,\Theta):=\\{u\in\mathcal{H}_{\frac{\partial}{\partial
n}}:\exists\;a_{1},\cdots,a_{k}\in\partial
M,\alpha_{1},\cdots,\alpha_{k}>0,\;\;\lambda_{1},\cdots,\lambda_{k}>0,\beta_{1},\cdots,\beta_{\bar{k}}\in\mathbb{R},\\\
&\;\;||u-\overline{u}_{(Q,T)}-\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}-\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})||_{\mathbb{P}^{4,3}}<\epsilon,\;\;\sum_{i=1}^{k}\alpha_{i}=k\;\;\;\alpha_{i}\geq
1-\nu,\\\
&\;\lambda_{i}\geq\frac{1}{\epsilon},i=1,\cdots,k,\;\;\frac{2}{\Lambda}\leq\frac{\lambda_{i}}{\lambda_{j}}\leq\frac{\Lambda}{2},i,j=1,\cdots,k,\;|\beta_{r}|\leq\Theta,\;r=1,\cdots,\bar{k},\\\
&\;\;\text{and}\;\;\lambda_{i}d_{\hat{g}}(a_{i},a_{j})\geq
4\overline{C}R\;\;\text{for}\;i\neq j\\},\end{split}$

where $\overline{C}$ is as in (39), the $\varphi_{a_{i},\lambda_{i}}$’s are as
in (43), $\bar{k}$ is as in (29), the $v_{r}$’s are defined as in (30), the
$\overline{(v_{r})}_{(Q,t)}$’s are as in (27), and
$||\cdot||_{\mathbb{P}^{4,3}}$ is defined as in (33).

As observed by Chen-Lin [26] for Liouville type problems, the minimization at
infinity of Bahri-Coron [13] for Yamabe type problems has the following
analogue for our problem. For $\Theta\geq 0$, there exists
$\epsilon_{0}=\epsilon_{0}(\Theta)$ small and positive such that
$\forall\;0<\epsilon\leq\epsilon_{0}$, we have

(21) $\forall u\in V(k,\epsilon,\Theta),\text{the minimization problem }\\\
\min_{B_{\epsilon}^{\Theta}}||u-\overline{u}_{(Q,T)}-\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}-\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})||_{\mathbb{P}^{4,3}}$

has a unique solution, up to permutations, where $B_{\epsilon}^{\Theta}$ is
defined as follows

(22)
$\begin{split}B_{\epsilon}^{\Theta}:=&\\{(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in\mathbb{R}^{k}_{+}\times(\partial
M)^{k}\times\mathbb{R}_{+}^{k}\times\mathbb{R}^{\bar{k}}:\sum_{i=1}^{k}\alpha_{i}=k,\;\alpha_{i}\geq
1-\nu,\;\lambda_{i}\geq\frac{1}{\epsilon},i=1,\cdots,k,\\\
&\;|\beta_{r}|\leq\Theta,r=1,\cdots,\bar{k},\;\lambda_{i}d_{\hat{g}}(a_{i},a_{j})\geq
4\overline{C}R,i\neq j,i,j=1,\cdots,k\\}.\end{split}$

The selection map $s_{k}$ is defined by
$s_{k}:V(k,\epsilon,\Theta)\longrightarrow(\partial M)^{k}/\sigma_{k}$ as
follows

(23) $s_{k}(u):=A,\;\;u\in
V(k,\epsilon,\Theta),\;\,\text{and}\,\;A\;\;\text{is given
by}\;\,\eqref{eq:mini},$

We denote the critical points at infinity of $\mathcal{E}_{g}$ by $z^{\infty}$
and use the notation $M_{\infty}(z^{\infty})$ for their Morse indices at
infinity, $W_{u}(z^{\infty})$ for their unstable manifolds and
$W_{s}(z^{\infty})$ for their stable manifolds, where $z$ is the corresponding
critical point of $\mathcal{F}_{K}$. Furthermore, we denote by $x^{\infty}$
the ”true” ones, namely $\mathcal{L}_{K}(x)<0$ and the $y^{\infty}$ the
”false” ones, namely $\mathcal{L}_{K}(y)>0$. Moreover, we define $S$ to be the
following invariant set

(24) $S:=\cup_{M_{\infty}(z_{1}^{\infty}),\;M_{\infty}(z_{2}^{\infty})\geq
4k-4+\bar{k}}\;W_{u}(z_{1}^{\infty})\cap W_{s}(z_{2}^{\infty}).$

We also define $S^{\infty}$ to be the part of $S$ at infinity, namely

(25)
$S^{\infty}:=\cup_{M_{\infty}(z_{1}^{\infty}),\;M_{\infty}(z_{2}^{\infty})\geq
4k-4+\bar{k}}\;W_{u}^{\infty}(z_{1}^{\infty})\cap W_{s}(z_{2}^{\infty}),$

where $W_{u}^{–\infty}(z_{1}^{\infty})$ denotes the restriction of
$W_{u}(z_{1}^{\infty})$ at infinity. Furthermore, we denote by
$S^{\infty}_{-}$ the exit set from $S^{\infty}$ starting from a false critical
point at infinity $y^{\infty}$.
Similarly, we denote by $S^{\infty}_{+}$ the entry set to $S^{\infty}$ after
having exited $S^{\infty}$ through a set contained in $S^{\infty}_{-}$ and
entering into $S^{\infty}$ through a true critical point at infinity
$x^{\infty}$. Finally to state the result of algebraic topological type
resulting from the Liouville version of the Barycenter technique of Bahri-
Coron developed in [46], we need the existence of

(26) $0\neq O^{*}_{\partial M}\in H^{3}(\partial M).$

Indeed, we prove:

######  Theorem 1.8.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that $H_{g}=0$,
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$, and
$\kappa_{(P^{4},P^{3})}=4k\pi^{2}$ with $k\geq 2$. Assuming that $K$ is a
smooth positive function on $\partial M$ such that $(ND)$ holds and either
there is no $x^{\infty}$ with $M_{\infty}(x^{\infty})=4k-4+\bar{k}$ or
$s_{k}^{*}(O^{*}_{\partial M})\neq 0$ in $H^{3}(S^{\infty})$ and
$s_{k}^{*}(O^{*}_{M})=0$ in $H^{3}(S_{+}^{\infty}\cup S_{-}^{\infty})$, then
$K$ is the $T$-curvature of a Riemannian metric conformally related to $g$
with zero $Q$-curvature in $M$ and zero mean curvature on $\partial M$.

As in [46], Theorem 1.8 implies the following collorary.

######  Corollary 1.9.

Let $(\overline{M},\;g)$ be a compact $4$-dimensional Riemannian manifold with
boundary $\partial M$ and interior $M$ such that
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$ and
$\kappa_{(P^{4},P^{3})}=4k\pi^{2}$ with $k\geq 2$. Assuming that $K$ is a
smooth positive function on $\partial M$ such that $(ND)$ holds and that every
critical point $x$ of $\mathcal{F}_{K}$ of Morse index $0$ or $1$ satisfies
$\mathcal{L}_{K}(x)<0$, then $K$ is the $T$-curvature of a Riemannian metric
conformally related to $g$ with zero $Q$-curvature in $M$ and zero mean
curvature $\partial M$.

.

##  2 Notation and preliminaries

In the following, for a Riemmanian metric $\bar{g}$ on $\partial M$ and
$p\in\partial M$ , we will use the notation $B^{\bar{g}}_{p}(r)$ to denote the
geodesic ball with respect to $\bar{g}$ of radius $r$ and center $p$. We also
denote by $d_{\bar{g}}(x,y)$ the geodesic distance with respect to $\bar{g}$
between two points $x$ and $y$ of $\partial M$, $exp_{x}^{\bar{g}}$ the
exponential map with respect to $\bar{g}$ at $x\in\partial M$.
$inj_{\bar{g}}(\partial M)$ stands for the injectivity radius of $(\partial
M,\bar{g})$, $dV_{\bar{g}}$ denotes the Riemannian measure associated to the
metric $\bar{g}$. Furthermore, we recall that $\nabla_{\bar{g}}$,
$\Delta_{\bar{g}}$, $R_{\bar{g}}$ will denote respectively the covariant
derivative, the Laplace-Beltrami operator, the scalar curvature and Ricci
curvature with respect to $\bar{g}$. For simplicity, we will use the notation
$B_{p}(r)$ to denote $B^{g}_{p}(r)$, namely $B_{p}(r)=B^{\hat{g}}_{p}(r)$.
$(\partial M)^{2}$ stands for the cartesian product $\partial M\times\partial
M$, while $Diag(\partial M)$ is the diagonal of $(\partial M)^{2}$.

Similarly, for a Riemmanian metric $\tilde{g}$ on $\overline{M}$ , we will use
the notation $B^{\tilde{g},+}_{p}(r)$ to denote the half geodesic ball with
respect to $\tilde{g}$ of radius $r$ and center $p\in\partial M$. We also
denote by $d_{\tilde{g}}(x,y)$ the geodesic distance with respect to
$\tilde{g}$ between two points $x$ and $y$ of $\overline{M}$,
$exp_{x}^{\tilde{g}}$ the exponential map with respect to $\tilde{g}$ at
$x\in\partial M$. $inj_{\tilde{g}}(\overline{M})$ stands for the injectivity
radius of $(\overline{M},g)$, $dV_{\tilde{g}}$ denotes the Riemannian measure
associated to the metric $g$, and $dS_{\tilde{g}}$ the Riemannian measure
associated to $\hat{\tilde{g}}:=\tilde{g}|_{\partial M}$, namely
$dS_{\tilde{g}}=dV_{\hat{\tilde{g}}}$. Furthermore, we recall that
$\nabla_{\tilde{g}}$, $\Delta_{\tilde{g}}$, $R_{\tilde{g}}$ will denote
respectively the covariant derivative, the Laplace-Beltrami operator, the
scalar curvature and Ricci curvature with respect to $g$. For simplicity, we
will use the notation $B^{+}_{p}(r)$ to denote $B^{g,+}_{p}(r)$, namely
$B_{p}^{+}(r)=B^{g,+}_{p}(r)$, $p\in\partial M$

For $1\leq p\leq\infty$ and $k\in\mathbb{N}$, $\theta\in]0,1[$, $L^{p}(M)$,
$W^{k,p}(M)$, $C^{k}(\overline{M})$, and $C^{k,\theta}(\overline{M})$ stand
respectively for the standard Lebesgue space, Sobolev space, $k$-continuously
differentiable space and $k$-continuously differential space of Hölder
exponent $\beta$, all with respect $g$. Similarly, $1\leq p\leq\infty$ and
$k\in\mathbb{N}$, $\theta\in]0,1[$, $L^{p}(\partial M)$, $W^{k,p}(\partial
M)$, $C^{k}(\partial M)$, and $C^{k,\theta}(\partial M)$ stand respectively
for the standard Lebesgue space, Sobolev space, $k$-continuously
differentiable space and $k$-continuously differential space of Hölder
exponent $\beta$, all with respect $\hat{g}$.

Given a function $u\in L^{1}(M)\cap L^{1}(\partial M)$, $\bar{u}_{\partial M}$
and $\overline{u}_{(Q,T)}$ are defined by

$\bar{u}_{\partial M}=\frac{\oint_{\partial M}u(x)dS_{g}}{Vol_{g}(\partial
M)},$

with $Vol_{g}(\partial M)=\oint_{\partial M}dS_{g}$ and

(27)
$\overline{u}_{(Q,T)}=\frac{1}{4k\pi^{2}}\left(\int_{M}Q_{g}udV_{g}+\oint_{\partial
M}T_{g}udS_{g}\right).$

Given a generic Riemannian metric $\bar{g}$ on $\partial M$ and a function
$F(x,y)$ defined on a open subset of $(\partial M)^{2}$ which is symmetric and
with $F(\cdot,\cdot)\in C^{2}$ with respect to $\bar{g}$, we define
$\frac{\partial F(a,a)}{\partial a}:=\frac{\partial F(x,a)}{\partial
x}|_{x=a}=\frac{\partial F(a,y)}{\partial y}|_{y=a}=$, and
$\Delta_{\bar{g}}F(a_{1},a_{2}):=\Delta_{\bar{g},x}F(x,a_{2})|_{x=a_{1}}=\Delta_{\bar{g},y}F(a_{2},y)|_{y=a_{1}}.$

For $\epsilon>0$ and small, $\lambda\in\mathbb{R}_{+}$,
$\lambda\geq\frac{1}{\epsilon}$, and $a\in\partial M$,
$O_{\lambda,\epsilon}(1)$ stands for quantities bounded uniformly in
$\lambda$, and $\epsilon$, and $O_{a,\epsilon}(1)$ stands for quantities
bounded uniformly in $a$ and $\epsilon$. For $l\in\mathbb{N}^{*}$, $O_{l}(1)$
stands for quantities bounded uniformly in $l$ and $o_{l}(1)$ stands for
quantities which tends to $0$ as $l\rightarrow+\infty$. For $\epsilon$
positive and small, $a\in\partial M$ and $\lambda\in\mathbb{R}_{+}$ large,
$\lambda\geq\frac{1}{\epsilon}$, $O_{a,\lambda,\epsilon}(1)$ stands for
quantities bounded uniformly in $a$, $\lambda$, and $\epsilon$. For $\epsilon$
positive and small, $p\in\mathbb{N}^{*}$,
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{p})\in(\mathbb{R}_{+})^{p}$,
$\lambda_{i}\geq\frac{1}{\epsilon}$ for $i=1,\cdots,p$, and
$A:=(a_{1},\cdots,a_{p})\in M^{p}$ (where $(\mathbb{R}_{+})^{p}$ and
$(\partial M)^{p}$ denotes respectively the cartesian product of $p$ copies of
$\mathbb{R}_{+}$ and $\partial M$), $O_{A,\bar{\lambda},\epsilon}(1)$ stands
for quantities bounded uniformly in $A$, $\bar{\lambda}$, and $\epsilon$.
Similarly for $\epsilon$ positive and small, $p\in\mathbb{N}^{*}$,
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{p})\in(\mathbb{R}_{+})^{p}$,
$\lambda_{i}\geq\frac{1}{\epsilon}$ for $i=1,\cdots,p$,
$\bar{\alpha}:=(\alpha_{1},\cdots,\alpha_{p})\in\mathbb{R}^{p}$, $\alpha_{i}$
close to $1$ for $i=1,\cdots,p$, and $A:=(a_{1},\cdots,a_{p})\in(\partial
M)^{p}$ (where $\mathbb{R}^{p}$ denotes the cartesian product of $p$ copies of
$\mathbb{R}$, $O_{\bar{\alpha},A,\bar{\lambda},\epsilon}(1)$ will mean
quantities bounded from above and below independent of $\bar{\alpha}$, $A$,
$\bar{\lambda}$, and $\epsilon$. For $x\in\mathbb{R}$, we will use the
notation $O(x)$ to mean $|x|O(1)$ where $O(1)$ will be specified in all the
contexts where it is used. Large positive constants are usually denoted by $C$
and the value of $C$ is allowed to vary from formula to formula and also
within the same line. Similarly small positive constants are also denoted by
$c$ and their value may varies from formula to formula and also within the
same line.

We say $\mu\in\mathbb{R}$ is an eigenvalue of the $P_{g}^{4}$ to $P^{3}_{g}$
operator on $\mathcal{H}_{\frac{\partial}{\partial n}}$ if there exists $0\neq
v\in W^{2,2}(M)$ such that

(28) $\left\\{\begin{split}P^{4}_{g}v&=0\;\;&\text{in}\;\;M,\\\
P^{3}_{g}v&=\mu v\;\;&\text{on}\;\;\partial M,\\\ \frac{\partial v}{\partial
n_{g}}&=0\;\;&\text{on}\;\;\partial M.\end{split}\right.$

By abuse of notation, we call $v$ in (28) an eigenfunction associated to
$\mu$. We call $\bar{k}$ the number of negative eigenvalues (counted with
multiplicity) of the $P_{g}^{4}$ to $P^{3}_{g}$ operator on
$\mathcal{H}_{\frac{\partial}{\partial n}}$. We point out that $\bar{k}$ can
be zero, but it is always finite. If $\bar{k}\geq 1$, then we will denote by
$E_{-}\subset\mathcal{H}_{\frac{\partial}{\partial n}}$ the direct sum of the
eigenspaces corresponding to the negative eigenvalues of the $P_{g}^{4}$ to
$P^{3}_{g}$ operator on $\mathcal{H}_{\frac{\partial}{\partial n}}$ . The
dimension of $E_{-}$ is of course $\bar{k}$, i.e

(29) $\bar{k}=\dim E_{-}.$

On the other hand, we have the existence of a basis of eigenfunctions
$v_{1},\cdots,v_{\bar{k}}$ of $E_{-}$ satisfying

(30) $\left\\{\begin{split}P^{4}_{g}v_{r}&=0\;\;&\text{in}\;\;M,\\\
P^{3}_{g}v_{r}&=\mu_{r}v_{r}\;\;&\text{on}\;\;\partial M,\\\ \frac{\partial
v_{r}}{\partial n_{g}}&=0\;\;&\text{on}\;\;\partial M.\end{split}\right.$ (31)
$\mu_{1}\leq\mu_{2}\leq\cdots\leq\mu_{\bar{k}}<0<\mu_{\bar{k}+1}\leq\cdots,$

where $\mu_{r}$’s are the eigenvalues of the operator $P_{g}^{4}$ to
$P^{3}_{g}$ on $\mathcal{H}_{\frac{\partial}{\partial n}}$ counted with
multiplicity. We define $\mathbb{P}^{4,3}_{g,+}$ by

(32)
$\mathbb{P}^{4,3}_{g,+}(u,v)=\mathbb{P}^{4,3}_{g}(u,v)-2\sum_{r=1}^{\bar{k}}\mu_{r}\left(\oint_{\partial
M}uv_{r}dS_{g}\right)\left(\oint_{\partial M}vv_{r}dS_{g}\right).$

$\mathbb{P}^{4,3}_{g,+}$ is obtained by just reversing the sign of the
negative eigenvalue of $\mathbb{P}^{4,3}_{g}$. We set also

(33)
$||u||_{\mathbb{P}^{4,3}}:=\sqrt{\mathbb{P}^{4,3}_{g,+}(u,u)},\;\;\;\text{and}\;\;\;\left<u,v\right>_{\mathbb{P}^{4,3}}=P^{4,3}_{g,+}(u,v),$

where $\mathbb{P}^{4,3}_{g,+}$ is defined as in (32). We can choose
$v_{1},\cdots,v_{\bar{k}}$ so that they constitute a
$\left<\cdot,\cdot\right>_{\mathbb{P}^{4,3}}$-orthonormal basis fo $E_{-}$. We
denote by $\nabla^{\mathbb{P}^{4,3}}$ the gradient with respect to
$\left<\cdot,\cdot\right>_{\mathbb{P}^{4,3}}$.

For $t>0$, we define the following perturbed functional

(34)
$(\mathcal{E}_{g})_{t}(u):=\mathbb{P}^{4,3}(u,u)+4t\int_{M}Q_{g}udV_{g}+4t\oint_{\partial
M}T_{g}udS_{g}-\frac{4}{3}t\kappa_{(P^{4},P^{3})}\log\oint_{M}Ke^{3u}dS_{g},\;\;u\in\mathcal{H}_{\frac{\partial}{\partial
n}}.$

$\bar{B}^{\bar{k}}_{r}$ will stand for the closed ball of center $0$ and
radius $r$ in $\mathbb{R}^{\bar{k}}$. $\mathbb{S}^{\bar{k}-1}$ will denote the
boundary of $\bar{B}^{\bar{k}}_{1}$. Given a set $X$, we define
$\widetilde{X\times\bar{B}^{\bar{k}}_{1}}$ to be the cartesian product
$X\times\bar{B}^{\bar{k}}_{1}$ where the tilde means that $X\times\partial
B^{\bar{k}}_{1}$ is identified with $\partial B_{1}^{\bar{k}}$.

In the sequel also, $(\mathcal{E}_{g})^{c}$ with $c\in\mathbb{R}$ will stand
for $(\mathcal{E}_{g})^{c}:=\\{u\in\mathcal{H}_{\frac{\partial}{\partial
n}}:\;\;\mathcal{E}_{g}(u)\leq c\\}$. For $X$ a topological space, $H_{*}(X)$
will denote the singular homology of $X$, $H^{*}(X)$ for the cohomology, and
$\chi(X)$ the Euler characteristic of $X$, all with $\mathbb{Z}_{2}$
coefficients.

As above, in the general case, namely $\bar{k}\geq 0$, for $\epsilon$ small
and positive,
$\bar{\beta}:=(\beta_{1},\cdots,\beta_{\bar{k}})\in\mathbb{R}^{\bar{k}}$ with
$\beta_{i}$ close to $0$, $i=1,\cdots,\bar{k}$) (where $\mathbb{R}^{\bar{k}}$
is the empty set when $\bar{k}=0$),
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{p})\in(\mathbb{R}_{+})^{p}$,
$\lambda_{i}\geq\frac{1}{\epsilon}$ for $i=1,\cdots,p$,
$\bar{\alpha}:=(\alpha_{1},\cdots,\alpha_{p})\in\mathbb{R}^{p}$, $\alpha_{i}$
close to $1$ for $i=1,\cdots,p$, and $A:=(a_{1},\cdots,a_{p})\in(\partial
M)^{p}$, $p\in\mathbb{N}^{*}$, $w\in\mathcal{H}_{\frac{\partial}{\partial n}}$
with $||w||_{\mathbb{P}^{4,3}}$ small,
$O_{\bar{\alpha},A,\bar{\lambda},\bar{\beta},\epsilon}(1)$ will stand
quantities bounded independent of $\bar{\alpha}$, $A$, $\bar{\lambda}$,
$\bar{\beta}$, and $\epsilon$, and
$O_{\bar{\alpha},A,\bar{\lambda},\bar{\beta},w,\epsilon}(1)$ will stand
quantities bounded independent of $\bar{\alpha}$, $A$, $\bar{\lambda}$,
$\bar{\beta}$, $w$ and $\epsilon$.

For point $b\in\mathbb{R}^{3}$ and $\lambda$ a positive real number, we define
$\delta_{b,\lambda}$ by

(35)
$\delta_{b,\lambda}(y):=\log\left(\frac{2\lambda}{1+\lambda^{2}|y-b|^{2}}\right),\;\;\;\;\;\;y\in\mathbb{R}^{3}.$

The functions $\delta_{b,\lambda}$ verify the following equation

(36)
$(-\Delta_{\mathbb{R}^{3}})^{\frac{3}{2}}\delta_{b,\lambda}=2e^{3\delta_{b,\lambda}}\;\;\;\text{in}\;\;\;\mathbb{R}^{3}.$

Using the existence of conformal Fermi coordinates, we have that, for
$a\in\partial M$ there exists a function $u_{a}\in C^{\infty}(\overline{M})$
such that

(37)
$g_{a}=e^{2u_{a}}g\;\;\text{verifies}\;\;detg_{a}(x)=1+O(d_{g_{a}}(x,a)^{m})\;\;\text{for}\;\;\;x\in
B^{g_{a},+}_{a}(\varrho_{a}).$

with $0<\varrho_{a}<\frac{inj_{g_{a}}(\overline{M})}{10}$. Moreover, we can
take the families of functions $u_{a}$, $g_{a}$ and $\varrho_{a}$ such that

(38) $\text{the maps}\;\;\;a\longrightarrow
u_{a},\;g_{a}\;\;\text{are}\;\;C^{1}\;\;\;\text{and}\;\;\;\;\varrho_{a}\geq\varrho_{0}>0,$

for some small positive $\varrho_{0}$ satisfying
$\varrho_{0}<\inf\\{\frac{inj_{g}(M)}{10},\frac{inj_{\hat{g}}(\partial
M)}{10}\\}$, and

(39)
$\begin{split}&||u_{a}||_{C^{4}(\overline{M})}=O_{a}(1),\;\;\frac{1}{\overline{C}^{2}}g\leq
g_{a}\leq\overline{C}^{2}g,\\\
\;&u_{a}(x)=O_{a}(d^{2}_{\hat{g}_{a}}(a,x))=O_{a}(d_{\hat{g}}^{2}(a,x))\;\;\text{for}\;\;x\in\;\;B_{a}^{\hat{g}_{a}}(\varrho_{0})\supset
B_{a}(\frac{\varrho_{0}}{2\overline{C}}),\;\;\text{and}\\\
&u_{a}(a)=0,\;\;\;R_{\hat{g}_{a}}(a)=0,\;\;\;\frac{\partial u_{a}}{\partial
n_{g}}(a)=0,\end{split}$

for some large positive constant $\overline{C}$ independent of $a$. For
$a\in\partial M$, and $r>0$, we set

(40)
$exp_{a}^{a}:=exp_{a}^{\hat{g}_{a}}\;\;\;\;\text{and}\;\;\;B_{a}^{a}(r):=B_{a}^{\hat{g}_{a}}(r).$

Now, for $0<\varrho<\min\\{\frac{inj_{g}(\partial
M)}{4},\frac{\varrho_{0}}{4}\\}$ where $\varrho_{0}$ is as in (38), we define
a smooth cut-off function satisfying the following properties:

(41) $\begin{cases}\chi_{\varrho}(t)=t\;\;&\text{ for }\;\;t\in[0,\varrho],\\\
\chi_{\varrho}(t)=2\varrho\;\;&\text{ for }\;\;t\geq 2\varrho,\\\
\chi_{\varrho}(t)\in[\varrho,2\varrho]\;\;\;&\text{ for
}\;\;t\in[\varrho,2\varrho].\end{cases}$

Using the cut-off function $\chi_{\varrho}$, we define for $a\in\partial M$
and $\lambda\in\mathbb{R}_{+}$ the function $\hat{\delta}_{a,\lambda}$ as
follows

(42)
$\hat{\delta}_{a,\lambda}(x):=\log\left(\frac{2\lambda}{1+\lambda^{2}\chi_{\varrho}^{2}(d_{\hat{g}_{a}}(x,a))}\right).$

For every $a\in\partial M$ and $\lambda\in\mathbb{R}_{+}$, we define
$\varphi_{a,\lambda}$ to be the unique the solution of

(43)
$\begin{cases}P^{4}_{g}\varphi_{a,\lambda}+\frac{2}{k}Q_{g}=0\;\;\;\text{in}\;\;\;M,\\\
P_{g}^{3}\varphi_{a,\lambda}\,+\,\frac{1}{k}T_{g}\,=4\pi^{2}\,\frac{e^{3(\hat{\delta}_{a,\lambda}\,+\,u_{a})}}{\oint_{M}e^{3(\hat{\delta}_{a,\lambda}\,+\,u_{a})}dS_{g}}\;\;\mbox{
in }\;\;\partial M,\\\ \frac{\partial\varphi_{a,\lambda}}{n}=0,\\\
(\phi_{a,\lambda})_{(Q,T)}=0.\end{cases}$

Next, let $S(a,x)$ ; $(a,x)\in\partial M\times\overline{M}$ be defined by

(44)
$\begin{cases}P_{g}^{4}S(a,\cdot)+\frac{2}{k}Q_{g}(\cdot)=0\;\;\text{in}\;\;M,\\\
P^{3}_{g}S(a,\cdot)+\frac{1}{k}T_{g}(\cdot)=4\pi^{2}\delta_{a}(\cdot),\;\;\text{on}\;\;\partial
M,\\\ \frac{\partial S(a,\cdot)}{\partial n}=0\;\;\;\text{on}\;\;\partial
M,\\\ \int_{M}S(a,x)Q_{g}(x)dV_{g}(x)=0.\end{cases}$

Then

(45) $G(a,\cdot)=S(a,\cdot)|_{\partial M}.$

is a Green’s function of the $P^{4}_{g}+\frac{2}{k}Q_{g}(\cdot)$ to
$P^{3}_{g}+\frac{1}{k}T_{g}(\cdot)$ operator on
$\mathcal{H}_{\frac{\partial}{\partial n}}$. Thus, we have the integral
representation: $\forall u\in\mathcal{H}_{\frac{\partial}{\partial n}}$ such
that $P^{4}_{g}u+\frac{2}{k}Q_{g}=0$,

(46) $u(x)-\overline{u}_{(Q,T)}=\frac{1}{4\pi^{2}}\oint_{\partial
M}G(x,y)P_{g}^{3}u(y),\;\;\;x\in\partial M.$

Moreover, $G$ decomposes as follows

(47)
$G(a,x)=\log\left(\frac{1}{\chi_{\varrho}^{2}(d_{{\hat{g}}_{a}}(a,x))}\right)+H(a,x),$

where $H$ is the regular par of $G$. Furthermore, we have

(48) $G\in C^{\infty}((\partial M)^{2}-Diag(\partial
M)),\;\;\;\;\text{and}\;\;\;H\in C^{3,\beta}((\partial
M)^{2})\;\;\;\forall\beta\in(0,1).$

B symmetry of $H$, we have

(49) $\frac{\partial\mathcal{F}(a_{1},\cdots,a_{k})}{\partial
a_{i}}=\frac{2}{3}\frac{\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})}{\mathcal{F}^{A}_{i}(a_{i})},\;\;\;i=1,\cdots,k.$

Next, setting

(50)
$l_{K}(A):=\sum_{i=1}^{k}\left(\frac{\Delta_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})}{(\mathcal{F}^{A}_{i}(a_{i}))^{\frac{1}{3}}}-\frac{3}{4}R_{\hat{g}}(a_{i})(\mathcal{F}^{A}_{i}(a_{i}))^{\frac{2}{3}}\right),$

we have

(51) $l_{K}(A)=6\mathcal{L}_{K}(A),\;\;\forall A\in Crit(\mathcal{F}_{K}).$

For $k\geq 2$, we denote by $B_{k}(\partial M)$ the set of formal barycenters
of $\partial M$ of order $k$, namely

(52) $B_{k}(\partial
M):=\\{\sum_{i=1}^{k}\alpha_{i}\delta_{a_{i}},a_{i}\in\partial
M,\alpha_{i}\geq 0,i=1,\cdots,k,\;\,\sum_{i=1}^{k}\alpha_{i}=k\\},$

Finally, we set

(53) $A_{k,\bar{k}}:=\widetilde{B_{k}(\partial
M)\times\bar{B}^{\bar{k}}_{1}},$

and

(54) $A_{k-1,\bar{k}}:=\widetilde{B_{k-1}(\partial
M)\times\bar{B}^{\bar{k}}_{1}}.$

##  3 Blow-up analysis and critical points at infinity

This section deals with the blowup analysis of sequences of vanishing
viscosity solutions of the type

(55)
$\left\\{\begin{split}P^{4}_{g}u_{l}+2t_{l}Q_{g}&=0\;\;&\text{in}\;\;M,\\\
P^{3}_{g}u_{l}+t_{l}T_{g}&=t_{l}Ke^{3u}\;\;&\text{on}\;\;\partial M,\\\
\frac{\partial u_{l}}{\partial n_{g}}&=0\;\;&\text{on}\;\;\partial
M.\end{split}\right.$

with $t_{l}\rightarrow 1$ under the assumption
$\ker\mathbb{P}_{g}^{4,3}\simeq\mathbb{R}$ and
$\kappa_{(P^{4},P^{3})}=4k\pi^{2}$ with $k\geq 1$ and their use to
characterize the critical points at infinity of $\mathcal{E}_{g}$.

###  3.1 Blow-up analysis

The local behaviour of blowing up sequences of solutions of (55) is
understood. In fact, in [44], we prove the following lemma.

######  Lemma 3.1.

Assuming that $(u_{l})$ is a blowing up sequence of solutions to (55), then up
to a subsequence, there exists $k$ converging sequence of points
$(x_{i,l})_{l\in\mathbb{N}},x_{i,l}\in\partial M$ with limits
$x_{i}\in\partial M$, $i=1,\cdots,k$, $k$ sequences
$(\mu_{i,l})_{l\in\mathbb{N}}\;\;i=1,\cdots,k$ of positive real numbers
converging to $0$ such that the following hold:

a)

$\frac{d_{\hat{g}}(x_{i,l},x_{j,l})}{\mu_{i,l}}\longrightarrow+\infty\;\;\;\;i\neq
j\;\;i,j=1,\cdots,k\;\;\;\;\;and\;\;\;\;t_{l}K(x_{i,l})\mu_{i.l}^{3}e^{3u_{l}(x_{i,l})}e^{-3\log
2}=2.$

b)

$v_{i,l}(x)=u_{l}(exp^{g}_{x_{i,l}}(\mu_{i,l}x))-u_{l}(x_{i,l})+\log
2\longrightarrow V_{0}(x)\;\;\;\;in\;\;\;C^{4}_{loc}(\mathbb{R}^{4}_{+}),$
$V_{0}|_{\mathbb{R}^{3}}(x):=\log\left(\frac{2}{1+|x|^{2}}\right).$

c)

$\text{There exists}\;\;C>0\;\;\text{such
that}\;\;\inf_{i=1,\cdots,k}d_{\hat{g}}(x_{i,l},x)^{3}e^{3u_{l}(x)}\leq
C\;\;\;\;\forall x\in\partial M,\;\;\forall l\in\mathbb{N}.$

d)

$\begin{split}t_{l}Ke^{3u_{l}}dS_{g}\rightarrow
4\pi^{2}\sum_{i=1}^{k}\delta_{x_{i}}\;\;\;\text{in the sense of
measure},\;\;\;\text{and}\\\ \lim_{l\rightarrow+\infty}\oint_{\partial
M}t_{l}Ke^{3u_{l}}dS_{g}=4\pi^{2}k.\end{split}$

e)

$u_{l}-\overline{(u_{l})}_{Q,T}\rightarrow\sum_{i=1}^{k}G(x_{i},\cdot)\;\;\;\text{in}\;\;\;C^{3}_{loc}(\partial
M-\\{x_{1},\cdots,x_{k}\\}),\;\;\;\;\overline{(u_{l})}_{Q,T}\rightarrow-\infty.$

As a Liouville type problem, the following Harnack type inequality is
sufficient to get the global description of blowing up sequences of solutions
needed to describe the critical points at infinity of $\mathcal{E}_{g}$.

######  Proposition 3.2.

Assuming that $u_{l}$ is a blowing up sequence of solutions to (55), then
Lemma 3.1 holds, and keeping the notations in Lemma 3.1, we have that the
points $x_{i,l}$ are uniformly isolated, namely there exists
$0<\eta_{k}<\frac{\varrho_{0}}{10}$ (where $\varrho_{0}$ is as in (38)) such
that for $l$ large eneough, there holds

(56) $d_{\hat{g}}(x_{i,l},x_{j,l})\geq 4\overline{C}\eta_{k},\;\;\;\forall
i\neq j=1,\cdots,k.$

Moreover, the scaling parameters $\lambda_{i,l}:=\mu_{i,l}^{-1}$ are
comparable, namely there exists a large positive constant $\Lambda_{0}$ such
that

(57)
$\Lambda_{0}^{-1}\lambda_{j,l}\leq\lambda_{i,l}\leq\Lambda_{0}\lambda_{j,l},\;\;\forall\;i,j$

Furthermore, we have that the following estimate around the blow up points
holds

(58)
$u_{l}(y)+\frac{1}{3}\log\frac{t_{t}K_{l}(x_{i,l})}{2}=\log\frac{2\lambda_{i,l}}{1+\lambda_{i,l}^{2}(d_{\hat{g}_{x_{i,l}}}(y,x_{i,l}))^{2}}+O(d_{\hat{g}}(y,x_{i,l})),\;\;\;\;\forall\;y\in\;B^{\hat{g}}_{x_{i,l}}(\eta).$

To prove Proposition 3.2, as it is standard for Louiville type problems, one
starts with the uniform isolation of blowing-up points. Indeed, we have

######  Lemma 3.3.

Assuming that $(u_{l})_{l\in\mathbb{N}}$ is a bubbling sequence of solutions
to BVP (55), then keeping the notations in Lemma 3.1, we have that the points
$x_{i,l}$ are uniformly isolated, namely there exists
$0<\eta_{k}<\frac{\varrho_{0}}{10}$ (where $\varrho_{0}$ is as in (38)) such
that for $l$ large eneough, there holds

(59) $d_{\hat{g}}(x_{i,l},x_{j,l})\geq 4\overline{C}\eta_{k},\;\;\;\forall
i\neq j=1,\cdots,k.$

Proof. The proof use the integral method of Step 4 in [42] and hence we will
be skectchy in many arguments. As in [42], we first fix
$\frac{1}{3}<\nu<\frac{2}{3}$, and for $i=1,\cdots,k$, we set

$\bar{u}_{i,l}(r)=Vol_{\hat{g}}(\partial B_{x_{i}}(r))^{-1}\int_{\partial
B_{x_{i}}(r)}u_{l}(x)d\sigma_{\hat{g}}(x),\;\;\;\forall\;0\leq
r<inj_{\hat{g}}(\partial M),$

and

$\psi_{i,l}(r)=r^{4\nu}exp(4\bar{u}_{i,l}(r)),\;\;\;\forall\;0\leq
r<inj_{g}(\partial M).$

Furthermore, as in [42], we define $r_{i,l}$ as follows

(60) $r_{i,l}:=\sup\\{R_{\nu}\mu_{i,l}\leq
r\leq\frac{R_{i,l}}{2}\;\;\text{such
that}\;\;\psi_{i,l}^{{}^{\prime}}(r)<0\;\;\text{in}\;[R_{\nu}\mu_{i,l},r[\\};$

where $R_{i,l}:=\min_{j\neq i}d_{\hat{g}}(x_{i,l},x_{j,l})$. Thus, by
continuity and the definition of $r_{i,l}$, we have that

(61) $\psi_{i,l}^{{}^{\prime}}(r_{i,l})=0$

Now, as in [42], to prove (59), it suffices to show that $r_{i,l}$ is bounded
below by a positive constant in dependent of $l$. Thus, we assume by
contradiction that (up to a subsequence) $r_{i,l}\rightarrow 0$ as
$l\rightarrow+\infty$ and look for a contradiction. In order to do that, we
use the integral representation formula (46) and argue as in Step 4 of [42] to
derive the following estimate

$\psi^{{}^{\prime}}_{i,l}(r_{i,l})\leq(r_{i,l})^{3\nu-1}exp(\bar{u}_{i,l}(r_{i,l}))\left(3\nu-2C+o_{l}(1)+O_{l}(r_{i,l})\right).$

with $C>1$. So from $\frac{1}{3}<\nu<\frac{2}{3}$, $C>1$ and
$r_{i,l}\longrightarrow 0$ as $l\rightarrow+\infty$, we deduce that for $l$
large enough, there holds

(62) $\psi_{i,l}^{{}^{\prime}}(r_{i,l})<0.$

Thus, (61) and (62) lead to a contradiction, thereby concluding the proof of
(59). Hence, the proof of the Lemma is complete.

The next step to derive Proposition 3.2 is to establish its weak $O(1)$
-version.

######  Lemma 3.4.

Assuming that $(u_{l})_{l\in\mathbb{N}}$ is a bubbling sequence of solutions
to BVP (55), then keeping the notations in Lemma 3.1 and Lemma 3.3, we have
that for $l$ large enough, there holds

(63)
$u_{l}(x)+\frac{1}{3}\log\frac{t_{l}K(x_{i})}{2}=\log\frac{2\lambda_{i,l}}{1+\lambda_{i,l}^{2}(d_{\hat{g}_{x_{i}}}(x,x_{i}))^{2}}+O(1),\;\;\;\;\forall\;x\in\;B^{x_{i}}_{x_{i}}(\eta_{k}),$

up to choosing $\eta_{k}$ smaller than in Lemma 3.3.

######  Remark 3.5.

We point out that the comparability of the scaling parameters
$\lambda_{i,l}$’s follows directly from Lemma 3.4.

Proof. We are going to use the method of [47], hence we will be sketchy in
many arguments. Like in [47], thanks to Lemma 3.3, we will focus only on one
blow-up point and called it $x\in\partial M$. Thus, we are in the situation
where there exists a sequence $x_{l}\in\partial M$ such that $x_{l}\rightarrow
x$ with $x_{l}$ local maximum point for $u_{l}$ on $\partial M$ and
$u_{l}(x_{l})\rightarrow+\infty$. Now, we recall $g_{x}=e^{2u_{x}}g$ and
choose $\eta_{1}$ such that $20\eta_{1}<\min\\{inj_{g}(\partial
M),\varrho_{0},\varrho_{k},d\\}$ with $4d\leq r_{i,l}$ where $r_{i,l}$ is as
in the proof of Lemma 3.3. Next, we let $\hat{w}_{x}$ be the unique solution
of the following boundary value problem

(64)
$\left\\{\begin{split}P^{4}_{g_{x}}\hat{w}_{x}&=P^{4}_{\hat{g}}u_{x}\;\;&\text{in}\;\;M,\\\
P^{3}_{g_{x}}\hat{w}_{x}&=P^{3}_{\hat{g}}u_{x}\;\;&\text{on}\;\;\partial M,\\\
\frac{\partial\hat{w}_{x}}{\partial n_{g_{x}}}&=0\;\;&\text{on}\;\;\partial
M,\\\ \overline{\hat{w}}_{(Q,T)}&=0.\end{split}\right.$

Using standard elliptic regularity theory and (39), we derive

(65)
$\hat{w}(y)=O(d_{g}(y,x))\;\;\;\text{in}\;\;\;\;B_{x}^{g_{x},+}(2\eta_{1}).$

On the other hand, using the conformal covariance properties of the Paneitz
operator and of the Chang-Qing one, see (1), we have that
$\hat{u}_{l}:=u_{l}-\hat{w}_{x}$ satisfies

$\left\\{\begin{split}P^{4}_{g_{x}}\hat{u}_{l}+2\hat{Q}_{l}&=0\;\;&\text{in}\;\;M,\\\
P^{3}_{g_{x}}\hat{u}_{l}+\hat{T}_{l}&=t_{l}Ke^{3\hat{u}_{l}}\;\;&\text{on}\;\;\partial
M,\\\ \frac{\partial\hat{u}_{l}}{\partial
n_{g_{x}}}&=0\;\;&\text{on}\;\;\partial M.\end{split}\right.$

with

$\hat{Q}_{l}=t_{l}e^{-4\hat{w}}Q_{g}+\frac{1}{2}P^{4}_{\hat{g}}\hat{w}\;\;\;\text{and}\;\;\;\hat{T}_{l}=t_{l}e^{-3\hat{w}}T_{g}+P^{3}_{\hat{g}}\hat{w}.$

Next, as in [47], we are going to establish the classical sup+inf-estimate for
$\hat{u}_{l}$ , since thanks (65) all terms coming from $\hat{w}_{x}$ can be
absorbed on the right hand side of (63). Now, we are going to rescale the
functions $\hat{u}_{l}$ around the points $x$. In order to do that, we define
$\varphi_{l}:B^{\mathbb{R}^{3}}_{0}(2\eta_{1}\mu_{l}^{-1})\longrightarrow
B^{\hat{g}_{x}}_{x}(2\eta_{1})$ by the formula $\varphi_{l}(z):=\mu_{l}z$ and
$\mu_{l}$ is the corresponding scaling parameter given by Lemma 3.1.
Furthermore, as in [47], we define the following rescaling of $\hat{u}_{l}$

$v_{l}:=\hat{u_{l}}\circ\varphi_{l}+\log\mu_{l}+\frac{1}{3}\log\frac{t_{l}K(x)}{2}.$

Using the Green’s representation formula for and the method of the method of
[47], we get

(66)
$v_{l}(z)+2\log|z|=O(1),\;\;\text{for}\;\;z\in\bar{B}^{\mathbb{R}^{3}}_{0}(\frac{\eta_{1}}{\mu_{l}})-B^{\mathbb{R}^{3}}_{0}(-\log\mu_{l}).$

Now, we are going to show that the estimate (66) holds also in
$\bar{B}^{\mathbb{R}^{3}}_{0}(-\log\mu_{l})$. To do so, we use Lemma 3.1and
the same arguments as in [47] to deduce

(67)
$v_{l}(z)+2\log|z|=O(1),\;\;\text{for}\;\;\;z\in\bar{B}^{\mathbb{R}^{3}}_{0}(-\log\mu_{l}).$

Now, combining (66) and (67), we obtain

(68)
$v_{l}(z)+2\log|z|=O(1),\;\;\text{for}\;\;\;z\in\bar{B}^{\mathbb{R}^{3}}_{0}(\frac{\eta_{1}}{\mu_{l}}).$

Thus scaling back, namely using $y=\mu_{l}z$ and the definition of $v_{l}$, we
obtain the desired $O(1)$-estimate. Hence the proof of the Lemma is complete.

Proof of formula (58) of Proposition 3.2
We are going to use the method of [47], hence we will be sketchy in many
arguments. Now, let $V_{0}$ be the unique solution of the following
conformally invariant integral equation

$V_{0}(z)=\frac{1}{2\pi^{2}}\int_{\mathbb{R}^{3}}\log\frac{|y|}{|z-y|}e^{3V_{0}(y)}dy+\log
2,\;\;\;V_{0}(0)=\log 2,\nabla V_{0}(0)=0.$

Next, we set $w_{l}(z)=v_{l}(z)-V_{0}(z)$ for $z\in
B^{\mathbb{R}^{3}}_{0}(\eta_{1}\mu_{l}^{-1})$, and use Lemma 3.4 to infer that

(69) $|w_{l}|\leq
C\;\;\;\,\text{in}\;\;\;\;B^{\mathbb{R}^{3}}_{0}(\eta_{1}\mu_{l}^{-1})$

On the other hand, it is easy to see that to achieve our goal, it is
sufficient to show

(70) $|w_{l}|\leq
C\mu_{l}|z|\,\,\,\,\text{in}\,\,\,\,B^{\mathbb{R}^{3}}_{0}(\eta_{1}\mu_{l}^{-1}).$

To show (70), we first set

$\Lambda_{l}:=\max_{z\in\Omega_{l}}\frac{|w_{l}(z)|}{\mu_{l}(1+|z|)}$

with

$\Omega_{l}=\overline{B}^{\mathbb{R}^{3}}_{0}(\eta_{1}\mu_{l}^{-1})$

We remark that to show (70), it is equivalent to prove that $\Lambda_{l}$ is
bounded. Now, let us suppose that $\Lambda_{l}\rightarrow+\infty$ as
$l\rightarrow+\infty$, and look for a contradiction. To do so, we will use the
method of [47]. For this, we first choose a sequence of points
$z_{l}\in\Omega_{l}$ such that
$\Lambda_{l}=\frac{|w_{l}(z_{l})|}{\mu_{l}(1+|z_{l}|)}$. Next, up to a
subsequence, we have that either $z_{l}\rightarrow z^{*}$ as
$l\rightarrow+\infty$ (with $z^{*}\in\mathbb{R}^{3}$) or
$|z_{l}|\rightarrow+\infty$ as $l\rightarrow+\infty$. Now, we make the
following definition

$\bar{w}_{l}(z):=\frac{w_{l}(z)}{\Lambda_{l}\mu_{l}(1+|z_{l}|)},$

and have

(71) $|\bar{w}_{l}(z)|\leq\left(\frac{1+|z|}{1+|z_{l}|}\right),$

and

(72) $|\bar{w}_{l}(z_{l})|=1.$

Now, we consider the case where the points $z_{l}$ escape to infinity.

Case 1 : $|z_{l}|\rightarrow+\infty$
In this case, using the integral representation (46) with respect to $g_{x}$
and the method of [47], we obtain

$\bar{w}_{l}(z_{l})=\frac{1}{2\pi^{2}}\int_{\Omega_{l}}\log\frac{|\xi|}{|z_{l}-\xi|}\left(\frac{O(1)(1+|\xi|)^{-5}}{(1+|z_{l}|)}+\frac{O(1)(1+|\xi|)^{-5}}{\Lambda_{l}(1+|z_{l}|)}\right)d\xi+o(1).$

Now, using the fact that $|z_{l}|\rightarrow+\infty$ as $l\rightarrow+\infty$,
one can easily check that

$\bar{w}_{l}(z_{l})=\frac{1}{2\pi^{2}}\int_{\Omega_{l}}\log\frac{|\xi|}{|z_{l}-\xi|}\left(\frac{O(1)(1+|\xi|)^{-5}}{(1+|z_{l}|)}+\frac{O(1)(1+|\xi|)^{-5}}{\Lambda_{l}(1+|z_{l}|)}\right)d\xi=o(1).$

Hence, we reach a contradiction to (72).
Now, we are going to show that, when the points $z_{l}\rightarrow z^{*}$ as
$l\rightarrow+\infty$, we reach a contradiction as well.

Case 2: $z_{l}\rightarrow z^{*}$
In this case, using the assumption $z_{l}\longrightarrow z^{*}$, the Green’s
representation formula, and the method of [47], we obtain that up to a
subsequence

(73) $\bar{w}_{l}\rightarrow
w\;\;\;\text{in}\;\;C^{1}_{loc}(\mathbb{R}^{3})\;\;\;\text{as}\;\;\;l\rightarrow+\infty,\;\;\>\
$

and

(74)
$\begin{split}\bar{w}_{l}(z)=\frac{1}{2\pi^{2}}\int_{\Omega_{l}}\log\frac{|\xi|}{|z-\xi|}\frac{K\circ\varphi_{l}(\xi)}{K\circ\varphi_{l}(0)}e^{3\vartheta_{l}(\xi)}\bar{w}(\xi)d\xi+\frac{1}{\Lambda_{l}\mu_{l}(1+|z_{l}|)2\pi^{2}}\int_{\Omega_{l}}\log\frac{|\xi|}{|z-\xi|}O(\mu_{l}(1+|\xi|)^{-5})d\xi\\\
+\frac{O(1)+O(|z|)}{\Lambda_{l}(1+|z_{l}|)},\end{split}$

where $e^{3\theta_{l}}:=\int_{0}^{1}e^{3(sv_{l}+(1-s)V_{0})}ds$. Thus,
appealing to (73) and (74), we infer that $w$ satisfies

(75)
$w(z)=\frac{1}{2\pi^{2}}\int_{\mathbb{R}^{3}}\log\frac{|\xi|}{|z-\xi|}e^{3V_{0}(\xi)}w(\xi)d\xi$

Now, using (71), we have that $w$ satisfies the following asymptotics

(76) $|w(z)|\leq C(1+|z|).$

On the other hand, from the definition of $v_{l}$, it is easy to see that

(77) $w(0)=0,\;\;\;\text{and}\;\;\;\;\nabla w(0)=0.$

So, using (75)-(77), and observing that Lemma 3.7 in [47] holds for dimension
$3$, , we obtain

$w=0.$

However, from (72), we infer that $w$ satisfies also

(78) $|w(z^{*})|=1$

So we reach a contradiction in the second case also. Hence the proof of the
lemma is complete.

Because of the lack of understanding of the blowing PS-sequences for Louiville
type problems, the role of the PS-sequences can be replaced by the vanishing
viscosity solutions of the type of (55) via the following Bahri-Lucia’s
deformation lemma.

######  Lemma 3.6.

Assuming that $a,\;b\in\mathbb{R}$ such that $a<b$ and there is no critical
values of $\mathcal{E}_{g}$ in $[a,b]$, then there are two possibilities
1) Either

$(\mathcal{E}_{g})^{a}\;\text{is a deformation retract
of}\;\;(\mathcal{E}_{g})^{b}.$

2) Or there exists a sequence $t_{l}\rightarrow 1$ as $l\rightarrow+\infty$
and a sequence of critical point $u_{l}$ of $(\mathcal{E}_{g})_{t_{l}}$
verifying $a\leq\mathcal{E}_{g}(u_{l})\leq b$ for all $l\in\mathbb{N}^{*}$,
where $(\mathcal{E}_{g})_{t_{l}}$ is as in (34) with $t$ replaced by $t_{l}$.

On the other hand, setting

(79)
$\begin{split}V_{R}(k,\epsilon,\eta):=\\{u\in\mathcal{H}_{\frac{\partial}{\partial
n}}:\;\;\exists a_{1},\cdots,a_{k}\in\partial
M,\;\;\lambda_{1},\cdots,\lambda_{k}>0,\;\;||u-\overline{u}_{Q,T}-\sum_{i=1}^{k}\varphi_{a_{i},\lambda_{i}}||_{\mathbb{P}^{4,3}}=\\\
O\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}}\right)\;\;\;\lambda_{i}\geq\frac{1}{\epsilon},\;\;\frac{2}{\Lambda}\leq\frac{\lambda_{i}}{\lambda_{j}}\leq\frac{\Lambda}{2},\;\;\text{and}\;\;d_{\hat{g}}(a_{i},a_{j})\geq
4\overline{C}\eta\;\;\text{for}\;i\neq j\\},\end{split}$

where $\overline{C}$ is as in (39), $L$ as in (20), ,
$O(1):=O_{A,\bar{\lambda},u,\epsilon}(1)$ meaning bounded uniformly in
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$, $A:=(a_{1},\cdots,a_{k})$,
$u$, $\epsilon$, we have as in [47] that Proposition 3.2 implies the following
one.

######  Lemma 3.7.

Let $\epsilon$ and $\eta$ be small positive real numbers with
$0<2\eta<\varrho$ where $\varrho$ is as in (41). Assuming that $u_{l}$ is a
sequence of blowing up critical point of $(\mathcal{E}_{g})_{t_{l}}$ with
$\overline{(u_{l})}_{Q,T}=0,l\in\mathbb{N}$ and $t_{l}\rightarrow 1$ as
$l\rightarrow+\infty$ , then there exists $l_{\epsilon,\eta}$ a large positive
integer such that for every $l\geq l_{\epsilon,\eta}$, we have $u_{l}\in
V_{R}(k,\epsilon,\eta)$, and for the definition of $V_{R}(k,\epsilon,\eta)$,
see (79).

Finally, as in [47], we have that Lemma 3.6 and Lemma 3.7 implies the
following one.

######  Lemma 3.8.

Assuming that $\epsilon$ and $\eta$ are small positive real numbers with
$0<2\eta<\varrho$, then for $a,b\in\mathbb{R}$ such that $a<b$, we have that
if there is no critical values of $\mathcal{E}_{g}$ in $[a,b]$, then there are
two possibilities
1) Either

$(\mathcal{E}_{g})^{a}\;\text{is a deformation retract
of}\;\;(\mathcal{E}_{g})^{b}.$

2) Or there exists a sequence $t_{l}\rightarrow 1$ as $l\rightarrow+\infty$
and a sequence of critical point $u_{l}$ of $(\mathcal{E}_{g})_{t_{l}}$ (for
its definition see (34)) verifying $a\leq\mathcal{E}_{g}(u_{l})\leq b$ for all
$l\in\mathbb{N}^{*}$ and $l_{\epsilon,\eta}$ a large positive integer such
that $u_{l}\in V_{R}(k,\epsilon,\eta)$ for all $l\geq l_{\epsilon,\eta}$, and
for the definition of $V_{R}(k,\epsilon,\eta)$, see (79).

###  3.2 Energy and gradient estimates at infinity

In this subsection, we present energy and gradient estimates needed to
characterize the critical points at infinity of $\mathcal{E}_{g}$. We start
with a parametrization of infinity. Indeed, as a Liouville type problem, we
have that for $\eta$ a small positive real number with $0<2\eta<\varrho$,
there exists $\epsilon_{0}=\epsilon_{0}(\eta)>0$ such that
$\forall\;0<\epsilon\leq\epsilon_{0}$, we have

(80) $\forall u\in V_{R}(k,\epsilon,\eta),\\\ \text{the minimization problem
}\min_{B_{\epsilon,\eta}}||u-\overline{u}_{Q,T}-\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}-\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{Q,T})||_{\mathbb{P}^{4,3}}$

has a unique solution, up to permutations, where $B_{\epsilon,\eta}$ is
defined as follows

(81)
$\begin{split}{B_{\epsilon,\eta}:=\\{(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in\mathbb{R}^{k}\times(\partial
M)^{k}\times(0,+\infty)^{k}\times\mathbb{R}^{\bar{k}}}:|\alpha_{i}-1|\leq\epsilon,\lambda_{i}\geq\frac{1}{\epsilon},i=1,\cdots,k,\\\
d_{\hat{g}}(a_{i},a_{j})\geq 4\overline{C}\eta,i\neq j,|\beta_{r}|\leq
R,r=1,\cdots,\bar{k}\\}.\end{split}$

Moreover, using the solution of (80), we have that every $u\in
V_{R}(k,\epsilon,\eta)$ can be written as

(82)
$u-\overline{u}_{Q,T}=\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{Q,T})+w,$

where $w$ verifies the following orthogonality conditions

(83)
$\begin{split}\overline{w}_{Q,T}=\left\langle\varphi_{a_{i},\lambda_{i}},w\right\rangle_{\mathbb{P}^{4,3}}=\left\langle\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial\lambda_{i}},w\right\rangle_{\mathbb{P}^{4,3}}=\left\langle\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial
a_{i}},w\right\rangle_{\mathbb{P}^{4,3}}=\left\langle
v_{r},w\right\rangle_{\mathbb{P}^{4,3}}=0,i=1,\cdots,k,\\\
r=1,\cdots,\bar{k}\end{split}$

and the estimate

(84)
$||w||_{\mathbb{P}^{4,3}}=O\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}}\right),$

where here
$O\left(1\right):=O_{\bar{\alpha},A,\bar{\lambda},\bar{\beta},w,\epsilon}\left(1\right)$.
Furthermore, the concentration points $a_{i}$, the masses $\alpha_{i}$, the
concentrating parameters $\lambda_{i}$ and the negativity parameter
$\beta_{r}$ in (82) verify also

(85) $\begin{split}d_{\hat{g}}(a_{i},a_{j})\geq 4\overline{C}\eta,\;i\neq
j=1,\cdots,k,\frac{1}{\Lambda}\leq\frac{\lambda_{i}}{\lambda_{j}}\leq\Lambda\;\;i,j=1,\cdots,k,\;\;\lambda_{i}\geq\frac{1}{\epsilon},\;\;\text{and}\\\
\;\;\;\sum_{r=1}^{\bar{k}}|\beta_{r}|+\sum_{i=1}^{k}|\alpha_{i}-1|\sqrt{\log\lambda_{i}}=O\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}}\right)\end{split}$

with still $O\left(1\right)$ as in (84).

Because of the translation invariant property of $\mathcal{E}_{g}$ and the
parametrization (82), to derive energy estimate in $V_{R}(k,\epsilon,\eta)$ we
start with the following lemma.

######  Lemma 3.9.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), then for $a_{i}\in M$ concentration points,
$\alpha_{i}$ masses, $\lambda_{i}$ concentration parameters ($i=1,\cdots,k$),
and $\beta_{r}$ negativity parameters ($r=1,\cdots,\bar{k}$) satisfying (85),
we have

$\begin{split}&\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=^{1}}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{Q,T})\right)=C^{k}_{0}-8\pi^{2}\mathcal{F}_{K}(a_{1},\dots,a_{k})+2\sum_{r=1}^{\bar{k}}\mu_{r}\beta_{r}^{2}\\\
&+\sum_{i=1}^{k}(\alpha_{i}-1)^{2}\left[16\pi^{2}\log\lambda_{i}+8\pi^{2}H(a_{i},a_{i})+C^{k}_{1}\right]\\\
&+8\pi^{2}\sum_{i=1}^{k}(\alpha_{i}-1)\left[\sum_{r=1}^{\bar{k}}2\beta_{r}(v_{r}-\overline{(v_{r})}_{Q,T})(a_{i})+\sum_{j=1,j\neq
i}^{k}(\alpha_{j}-1)G(a_{i},a_{j})\right]\\\
&-\frac{c^{1}8\pi^{2}}{9}\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}\left(\frac{\Delta_{\hat{g}_{a_{i}}}\mathcal{F}^{A}_{i}(a_{i})}{\mathcal{F}^{A}_{i}(a_{i})}-\frac{3}{4}R_{\hat{g}}(a_{i})\right)\\\
&+\frac{c^{1}8\pi^{2}}{9}\sum_{i=1}^{k}\frac{\tilde{\tau}_{i}}{\lambda_{i}^{2}}\left(\frac{\Delta_{\hat{g}_{a_{i}}}\mathcal{F}^{A}_{i}(a_{i})}{\mathcal{F}^{A}_{i}(a_{i})}-\frac{3}{4}R_{\hat{g}}(a_{i})\right)\\\
&+\frac{16\pi^{2}}{3}\sum_{i=1}^{k}\log(1-\tilde{\tau}_{i})+O\left(\sum_{i=1}^{k}|\alpha_{i}-1|^{3}+\sum_{r=1}^{\bar{k}}|\beta_{r}|^{3}+\sum_{i=1}^{k}\frac{1}{\lambda^{3}_{i}}\right),\end{split}$

where $O\left(1\right)$ means here
$O_{\bar{\alpha},A,\bar{\lambda},\bar{\beta},\epsilon}\left(1\right)$ with
$\bar{\alpha}=(\alpha_{1},\cdots,\alpha_{k})$, $A:=(a_{1},\cdots,a_{k})$,
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$,
$\bar{\beta}:=(\beta_{1},\cdots,\beta_{\bar{k}})$ and for $i=1,\cdots,k$,

$\tilde{\tau}_{i}:=1-\frac{k\tilde{\gamma}_{i}}{\Gamma},\;\;\;\;\Gamma:=\sum_{i=1}^{k}\tilde{\gamma}_{i},\;\;\;\tilde{\gamma}_{i}:=\tilde{c}_{i}\lambda_{i}^{6\alpha_{i}-3}\mathcal{F}^{A}_{i}(a_{i})\mathcal{G}_{i}(a_{i}),$

with

$\tilde{c}_{i}:=\int_{\mathbb{R}^{3}}\frac{1}{(1+|y|^{2})^{3\alpha_{i}}}dy$
$\begin{split}\mathcal{G}_{i}(a_{i}):=e^{3((\alpha_{i}-1)H(a_{i},a_{i})+\sum_{j=1,j\neq
i}^{k}(\alpha_{j}-1)G(a_{j},a_{i}))}e^{\frac{3}{2}\sum_{j=1,j\neq
i}^{k}\frac{\alpha_{j}}{\lambda_{j}^{2}}\Delta_{g_{a_{j}}}G(a_{j},a_{i})}e^{\frac{3}{2}\frac{\alpha_{i}}{\lambda_{i}^{2}}\Delta_{g_{a_{i}}}H(a_{i},a_{i})}\\\
\times e^{3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}(a_{i})},\end{split}$

$C_{0}^{k}$ is a real number depending only on $k$, $C_{1}^{k}$ is a real
number and $c^{1}$ is a positive real number and for the meaning of
$O_{\bar{\alpha},A,\bar{\lambda},\bar{\beta},\epsilon}\left(1\right)$.

Proof. The proof is the same as the one Lemma 4.1 in [1] replacing Lemma 10.1-
Lemma 10.4 in [1] by Lemma 5.1-Lemma 5.4.

Concerning the gradient estimates of $\mathcal{E}_{g}$ in
$V_{R}(k,\epsilon,\eta)$, we have in the directions of the scaling parameters
:

######  Lemma 3.10.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), then for $a_{i}\in\partial M$ concentration
points, $\alpha_{i}$ masses , $\lambda_{i}$ concentration parameters
($i=1,\cdots,k$) and $\beta_{r}$ negativity parameters ($r=1,\cdots,\bar{k}$)
satisfying (85), we have that for every $r=1,\cdots,k$, there holds

$\begin{split}&\left\langle\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{Q,T})\right),\lambda_{j}\frac{\partial\varphi_{a_{j},\lambda_{j}}}{\partial\lambda_{j}}\right\rangle_{\mathbb{P}^{4,3}}=16\pi^{2}\alpha_{j}\tau_{j}\\\
&-\frac{c^{2}8\pi^{2}}{3\lambda_{j}^{2}}\left(\frac{\Delta_{\hat{g}_{a_{j}}}\mathcal{F}^{A}_{j}(a_{j})}{\mathcal{F}^{A}_{j}(a_{j})}-\frac{3}{4}R_{\hat{g}}(a_{j})\right)-\frac{16\pi^{2}}{\lambda_{j}^{2}}\tau_{j}\Delta_{\hat{g}_{a_{j}}}H(a_{j},a_{j})\\\
&-\frac{16\pi^{2}}{\lambda_{j}^{2}}\sum_{i=1,i\neq
j}^{k}\tau_{i}\Delta_{\hat{g}_{a_{j}}}G(a_{j},a_{i})+\frac{c^{2}8\pi^{2}}{\lambda_{j}^{2}}\tau_{j}\left(\frac{\Delta_{\hat{g}_{a_{j}}}\mathcal{F}^{A}_{j}(a_{j})}{\mathcal{F}^{A}_{j}(a_{j})}-\frac{3}{4}R_{\hat{g}}(a_{j})\right)\\\
&+O\left(\sum_{i=1}^{k}|\alpha_{i}-1|^{2}+\sum_{r=1}^{\bar{k}}|\beta_{r}|^{3}+\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{3}}\right),\end{split}$

where $A:=(a_{1},\cdots,a_{k})$, $O\left(1\right)$ is as in Lemma 3.9, $c^{2}$
is a positive real number, and for $i=1,\cdots,k$,

$\tau_{i}:=1-\frac{k\tilde{\gamma}_{i}}{D},\;\;\;\;\;D:=\oint_{\partial
M}K(x)e^{3(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}(x)+\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}(x))}dS_{g}(x),$

with $\tilde{\gamma}_{i}$ as in Lemma 3.9.

Proof. The proof is the same as the one Lemma 5.1 in [1] replacing Lemma 10.1-
Lemma 10.4 in [1] by Lemma 5.1-Lemma 5.4.

As in [1], Lemma 3.10 implies the following corollary.

######  Corollary 3.11.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), then for $a_{i}\in\partial M$ concentration
points, $\alpha_{i}$ masses, $\lambda_{i}$ concentration parameters
($i=1,\cdots,k$), and $\beta_{r}$ negativity parameters ($r=1,\cdots,\bar{k}$)
satisfying (85), we have

$\begin{split}&\left\langle\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\right),\sum_{i=1}^{k}\frac{\lambda_{i}}{\alpha_{i}}\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial\lambda_{i}}\right\rangle_{\mathbb{P}^{4,3}}=\\\
&\sum_{i=1}^{k}\frac{c^{3}8\pi^{2}}{\lambda_{i}^{2}}\left(\frac{\Delta_{\hat{g}_{a_{i}}}\mathcal{F}^{A}_{i}(a_{i})}{\mathcal{F}^{A}_{i}(a_{i})}-\frac{3}{4}R_{\hat{g}}(a_{i})\right)\\\
&+O\left(\sum_{i=1}^{k}|\alpha_{i}-1|^{2}+\sum_{r=1}^{\bar{k}}|\beta_{r}|^{3}+\sum^{k}_{i=1}\tau^{3}_{i}+\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{3}}\right),\end{split}$

where $A:=(a_{1},\cdots,a_{k})$, $O\left(1\right)$ as as in Lemma 3.9, $c^{3}$
is a positive real number, and for $i=1,\cdots,k$, $\tau_{i}$ is as in Lemma
3.10.

Proof. The proof uses the strategy of the proof of Corollary 5.2 in [1]
replacing Lemma 5.1 in [1] by its counterpart Lemma 3.10.

For the gradient estimate in the directions of mass concentrations, we have:

######  Lemma 3.12.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), then for $a_{i}\in\partial M$ concentration
points, $\alpha_{i}$ masses, $\lambda_{i}$ concentration parameters
($i=1,\cdots,k$), and $\beta_{r}$ negativity parameters ($r=1,\cdots,\bar{k}$)
satisfying (85), we have that for every $j=1,\cdots,k$, there holds

$\begin{split}&\left\langle\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\right),\varphi_{a_{j},\lambda_{j}}\right\rangle_{\mathbb{P}^{4,3}}=\\\
&\left(2\log\lambda_{j}+H(a_{j},a_{j})-C_{2}\right)\frac{1}{\alpha_{j}}\left\langle\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\right),\lambda_{j}\frac{\partial\varphi_{a_{j},\lambda_{j}}}{\partial\lambda_{j}}\right\rangle_{\mathbb{P}^{4,3}}\\\
&+\sum_{i=1,i\neq
j}^{k}G(a_{j},a_{i})\left\langle\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\right),\lambda_{i}\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial\lambda_{i}}\right\rangle_{\mathbb{P}^{4,3}}\\\
&+32\pi^{2}(\alpha_{j}-1)\log\lambda_{j}+O\left(\log\lambda_{j}\left(\sum_{i=1}^{k}\frac{|\alpha_{i}-1|}{\log\lambda_{i}}+(\sum_{r=1}^{\bar{k}}|\beta_{r}|)(\sum_{i=1}^{k}\frac{1}{\log\lambda_{i}})+\sum_{i=1}^{k}\frac{1}{\lambda^{2}_{i}}\right)\right),\end{split}$

where $O\left(1\right)$ as as in Lemma 3.9 and $C_{2}$ is a real number.

Proof. It follows from the same arguments as in Lemma 5.3 in [1].

Concerning the gradient estimate in the directions of points of
concentrations, we have:

######  Lemma 3.13.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), then for $a_{i}\in\partial M$ concentration
points, $\alpha_{i}$ masses, $\lambda_{i}$ concentration parameters
($i=1,\cdots,k$), and $\beta_{r}$ negativity parameters ($r=1,\cdots,\bar{k}$)
satisfying (85), we have that for every $j=1,\cdots,k$, there holds

$\begin{split}\left\langle\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\right),\frac{1}{\lambda_{j}}\frac{\partial\varphi_{a_{j},\lambda_{j}}}{\partial
a_{j}}\right\rangle_{\mathbb{P}^{4,3}}&=-\frac{c^{2}32\pi^{2}}{\lambda_{j}}\frac{\nabla_{\hat{g}}\mathcal{F}_{j}^{A}(a_{j})}{\mathcal{F}_{j}^{A}(a_{j})}\\\
&+O\left(\sum_{i=1}^{k}|\alpha_{i}-1|^{2}\right)\\\
&+O\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}+\sum_{r=1}^{\bar{k}}|\beta_{r}|^{2}+\sum_{i=1}^{k}\tau_{i}^{2}\right),\end{split}$

where $A:=(a_{1},\cdots,a_{k})$, $O(1)$ is as in Lemma 3.9, $c^{2}$ is as in
Lemma 3.10 and for $i=1,\cdots,k$, $\tau_{i}$ is as in Lemma 3.10.

Proof. The proof is the same as the one of Lemma 5.4 in [1].

Concerning the gradient estimate in the directions of the negativity
parameters, we have:

######  Lemma 3.14.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), then for $a_{i}\in\partial M$ concentration
points, $\alpha_{i}$ masses, $\lambda_{i}$ concentration parameters
($i=1,\cdots,k$), ad $\beta_{r}$ negativity parameters ($r=1,\cdots,\bar{k}$)
satisfying (85), we have that for every $l=1,\cdots,\bar{k}$, there holds

$\begin{split}\left\langle\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\right),v_{l}-\overline{(v_{l})}_{Q,T}\right\rangle_{\mathbb{P}^{4,3}}=&4\mu_{l}\beta_{l}+O\left(\sum_{i=1}^{k}|\alpha_{i}-1|+\sum_{i=1}^{k}|\tau_{i}|\right)\\\
&+O\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}\right),\end{split}$

where $O(1)\;$ is as in Lemma 3.9 and for $i=1,\cdots,k$, $\tau_{i}$ is as in
Lemma 3.10

Proof. It follows from the same arguments as in the proof of Lemma 5.5 in [1].

###  3.3 Finite-dimensional reduction

In this subsection, we complete the energy estimate of $\mathcal{E}_{g}$ on
$V_{R}(k,\epsilon,\eta)$ via Lyapunov finite dimensional type reduction and
second variation arguments. First of all, we have:

######  Proposition 3.15.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80) and
$u=\overline{u}_{Q,T}+\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+w\in
V_{R}(k,\epsilon,\eta)$ with $w$, the concentration points $a_{i}$, the masses
$\alpha_{i}$, the concentrating parameters $\lambda_{i}$ ($i=1,\cdots,k$), and
the negativity parameters $\beta_{r}$ ($r=1,\cdots,\bar{k}$) verifying
(83)-(85), then we have

(86)
$\mathcal{E}_{g}(u)=\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\right)-f(w)+Q(w)+o(||w||_{\mathbb{P}^{4,3}}^{2}),$

where

(87) $f(w):=16\pi^{2}\frac{\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}wdS_{g}}{\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}dS_{g}},$

and

(88) $Q(w):=||w||_{\mathbb{P}^{4,3}}^{2}-24\pi^{2}\frac{\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}w^{2}dS_{g}}{\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}dS_{g}}.$

Moreover, setting

(89)
$\begin{split}E_{a_{i},\lambda_{i}}:=\\{w\in\mathcal{H}_{\frac{\partial}{\partial
n}}:\;\;\langle\varphi_{a_{i},\lambda_{i}},w\rangle_{\mathbb{P}^{4,3}}=\langle\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial\lambda_{i}},w\rangle_{\mathbb{P}^{4,3}}=\langle\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial
a_{i}},w\rangle_{\mathbb{P}^{4,3}}=0,\\\ \,\overline{w}_{(Q,T)}=\langle
v_{r},w\rangle_{\mathbb{P}^{4,3}}=0,\;r=1,\cdots,\bar{k},\,\;\text{and}\;\;||w||_{\mathbb{P}^{4,3}}=O\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}}\right)\\},\end{split}$

and

(90)
$A:=(a_{1},\cdots,a_{k}),\;\;\bar{\lambda}=(\lambda_{1},\cdots,\lambda_{k}),\;\;E_{A,\bar{\lambda}}:=\cap_{i=1}^{k}E_{a_{i},\lambda_{i}},$

we have that, the quadratic form $Q$ is positive definite in
$E_{A,\bar{\lambda}}$. Furthermore, the linear part $f$ verifies that, for
every $w\in E_{A,\bar{\lambda}}$, there holds

(91)
$f(w)=O\left[||w||_{\mathbb{P}^{4,3}}\left(\sum_{i=1}^{k}\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}+\sum_{i=1}^{k}|\alpha_{i}-1|\log\lambda_{i}+\sum_{r=1}^{\bar{k}}|\beta_{r}|+\sum_{i=1}^{k}\frac{\log\lambda_{i}}{\lambda_{i}^{2}}\right)\right].$

where here $o(1)=o_{\bar{\alpha},A,\bar{\beta},\bar{\lambda},w,\epsilon}(1)$
and
$O\left(1\right):=O_{\bar{\alpha},A,\bar{\beta},\bar{\lambda},w,\epsilon}\left(1\right)$.

As in [1], to prove Proposition 3.15, we will need the following three coming
lemmas. We start with the following one:

######  Lemma 3.16.

Assuming the assumptions of Proposition 3.15 and $\gamma\in(0,1)$ small, then
for every $q\geq 1$, there holds the following estimates

(92) $\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}|w|^{q}=O\left(||w||_{\mathbb{P}^{4,3}}^{q}(\sum_{i=1}^{k}\lambda_{i}^{3+\gamma})\right),$
(93) $\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\hat{\delta}_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}|w|^{q}=O\left(||w||_{\mathbb{P}^{4,3}}^{q}(\sum_{i=1}^{k}\lambda_{i}^{\gamma})\right),$
(94) $\oint_{\partial
M}e^{3\hat{\delta}_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}d_{\hat{g}_{a_{i}}}(a_{i},\cdot)|w|^{q}=O\left(||w||_{\mathbb{P}^{4,3}}^{q}\frac{1}{\lambda_{i}^{1-\gamma}}\right),\;\;i=1,\cdots,k,$
(95) $\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}e^{3\theta_{w}w}|w|^{q}=O\left(||w||_{\mathbb{P}^{4,3}}^{q}(\sum_{i=1}^{k}\lambda_{i}^{3+\gamma})\right),$

where $\theta_{w}\in[0,1]$, and

(96) $\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}\left(e^{3w}-1-3w-\frac{9}{2}w^{2}\right)dV_{g}=o\left(||w||_{\mathbb{P}^{4,3}}^{2}(\sum_{i=1}^{k}\lambda_{i}^{3})\right).$

where here $o(1)$ and $O\left(1\right)$ are as in Proposition 3.15.

Proof. The proof is the same as the one of Lemma 6.2 in [1] replacing Lemma
10.1 by its counterpart Lemma 5.1.

Still as in [1], the second lemma that we need for the proof of Proposition
3.15 read as follows:

######  Lemma 3.17.

Assuming the assumptions of Proposition 3.15, then there holds the following
estimate

(97) $\begin{split}&\frac{\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}wdS_{g}}{\oint_{\partial
M}Ke^{3\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+3\sum_{r=1}^{\bar{k}}\beta_{r}v_{r}}dS_{g}}=\\\
&O\left(||w||_{\mathbb{P}^{4,3}}\left(\sum_{i=1}^{k}\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}+\sum_{i=1}^{k}|\alpha_{i}-1|\log\lambda_{i}+\sum_{r=1}^{\bar{k}}|\beta_{r}|+\sum_{i=1}^{k}\frac{\log\lambda_{i}}{\lambda_{i}^{2}}\right)\right).\end{split}$

Proof. If follows from the same arguments as in the proof of Lemma 6.3 in [1]
replacing Lemma 10.1 by its counterpart Lemma 5.1.

Finally, as in [1], the third and last lemma that we need for the proof of
Proposition 3.15 is the following one.

######  Lemma 3.18.

Assuming the assumptions of Proposition 3.15, then for every $i=1,\cdots,k$,
there holds

(98) $\tau_{i}=O\left(\sum_{j=1}^{k}\frac{1}{\lambda_{j}}\right).$

Proof. The proof is the same as the one Lemma 6.4 in [1] replacing Lemma 5.1
by Lemma 3.10.

Proof of Proposition 3.15
It follows from the same arguments as in the proof of Lemma 6.1 in [1]
replacing Lemma 6.2-Lemma 6.4 in [1] by Lemma 3.16-lemma 3.18 and Lemma 10.1
in [1] by Lemma 5.1. Furthermore, Lemma 10.6 and Lemma 10.7 in [1] are
replaced by Lemma 5.9 and Lemma 5.10.

Now, as in [1], we have that Proposition 3.15 implies the following direct
corollaries.

######  Corollary 3.19.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80) and
$u:=\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})$
with the concentration points $a_{i}$, the masses $\alpha_{i}$, the
concentrating parameters $\lambda_{i}$ ($i=1,\cdots,k$) and the negativity
parameters $\beta_{r}$ ($r=1,\cdots,\bar{k}$) satisfying (85), then there
exists a unique $\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in
E_{A,\bar{\lambda}}$ such that

(99)
$\mathcal{E}_{g}\left(u+\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\right)=\min_{w\in
E_{A,\bar{\lambda}},u+w\in V_{R}(k,\epsilon,\eta)}\mathcal{E}_{g}(u+w),$

where $\bar{\alpha}:=(\alpha_{1},\cdots,\alpha_{k})$,
$A:=(a_{1},\cdots,a_{k})$, $\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$
and $\bar{\beta}:=(\beta_{1},\cdots,\beta_{k})$.
Furthermore,
$(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\longrightarrow\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in
C^{1}$ and satisfies the following estimate

(100)
$\frac{1}{C}||\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})||_{\mathbb{P}^{4,3}}^{2}\leq|f\left(\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\right)|\leq
C||\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})||_{\mathbb{P}^{4,3}}^{2},$

for some large positive constant $C$ independent of $\bar{\alpha}$, $A$,
$\bar{\lambda}$, and $\bar{\beta}$, hence

(101)
$||\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})||_{\mathbb{P}^{4,3}}=O\left(\sum_{i=1}^{k}\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}+\sum_{i=1}^{k}|\alpha_{i}-1|\log\lambda_{i}+\sum_{r=1}^{\bar{k}}|\beta_{r}|+\sum_{i=1}^{k}\frac{\log\lambda_{i}}{\lambda_{i}^{2}}\right).$

######  Corollary 3.20.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), and
$u_{0}:=\sum_{i=1}^{k}\alpha_{i}^{0}\varphi_{a_{i}^{0},\lambda_{i}^{0}}+\sum_{r=1}^{\bar{k}}\beta_{r}^{0}(v_{r}-\overline{(v_{r})}_{(Q,T)})$
with the concentration points $a_{i}^{0}$, the masses $\alpha_{i}^{0}$, the
concentrating parameters $\lambda_{i}^{0}$ ($i=1,\cdots,k$) and the negativity
parameters $\beta_{r}^{0}$ ($r=1,\cdots,\bar{k}$) satisfying (85), then there
exists an open neighborhood $U$ of
$(\bar{\alpha}^{0},A^{0},\bar{\lambda}^{0},\bar{\beta}^{0})$ (with
$\bar{\alpha}^{0}:=(\alpha^{0}_{1},\cdots,\alpha^{0}_{k})$,
$A^{0}:=(a_{1}^{0},\cdots,a^{0}_{k})$,
$\bar{\lambda}:=(\lambda_{1}^{0},\cdots,\lambda_{k}^{0})$ and
$\bar{\beta}^{0}:=(\beta_{1}^{0},\cdots,\beta_{\bar{k}}^{0})$) such that for
every $(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in U$ with
$\bar{\alpha}:=(\alpha_{1},\cdots,\alpha_{k})$, $A:=(a_{1},\cdots,a_{k})$,
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$,
$\bar{\beta}:=(\beta_{1},\cdots,\beta_{\bar{k}})$, and the $a_{i}$, the
$\alpha_{i}$, the $\lambda_{i}$ ($i=1,\cdots,k$) and the $\beta_{r}$
($r=1,\cdots,\bar{k}$) satisfying (85), and $w$ satisfying (85) with
$\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+w\in
V_{R}(k,\epsilon,\eta)$, we have the existence of a change of variable

(102) $w\longrightarrow V$

from a neighborhood of $\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ to
a neighborhood of $0$ such that

(103)
$\begin{split}&\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+w\right)=\\\
&\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\right)\\\
&+\frac{1}{2}\partial^{2}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}^{0}\varphi_{a_{i}^{0},\lambda_{i}^{0}}+\sum_{r=1}^{\bar{k}}\beta_{r}^{0}(v_{r}-\overline{(v_{r})}_{(Q,T)})+\bar{w}(\bar{\alpha}^{0},A^{0},\bar{\lambda}^{0},\bar{\beta}^{0})\right)(V,V),\end{split}$

Thus, as in [1], with this new variable, it is easy to see that in
$V_{R}(k,\epsilon,\eta)$ we have a splitting of the variables
$(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ and $V$, namely that one can
decrease the Euler-Lagrange functional $\mathcal{E}_{g}$ in the variable $V$
without touching the variable $(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ by
considering just the flow

(104) $\frac{dV}{dt}=-V.$

So, as in [1], and for the same reasons, to develop a Morse theory for
$\mathcal{E}_{g}$ is equivalent to do one for the functional

(105)
$\bar{\mathcal{E}_{g}}(\bar{\alpha},A,\bar{\lambda},\bar{\beta}):=\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\right),$

where $\bar{\alpha}=(\alpha_{1},\cdots,\alpha_{k})$, $A=(a_{1},\cdots,a_{k})$,
$\bar{\lambda}=(\lambda_{1},\cdots,\lambda_{k})$ and
$\bar{\beta}=\beta_{1},\cdots,\beta_{\bar{k}}$ with the concentration points
$a_{i}$, the masses $\alpha_{i}$, the concentrating parameters $\lambda_{i}$
($i=1,\cdots,k$) and the negativity parameters $\beta_{r}$
($r=1,\cdots,\bar{k}$) satisfying (85), and
$\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ is as in Corollary 3.19.

Finally, we have the following energy estimate of $\mathcal{E}_{g}$ on
$V_{R}(k,\epsilon,\eta)$ .

######  Lemma 3.21.

Under the assumptions of Proposition 3.15, $\forall
u=\overline{u}_{(Q,T)}+\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+w\in
V_{R}(k,\epsilon,\eta)$, we have

$\begin{split}&\mathcal{E}_{g}(u)=\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=^{1}}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+w\right)=C^{k}_{0}-8\pi^{2}\mathcal{F}_{K}(a_{1},\dots,a_{k})+2\sum_{r=1}^{\bar{k}}\mu_{r}\beta_{r}^{2}\\\
&+\sum_{i=1}^{k}(\alpha_{i}-1)^{2}\left[16\pi^{2}\log\lambda_{i}+8\pi^{2}H(a_{i},a_{i})+C^{k}_{1}\right]-\frac{c^{1}8\pi^{2}}{9}\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}\left(\frac{\Delta_{\hat{g}_{a_{i}}}\mathcal{F}^{A}_{i}(a_{i})}{\mathcal{F}^{A}_{i}(a_{i})}-\frac{3}{4}R_{\hat{g}}(a_{i})\right)\\\
&+\frac{1}{2}\partial^{2}\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}^{0}\varphi_{a_{i}^{0},\lambda_{i}^{0}}+\sum_{r=1}^{\bar{k}}\beta_{r}^{0}(v_{r}-\overline{(v_{r})}_{(Q,T)})+\bar{w}(\bar{\alpha}^{0},A^{0},\bar{\lambda}^{0},\bar{\beta}^{0})\right)(V,V)\\\
&+8\pi^{2}\sum_{i=1}^{k}(\alpha_{i}-1)\left[\sum_{r=1}^{\bar{k}}2\beta_{r}\left(v_{r}-\overline{(v_{r})}_{(Q,T)}\right)(a_{i})-\sum_{j=1,j\neq
i}^{k}(\alpha_{j}-1)G(a_{i},a_{j})\right]\\\
&+\frac{c^{1}8\pi^{2}}{9}\sum_{i=1}^{k}\frac{\tilde{\tau}_{i}}{\lambda_{i}^{2}}\left(\frac{\Delta_{\hat{g}_{a_{i}}}\mathcal{F}^{A}_{i}(a_{i})}{\mathcal{F}^{A}_{i}(a_{i})}-\frac{3}{4}R_{\hat{g}}(a_{i})\right)\\\
&+\frac{16\pi^{2}}{3}\sum_{i=1}^{k}\log(1-\tilde{\tau}_{i})+O\left(\sum_{i=1}^{k}|\alpha_{i}-1|^{3}+\sum_{r=1}^{\bar{k}}|\beta_{r}|^{3}+\sum_{i=1}^{k}\frac{1}{\lambda^{3}_{i}}+||\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})||^{2}_{\mathbb{P}^{4,3}}\right),\end{split}$

where $O\left(1\right)$ means here
$O_{\bar{\alpha},A,\bar{\lambda},\bar{\beta},\epsilon}\left(1\right)$ with
$\bar{\alpha}=(\alpha_{1},\cdots,\alpha_{k})$, $A:=(a_{1},\cdots,a_{k})$,
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$,
$\bar{\beta}:=(\beta_{1},\cdots,\beta_{\bar{k}})$ and for $i=1,\cdots,k$,
$\tilde{\tau}_{i}$ is as in Lemma 3.9. where
$\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ is as in Corollary 3.19.

Proof. It follows directly from Lemma 3.9, formula (103) and Proposition 3.15.

###  3.4 Morse lemma at infinity

In this subsection, we derive a Morse Lemma at infinity for $\mathcal{E}_{g}$.
As in [1], in order to do that, we first construct a pseudo-gradient for
$\bar{\mathcal{E}_{g}}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$, where
$\bar{\mathcal{E}_{g}}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ is defined
as in (105) exploiting the gradient estimates derived previously. Indeed, we
have:

######  Proposition 3.22.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), and $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80), then there exists a pseudogradient
$\mathcal{W}_{g}$ of
$\bar{\mathcal{E}_{g}}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ such that
1) For every
$u:=\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})\in
V_{R}(k,\epsilon,\eta)$ with the concentration points $a_{i}$, the masses
$\alpha_{i}$, the concentrating parameters $\lambda_{i}$ ($i=1,\cdots,k$) and
the negativity parameters $\beta_{r}$ ($r=1,\cdots,\bar{k}$) satisfying (85),
there holds

(106)
$\large\left\langle-\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}(u),\;\mathcal{W}_{g}\large\right\rangle_{\mathbb{P}^{4,3}}\geq
c\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}+\sum_{i=1}^{k}\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}+\sum_{i=1}^{k}|\alpha_{i}-1|+\sum_{i=1}^{k}|\tau_{i}|+\sum_{r=1}^{\bar{k}}|\beta_{r}|)\right),$

and for every
$u:=\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}\lambda(v_{r}-\overline{(v_{r})}_{(Q,T)})+\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in
V_{R}(k,\epsilon,\eta)$ with the concentration points $a_{i}$, the masses
$\alpha_{i}$, the concentrating parameters $\lambda_{i}$ ($i=1,\cdots,k$) and
the negativity parameters $\beta_{r}$ ($r=1,\cdots,\bar{k}$) satisfying (85),
and $\bar{w}(\bar{\alpha},A,\bar{\lambda},\bar{\beta})$ is as in (99), there
holds

(107)
$\left\langle-\nabla^{\mathbb{P}^{4,3}}\mathcal{E}_{g}(u+\bar{w}),\;\mathcal{W}_{g}+\frac{\partial\bar{w}}{\partial(\bar{\alpha},A,\bar{\lambda},\bar{\beta})}\right\rangle_{{}_{\mathbb{P}^{4,3}}}\geq
c\left(\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}+\sum_{i=1}^{k}\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}+\sum_{i=1}^{k}|\alpha_{i}-1|+\sum_{i=1}^{k}|\tau_{i}|+\sum_{r=1}^{\bar{k}}|\beta_{r}|\right),$

where $c$ is a small positive constant independent of
$A:=(a_{1},\cdots,a_{k})$, $\bar{\alpha}=(\alpha_{1},\cdots,\alpha_{k})$,
$\bar{\lambda}=(\lambda_{1},\cdots,\lambda_{k})$,
$\bar{\beta}=(\beta_{1},\cdots,\beta_{\bar{k}})$ and $\epsilon$.
2) $\mathcal{W}_{g}$ is a $||\cdot||_{\mathbb{P}^{4,3}}$-bounded vector field
and is compactifying outside the region where $A$ is very close to a critical
point $B$ of $\mathcal{F}_{K}$ satisfying $\mathcal{L}_{K}(B)<0$.

Proof. It follows from the same arguments as in the proof of Proposition 8.1
in [1] replacing formulas (52)-(54), Lemma 5.1, Corollary 5.2 and Lemma
5.3-Lemma 5.5 in [1] with (49)-(51), Lemma 3.10, Corollary 3.11 and Lemma
3.12-Lemma 3.14. Furthermore, Lemma 4.1, Lemma 7.1, and Lemma 10.5 in [1] are
replaced by Lemma 3.9, Lemma 3.15 and Lemma 5.5

Now, as in [1], we have that Proposition 3.22 implies the following
characterization of the critical points at infinity of $\mathcal{E}_{g}$.

######  Corollary 3.23.

1) The critical points at infinity of $\mathcal{E}_{g}$ correspond to the
”configurations” $\alpha_{i}=1$, $\lambda_{i}=+\infty$, $\tau_{i}=0$
$i=1,\cdots,k$, $\beta_{r}=0$, $r=1,\cdots,\bar{k}$, $A$ is a critical point
of $\mathcal{F}_{K}$ and $V=0$, and we denote them by $z^{\infty}$ with $z$
being the corresponding critical point of $\mathcal{F}_{K}$ .

2) The “true” critical points at infinity of $\mathcal{E}_{g}$ are the
$z^{\infty}$ satisfying $\mathcal{L}_{K}(z)<0$ and we denote them by
$x^{\infty}$ with $x$ being the corresponding critical point of
$\mathcal{F}_{K}$.

3) The “false” critical points at infinity of $\mathcal{E}_{g}$ are the
$z^{\infty}$ satisfying $\mathcal{L}_{K}(z)>0$ and we denote them by
$y^{\infty}$ with $y$ being the corresponding critical point of
$\mathcal{F}_{K}$.
4) The $\mathcal{E}_{g}$-energy of a critical point at infinity $z^{\infty}$
denoted by $\mathcal{J}_{g}(z^{\infty})$ is given by

(108)
$\mathcal{J}_{g}(z^{\infty})=C_{0}^{k}-8\pi^{2}\mathcal{F}_{K}(z_{1},\dots,z_{k})$

where $z=(z_{1},\cdots,z_{k})$ and $C_{0}^{k}$ is as in Lemma 3.9.

Proof. Point 1)- Point 3) follow from (51), Lemma 3.8, the discussions right
after (103), and Proposition 3.22, while Point 4) follows from Point 1)
combined with (101) and Lemma 3.21.

Finally, we are going to conclude this subsection by establishing an analogue
of the classical Morse lemma for both “true” and “false” critical points at
infinity. In order to do that, we first remark that, as in [1], the arguments
of Proposition 3.22 implies that $V_{-}:=\\{u\in
V_{R}(k,\epsilon,\eta):\;l_{K}(A)<0,\;\forall
r\in\\{1,\cdots,\bar{k}\\}\,\,\;\,|\beta_{r}|\leq
2\tilde{C}_{0}\left(\sum_{i=1}^{k}\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}(a_{i})|}{\lambda_{i}}+\sum_{i=1}^{k}|\alpha_{i}-1|+\sum_{i=1}^{k}|\tau_{i}|+\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}\right),\;\;\forall\;i\in\\{1,\cdots,k\\}\;\,\;|\tau_{i}|\leq
2\frac{\hat{C}_{0}}{\lambda_{i}^{2}},\;\;\text{and}\;\;\;\forall
i\in\\{1,\cdots,k\\}\;\;\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}\leq
4\frac{C_{0}}{\lambda_{i}^{2}}\;\\}$ and $V_{+}:=\\{u\in
V_{R}(k,\epsilon,\eta):l_{K}(A)>0,\forall
r\in\\{1,\cdots,\bar{k}\\}\,\,\;\,|\beta_{r}|\leq
2\tilde{C}_{0}\left(\sum_{i=1}^{k}\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}(a_{i})|}{\lambda_{i}}+\sum_{i=1}^{k}|\alpha_{i}-1|+\sum_{i=1}^{k}|\tau_{i}|+\sum_{i=1}^{k}\frac{1}{\lambda_{i}^{2}}\right),\\\
\;\;\forall\;i\in\\{1,\cdots,k\\}\;\,\;|\tau_{i}|\leq
2\frac{\hat{C}_{0}}{\lambda_{i}^{2}},\;\;\text{and}\;\;\;\forall
i\in\\{1,\cdots,k\\}\;\;\frac{|\nabla_{\hat{g}}\mathcal{F}^{A}_{i}(a_{i})|}{\lambda_{i}}\leq
4\frac{C_{0}}{\lambda_{i}^{2}}\;\\}$ (where $\tilde{C}_{0}$, $\hat{C}_{0}$ and
$C_{0}$ are large positive constants) are respectively a neighborhood of the
“true” and “false” critical points at infinity of the variational problem.
Hence, as in [1], (101), Corollary3.20, Lemma 3.21 and classical Morse lemma
imply the following Morse type lemma for a “true” critical point at infinity.

######  Lemma 3.24.

(Morse lemma at infinity near a “true” one)
Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80) and
$u_{0}:=\sum_{i=1}^{k}\alpha_{i}^{0}\varphi_{a_{i}^{0},\lambda_{i}^{0}}+\sum_{r=1}^{\bar{k}}\beta_{r}^{0}(v_{r}-\overline{(v_{r})}_{(Q,T)})+\bar{w}((\bar{\alpha}^{0},A^{0},\bar{\lambda}^{0},\bar{\beta}^{0}))\in
V_{-}(k,\epsilon,\eta)$ (where
$\bar{\alpha}^{0}:=(\alpha^{0}_{1},\cdots,\alpha^{0}_{k})$,
$A^{0}:=(a_{1}^{0},\cdots,a^{0}_{k})$,
$\bar{\lambda}:=(\lambda_{1}^{0},\cdots,\lambda_{k}^{0})$ and
$\bar{\beta}^{0}:=(a_{1}^{0},\cdots,\beta_{\bar{k}}^{0})$) with the
concentration points $a_{i}^{0}$, the masses $\alpha_{i}^{0}$, the
concentrating parameters $\lambda_{i}^{0}$ ($i=1,\cdots,k$) and the negativity
parameters $\beta_{r}^{0}$ ($r=1,\cdots,\bar{k}$) satisfying (85) and
furthermore $A^{0}\in Crit(\mathcal{F}_{K})$, then there exists an open
neighborhood $U$ of
$(\bar{\alpha}^{0},A^{0},\bar{\lambda}^{0},\bar{\beta}^{0})$ such that for
every $(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in U$ with
$\bar{\alpha}:=(\alpha_{1},\cdots,\alpha_{k})$, $A:=(a_{1},\cdots,a_{k})$,
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$,
$\bar{\beta}:=(\beta_{1},\cdots,\beta_{\bar{k}})$, and the $a_{i}$, the
$\alpha_{i}$, the $\lambda_{i}$ ($i=1,\cdots,k$) and the $\beta_{r}$
($r=1,\cdots,\bar{k}$) satisfying (85), and $w$ satisfying (85) with
$u=\overline{u}_{(Q,T)}\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)})+w\in
V_{-}(k,\epsilon,\eta)$, we have the existence of a change of variable

(109) $\begin{split}&\alpha_{i}\longrightarrow s_{i},i=1,\cdots,k,\\\
&A\longrightarrow\tilde{A}=(\tilde{A}_{-},\tilde{A}_{+})\\\
&\lambda_{1}\longrightarrow\theta_{1},\\\
&\tau_{i}\longrightarrow\theta_{i},i=2,\cdots,k,\\\
&\beta_{r}\longrightarrow\tilde{\beta}_{r}\\\
&V\longrightarrow\tilde{V},\end{split}$

such that

(110)
$\mathcal{E}_{g}(u)=\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}\left(v_{r}-\overline{(v_{r})}_{(Q,T)}\right)+w\right)=-|\tilde{A}_{-}|^{2}+|\tilde{A}_{+}|^{2}+\sum_{i=1}^{k}s_{i}^{2}-\sum_{r=1}^{\bar{k}}\tilde{\beta}_{r}^{2}+\theta_{1}^{2}-\sum_{i=2}^{k}\theta_{i}^{2}+||\tilde{V}||^{2}$

where $\tilde{A}=(\tilde{A}_{-},\;\tilde{A}_{+})$ is the Morse variable of the
map $\mathcal{J}_{g}:(\partial M)^{k}\setminus F((\partial
M)^{k})\longrightarrow\mathbb{R}$ which is defined by the right hand side of
(108). Hence a “true“critical point at infinity $x^{\infty}$ of
$\mathcal{E}_{g}$ has Morse index at infinity
$\mathcal{M}_{\infty}(x^{\infty})=i_{\infty}(x)+\bar{k}$.

Similarly, and for the same reasons as above, we have the following analogue
of the classical Morse lemma for a ”false“ critical point at infinity.

######  Lemma 3.25.

(Morse lemma at infinity near a “false” one)
Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), $0<\epsilon\leq\epsilon_{0}$ where
$\epsilon_{0}$ is as in (80) and
$u_{0}:=\sum_{i=1}^{k}\alpha_{i}^{0}\varphi_{a_{i}^{0},\lambda_{i}^{0}}+\sum_{r=1}^{\bar{k}}\beta_{r}^{0}(v_{r}-\overline{(v_{r})}_{(Q,T)})+\bar{w}((\bar{\alpha}^{0},A^{0},\bar{\lambda}^{0},\bar{\beta}^{0}))\in
V_{+}(k,\epsilon,\eta)$ (where
$\bar{\alpha}^{0}:=(\alpha^{0}_{1},\cdots,\alpha^{0}_{k})$,
$A^{0}:=(a_{1}^{0},\cdots,a^{0}_{k})$,
$\bar{\lambda}:=(\lambda_{1}^{0},\cdots,\lambda_{k}^{0})$ and
$\bar{\beta}^{0}:=(a_{1}^{0},\cdots,\beta_{\bar{k}}^{0})$) with the
concentration points $a_{i}^{0}$, the masses $\alpha_{i}^{0}$, the
concentrating parameters $\lambda_{i}^{0}$ ($i=1,\cdots,k$) and the negativity
parameters $\beta_{r}^{0}$ ($r=1,\cdots,\bar{k}$) satisfying (85) and
furthermore $A^{0}\in Crit(\mathcal{F}_{K})$, then there exists an open
neighborhood $U$ of
$(\bar{\alpha}^{0},A^{0},\bar{\lambda}^{0},\bar{\beta}^{0})$ such that for
every $(\bar{\alpha},A,\bar{\lambda},\bar{\beta})\in U$ with
$\bar{\alpha}:=(\alpha_{1},\cdots,\alpha_{k})$, $A:=(a_{1},\cdots,a_{k})$,
$\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$,
$\bar{\beta}:=(\beta_{1},\cdots,\beta_{\bar{k}})$, and the $a_{i}$, the
$\alpha_{i}$, the $\lambda_{i}$ ($i=1,\cdots,k$) and the $\beta_{r}$
($r=1,\cdots,\bar{k}$) satisfying (85), and $w$ satisfying (85) with
$u=\overline{u}_{(Q,T)}+\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{k}}\beta_{r}(v_{r}-\overline{(v_{r})}_{Q^{n}})+w\in
V_{+}(k,\epsilon,\eta)$, we have the existence of a change of variable

(111) $\begin{split}&\alpha_{i}\longrightarrow s_{i},i=1,\cdots,k,\\\
&A\longrightarrow\tilde{A}=(\tilde{A}_{-},\tilde{A}_{+})\\\
&\lambda_{1}\longrightarrow\theta_{1},\\\
&\tau_{i}\longrightarrow\theta_{i},i=2,\cdots,k,\\\
&\beta_{r}\longrightarrow\tilde{\beta}_{r}\\\
&V\longrightarrow\tilde{V},\end{split}$

such that

(112)
$\mathcal{E}_{g}(u)=\mathcal{E}_{g}\left(\sum_{i=1}^{k}\alpha_{i}\varphi_{a_{i},\lambda_{i}}+\sum_{r=1}^{\bar{m}}\beta_{r}\left(v_{r}-\overline{(v_{r})}_{(Q,T)}\right)+w\right)=-|\tilde{A}_{-}|^{2}+|\tilde{A}_{+}|^{2}+\sum_{i=1}^{k}s_{i}^{2}-\sum_{r=1}^{\bar{k}}\tilde{\beta}_{r}^{2}-\sum_{i=1}^{k}\theta_{i}^{2}+||\tilde{V}||^{2},$

where $\tilde{A}=(\tilde{A}_{-},\tilde{A}_{+})$ is the Morse variable of the
map $\mathcal{J}_{g}:(\partial M)^{k}\setminus F((\partial
M)^{k})\longrightarrow\mathbb{R}$ which is defined by the right hand side of
(108). Hence a “false“critical point at infinity $y^{\infty}$ of
$\mathcal{E}_{g}$ has Morse index at infinity
$M_{\infty}(y^{\infty})=i_{\infty}(y)+1+\bar{k}$.

##  4 Proof of existence theorems

In this section, we show how the Morse lemma at infinity implies the main
existence results via strong Morse type inequalities or Barycenter technique
of Bahri-Coron.

###  4.1 Topology of vey high and negative sublevels of $\mathcal{E}_{g}$

We study the topology of very high sublevels of $\mathcal{E}_{g}$ and its
every negative ones. We start with the very high sublevels of
$\mathcal{E}_{g}$ and first derive the following Lemma.

######  Lemma 4.1.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), then there exists
$\hat{C}_{0}^{k}:=\hat{C}^{k}_{0}(\eta)$ such that for every
$0<\epsilon\leq\epsilon_{0}$ where $\epsilon_{0}$ is as in (80), there holds

$V(k,\epsilon,\eta)\subset(\mathcal{E}_{g})^{\hat{C}_{0}^{k}}-(\mathcal{E}_{g})^{-\hat{C}_{0}^{k}}.$

Proof. It follows directly from (82)-(85), Proposition 3.15, Lemma 3.18, and
Lemma 3.21.

Next, combining Proposition 3.7 and the latter lemma, we have the following
corollary.

######  Corollary 4.2.

There exists a large positive constant $\hat{C}_{1}^{k}$ such that

$Crit(\mathcal{E}_{g})\subset(\mathcal{E}_{g})^{\hat{C}_{1}^{k}}-(\mathcal{E}_{g})^{-\hat{C}_{1}^{k}}.$

Proof. It follows, via a contradiction argument, from the the fact that
$\mathcal{E}_{g}$ is invariant by translation by constants, Proposition 3.7,
and Lemma 4.1.

Now, we are ready to characterize the topology of very high sublevels of
$\mathcal{E}_{g}$. Indeed, as in [1] and for the same reasons, we have that
Lemma 3.8, Lemma 4.1 and Corollary 4.2 imply the following one which describes
the topology of very high sublevels of the Euler-Lagrange functional
$\mathcal{E}_{g}$.

######  Lemma 4.3.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), then there exists a large positive constant
$L^{k}:=L^{k}(\eta)$ with $L^{k}>2\max\\{\hat{C}_{0}^{k},\hat{C}_{1}^{k}\\}$
such that for every $L\geq L^{k}$, we have that $(\mathcal{E}_{g})^{L}$ is a
deformation retract of $\mathcal{H}_{\frac{\partial}{\partial n}}$, and hence
it has the homology of a point, where $\hat{C}^{k}_{0}$ is as in Lemma 4.1 and
$\hat{C}^{k}_{1}$ is as in Lemma 4.2.

Next, we turn to the study of the topology of very negative sublevels of
$\mathcal{E}_{g}$ when $k\geq 2$ or $\bar{k}\geq 1$. Indeed, as in [1] and for
the same reasons, we have that the well-know topology of very negative
sublevels in the nonresont case (see [44]), Proposition 3.7, Lemma 4.1 and
Corollary 4.2 imply the following lemma which gives the homotopy type of the
very negative sublevels of the Euler-Lagrange functional $\mathcal{E}_{g}$.

######  Lemma 4.4.

Assuming that $k\geq 2$ or $\bar{k}\geq 1$, and $\eta$ is a small positive
real number with $0<2\eta<\varrho$ where $\varrho$ is as in (41), then there
exists a large positive constant $L_{k,\bar{k}}:=L_{k,\bar{k}}(\eta)$ with
$L_{k,\bar{k}}>2\max\\{\hat{C}_{0}^{k},\hat{C}_{1}^{k}\\}$ such that for every
$L\geq L_{k,\bar{k}}$, we have that $(\mathcal{E}_{g})^{-L}$ has the same
homotopy type as $B_{k-1}(\partial M)$ if $k\geq 2$ and $\bar{k}=0$, as
$A_{k-1,\bar{k}}$ if $k\geq 2$ and $\bar{k}\geq 1$ and as $S^{\bar{k}-1}$ if
$k=1$ and $\bar{k}\geq 1$, where $\hat{C}^{k}_{0}$ is as in Lemma 4.1 and
$\hat{C}^{k}_{1}$ as in Lemma 4.2 .

However, as in [46], to prove Theorem 1.8, we need a further information about
the topology of very negative sublevels of $\mathcal{E}_{g}$. In order to
derive that, we first make some definitions. For $p\in\mathbb{N}^{*}$ and
$\lambda>0$, we define

$f_{p}(\lambda):B_{p}(\partial
M)\longrightarrow\mathcal{H}_{\frac{\partial}{\partial n}}$

as follows

(113)
$f_{p}(\lambda)(\sum_{i=1}^{p}\alpha_{i}\delta_{a_{i}}):=\sum_{i=1}^{p}\alpha_{i}\varphi_{a_{i},\lambda},\;\;\;\;\sigma=\sum_{i=1}^{p}\alpha_{i}\delta_{a_{i}}\in
B_{p}(\partial M),$

with the $\varphi_{a_{i},\lambda}$’s defined by (43). Furthermore, when
$\bar{k}\geq 1$, for $\Theta>0$, we define

(114)
$\Psi_{p,\bar{k}}(\lambda,\Theta):A_{p,\bar{k}}\longrightarrow\mathcal{H}_{\frac{\partial}{\partial
n}}$

as follows

(115)
$\Psi_{p,\bar{k}}(\lambda,\theta)(\sigma,s):=\begin{cases}\varphi_{s}+f_{p}(\lambda)(\sigma)\;\;\;&\text{for}\;\;|s|\leq\frac{1}{4},\;\;\;\sigma\in
B_{p}(\partial M),\\\
\varphi_{s}+f_{p}(2\lambda-1+4(1-\lambda)|s|)(\sigma)\;\;\;&\text{for}\;\;\frac{1}{4}\leq|s|\leq\frac{1}{2},\;\;\;\sigma\in
B_{p}(\partial M),\\\
\varphi_{s}+2(1-f_{p}(1)(\sigma))|s|+2f_{p}(1)-1\;\;\;&\text{for}\;\;\;|s|\geq\frac{1}{2},\;\;\;\sigma\in
B_{p}(\partial M),\end{cases}$

where $\varphi_{s}$ is defined by the following formula

(116)
$\varphi_{s}=\Theta\sum_{r=1}^{\bar{k}}s_{r}(v_{r}-\overline{(v_{r})}_{(Q,T)}),$

with $s=(s_{1},\cdots,s_{\bar{k}})$. As in [46], concerning the
$f_{p}(\lambda)$’s, we have the following estimates.

######  Lemma 4.5.

Assuming that $p\in\mathbb{N}^{*}$, then we have
1) If $p<k$, then for every $L>0$, there exists $\lambda^{L}_{p}>0$ such that
for all $\lambda\geq\lambda^{L}_{p}$, we have

$f_{p}(\lambda)(B_{p}(\partial M))\subset(\mathcal{E}_{g})^{-L}.$

2) If $p=k$, then there exist $\hat{C}_{k}>0$ and $\lambda_{k}>0$ such that
for all $\lambda\geq\lambda_{k}$, we have

$f_{k}(\lambda)(B_{k}(\partial M))\subset(\mathcal{E}_{g})^{\hat{C}_{k}}.$

3) There exists $\hat{C}^{k}>0$ such that up to taking $\;\epsilon_{0}$
smaller, where $\epsilon_{0}$ is given by (80), we have that for every
$0<\epsilon\leq\epsilon_{0}$, there holds

$V(k,\epsilon)\subset(\mathcal{E}_{g})^{\hat{C}^{k}}.$

Proof. It follows from the same arguments as in the proof of Lemma 3.1 in [46]
by using Lemma 5.1, Lemma 5.3 and Lemma 5.6-Lemma 5.8.

Still, as in [46], we have the following estimates for the
$\Psi_{p}(\lambda,\Theta)$’s when $\bar{k}\geq 1$.

######  Lemma 4.6.

Assuming that $p\in\mathbb{N}^{*}$, then we have
1) If $1\leq p<k$, then for every $L>0$, there exists
$\lambda^{L}_{p,\bar{k}}>0$ and $\Theta_{p,\bar{k}}^{L}>0$ such that for all
$\lambda\geq\lambda^{L}_{p,\bar{k}}$, we have

$\Psi_{p,\bar{k}}(\lambda,\Theta_{p,\bar{k}}^{L})(A_{p,\bar{k}})\subset(\mathcal{E}_{g})^{-L}.$

2) If $p=k$ and $\Theta>0$, then there exists $C_{k,\bar{k}}^{\Theta}>0$,
$\lambda_{k,\bar{k}}^{\Theta}>0$, such that for every
$\lambda\geq\lambda^{\Theta}_{k,\bar{k}}$, we have

$\Psi_{k,\bar{k}}(\lambda,\Theta)(A_{k,\bar{k}})\subset(\mathcal{E}_{g})^{C_{k,\bar{k}}^{\Theta}}.$

3) If $\Theta>0$ , then there exists $C^{k,\bar{k}}_{\Theta}>0$ such that up
to taking $\epsilon_{0}$ smaller, where $\epsilon_{0}$ is given by (80), we
have that for every $0<\epsilon\leq\epsilon_{0}$, there holds

$V(k,\epsilon,\Theta)\subset(\mathcal{E}_{g})^{C^{k,\bar{k}}_{\Theta}}.$

Proof. It follows from the same arguments as in the proof of Lemma 4.1 in [46]
by using Lemma 4.5.

On the other hand, as in [46], Lemma 4.4 and Lemma 4.5 imply the following
one:

######  Lemma 4.7.

Assuming that $k\geq 2$, $\bar{k}=0$, and $L\geq L_{k.0}$, then there exists
$\lambda^{L}_{k-1}$ such that for all ;$\lambda\geq\lambda^{L}_{k-1}$, we have

$f_{k-1}(\lambda):B_{k-1}(\partial M)\longrightarrow(\mathcal{E}_{g})^{-L}$

is well defined and induces an isomorphism in homology.

Furthermore, still as in [46], we have also that Lemma 4.4 and Lemma 4.5 imply
the following one:

######  Lemma 4.8.

Assuming that $k\geq 2$, $\bar{k}\geq 1$, $L\geq L_{k,,\bar{k}}$, then there
exists $\lambda^{L}_{k-1,\bar{k}}>0$ and $\Theta_{k-1,\bar{k}}^{L}>0$ such
that for all $\lambda\geq\lambda^{L}_{k-1,\bar{k}}$, we have

$\Psi_{k-1,\bar{k}}(\lambda,\Theta_{k-1,\bar{k}}^{L}):A_{k-1,\bar{k}}\longrightarrow(\mathcal{E}_{g})^{-L}$

is well defined and induces an isomorphism in homology.

###  4.2 Morse theoretical type results

Proof of Theorem 1.1-Theorem 1.7
The proof is the same as the one of Theorem 1.1-Theorem 1.6 in [1] by using
Lemma 3.8, Proposition 3.22, Corollary 3.23, Lemma 3.24, Lemma 4.1, Corollary
4.2, Lemma 4.3, and Lemma 4.4 combined with the works of Bahri-Rabinowitz[12],
Karell-Karoui[34] and Malchiodi[39].

###  4.3 Algebraic topological type results

In order to carry the algebraic topological argument for existence, as in
[46], we need the following lemma.

######  Lemma 4.9.

Assuming that $(ND)$ holds, $s_{k}^{*}(O^{*}_{\partial M})\neq 0$ in
$H^{3}(S^{\infty})$ and $s_{k}^{*}(O^{*}_{\partial M})=0$ in
$H^{3}(S_{+}^{\infty}\cup S_{-}^{\infty})$, then there exists
$0\neq\tilde{O}^{*}_{\partial M}\in H^{3}(S)$ such that

$i^{*}(\tilde{O}^{*}_{\partial M})=s_{k}^{*}(O^{*}_{\partial M}),$

where $i:S^{\infty}\longrightarrow S$ is the canonical injection.

Proof. It follows from the same arguments as in the proof of Lemma 3.6 in [46]
by using the analysis of Section 3.

Proof of Theorem 1.8
The proof is the same as the one Theorem in [46] by using the algebraic
topological tools (52)-(54), characterization of the critical points at
infinity of $\mathcal{E}_{g}$ established in Section 3, and Lemma 4.9.

##  5 Appendix

######  Lemma 5.1.

Assuming that $\epsilon$ is positive and small, $a\in\partial M$ and
$\lambda\geq\frac{1}{\epsilon}$, then
1)

$\varphi_{a,\lambda}(\cdot)=\hat{\delta}_{a,\lambda}(\cdot)+\log\frac{\lambda}{2}+H(a,\cdot)+\frac{1}{2\lambda^{2}}\Delta_{\hat{g}_{a}}H(a,\cdot)+O\left(\frac{1}{\lambda^{3}}\right)\;\;\;\text{on}\;\;\;\partial
M$

2)

$\lambda\frac{\partial\varphi_{a,\lambda}(\cdot)}{\partial\lambda}=\frac{2}{1+\lambda^{2}\chi_{\varrho}^{2}(d_{\hat{g}_{a}}(a,\cdot))}-\frac{1}{\lambda^{2}}\Delta_{\hat{g}_{a}}H(a,\cdot)+O\left(\frac{1}{\lambda^{3}}\right)\;\;\;\text{on}\;\;\;\partial
M,$

3)

$\frac{1}{\lambda}\frac{\partial\varphi_{a,\lambda}(\cdot)}{\partial
a}=\frac{\chi_{\varrho}(d_{\hat{g}_{a}}(a,\cdot))\chi_{\varrho}^{{}^{\prime}}((d_{\hat{g}_{a}}(a,\cdot))}{d_{\hat{g}_{a}}(a,\cdot)}\frac{2\lambda
exp_{a}^{-1}(\cdot)}{1+\lambda^{2}\chi_{\varrho}^{2}(d_{\hat{g}_{a}}(a,\cdot))}+\frac{1}{\lambda}\frac{\partial
H(a,\cdot)}{\partial
a}+O\left(\frac{1}{\lambda^{3}}\right);\;\;\text{on}\;\;\;\partial M,$

where $O(1)$ means $O_{a,\lambda,\epsilon}(1)$ and for it meaning see Section
2.

######  Lemma 5.2.

Assuming that $\epsilon$ is small and d positive, $a\in M$,
$\lambda\geq\frac{1}{\epsilon}$, and $0<2\eta<\varrho$ with $\varrho$ as in
(41), then there holds

$\varphi_{a,\lambda}(\cdot)=G(a,\cdot)+\frac{1}{2\lambda^{2}}\Delta_{\hat{g}_{a}}G(a,\cdot)+O\left(\frac{1}{\lambda^{3}}\right)\;\;\text{on}\;\;\;\partial
M\setminus B^{a}_{a}(\eta),$
$\lambda\frac{\partial\varphi_{a,\lambda}(\cdot)}{\partial\lambda}=-\frac{1}{\lambda^{2}}\Delta_{\hat{g}_{a}}G_{(}a,\cdot)+O\left(\frac{1}{\lambda^{3}}\right)\;\;\text{on}\;\;\;\partial
M\setminus B^{a}_{a}(\eta),$

and

$\frac{1}{\lambda}\frac{\partial\varphi_{a,\lambda}(\cdot)}{\partial
a}=\frac{1}{\lambda}\frac{\partial G(a,\cdot)}{\partial
a}+O\left(\frac{1}{\lambda^{3}}\right)\;\;\text{on}\;\;\;\partial M\setminus
B^{a}_{a}(\eta),$

where $O(1)$ means $O_{a,\lambda,\epsilon}(1)$ and for it meaning see Section
2.

######  Lemma 5.3.

Assuming that $\epsilon$ is small and positive, $a\in\partial M$ and
$\lambda\geq\frac{1}{\epsilon}$, then there holds

$\mathbb{P}_{g}^{4,3}\left(\varphi_{a,\lambda},\;\varphi_{a,\lambda}\right)=16\pi^{2}\log\lambda-8\pi^{2}C_{0}+8\pi^{2}H(a,a)+\frac{8\pi^{2}}{\lambda^{2}}\Delta_{\hat{g}_{a}}H(a,a)+O\left(\frac{1}{\lambda^{3}}\right),$
$\mathbb{P}_{g}^{4,3}\left(\varphi_{a,\lambda},\;\lambda\frac{\varphi_{a,\lambda}}{\partial\lambda}\right)=8\pi^{2}-\frac{8\pi^{2}}{\lambda^{2}}\Delta_{\hat{g}_{a}}H(a,a)+O\left(\frac{1}{\lambda^{3}}\right),$
$\mathbb{P}_{g}^{4,3}\left(\varphi_{a,\lambda},\;\frac{1}{\lambda}\frac{\varphi_{a,\lambda}}{\partial
a}\right)=\frac{8\pi^{2}}{\lambda}\frac{\partial H(a,a)}{\partial
a}+O\left(\frac{1}{\lambda^{3}}\right),$

where $C_{0}$ is a positive constant depending only on $n$, $O(1)$ means
$O_{a,\lambda,\epsilon}(1)$ and for its meaning see Section 2.

######  Lemma 5.4.

Assuming that $\epsilon$ is small and positive $a_{i},a_{j}\in\partial M$,
$d_{\hat{g}}(a_{i},a_{j})\geq 4\overline{C}\eta$, $0<2\eta<\varrho$,
$\frac{1}{\Lambda}\leq\frac{\lambda_{i}}{\lambda_{j}}\leq\Lambda$, and
$\lambda_{i},\lambda_{j}\geq\frac{1}{\epsilon}$, $\overline{C}$ as in (39),
and $\varrho$ as in (41), then there hold

$\begin{split}\mathbb{P}_{g}^{4,3}\left(\varphi_{a_{i},\lambda_{i}},\;\varphi_{a_{j},\lambda_{j}}\right)=&8\pi^{2}G(a_{j},a_{i})+\frac{4\pi^{2}}{\lambda_{i}^{2}}\Delta_{\hat{g}_{a_{i}}}G(a_{i},a_{j})+\frac{4\pi^{2}}{\lambda_{j}^{2}}\Delta_{\hat{g}_{a_{j}}}G(a_{j},a_{i})\\\
&+O\left(\frac{1}{\lambda^{3}_{i}}+\frac{1}{\lambda_{j}^{3}}\right),\end{split}$
$\mathbb{P}_{g}^{4,3}\left(\varphi_{a_{i},\lambda_{i}},\;\lambda_{j}\frac{\partial\varphi_{a_{j},\lambda_{j}}}{\partial\lambda_{j}}\right)=-\frac{8\pi^{2}}{\lambda_{j}^{2}}\Delta_{\hat{g}_{a_{j}}}G(a_{j},a_{i})+O\left(\frac{1}{\lambda^{3}_{j}}\right),$

and

$\mathbb{P}_{g}^{4,3}\left(\varphi_{a_{i},\lambda_{i}},\;\frac{1}{\lambda_{j}}\frac{\partial\varphi_{a_{j},\lambda_{j}}}{\partial
a_{j}}\right)=\frac{8\pi^{2}}{\lambda_{j}}\frac{\partial
G(a_{j},a_{i})}{\partial a_{j}}+O\left(\frac{1}{\lambda^{3}_{j}}\right),$

where $O(1)$ means here $O_{A,\bar{\lambda},\epsilon}(1)$ with
$A=(a_{i},a_{j})$ and $\bar{\lambda}=(\lambda_{i},\lambda_{j})$ and for the
meaning of $O_{A,\bar{\lambda},\epsilon}(1)$, see Section 2.

######  Lemma 5.5.

Assuming that $\epsilon>0$ is very small, we have that for $a\in\partial M$,
$\lambda\geq\frac{1}{\epsilon}$, there holds

(117)
$||\lambda\frac{\partial\varphi_{a,\lambda}}{\partial\lambda}||_{\mathbb{P}^{4,3}}=\tilde{O}(1),$
(118) $||\frac{1}{\lambda}\frac{\partial\varphi_{a,\lambda}}{\partial
a}||_{\mathbb{P}^{4,3}}=\tilde{O}(1),$

and

(119)
$||\frac{1}{\sqrt{\log\lambda}}\varphi_{a,\lambda}||_{\mathbb{P}^{4,3}}=\tilde{O}(1),$

where here $\tilde{O}(1)$ means bounded by positive constants form below and
above independent of $\epsilon$, $a$, and $\lambda$.

######  Lemma 5.6.

1) If $\epsilon$ is small and positive, $a\in\partial M$,
$p\in\mathbb{N}^{*}$, and $\lambda\geq\frac{1}{\epsilon}$ , then there holds

(120) $C^{-1}\lambda^{6p-3}\leq\oint_{\partial
M}e^{3p\varphi_{a,\lambda}}dS_{g}\leq C\lambda^{6p-3},$

where $C$ is independent of $a$, $\lambda$, and $\epsilon$.

2) If $\epsilon$ is positive and small, $a_{i},a_{j}\in\partial M$,
$\lambda\geq\frac{1}{\epsilon}$ and $\lambda d_{\hat{g}}(a_{i},a_{j})\geq
4\overline{C}R$, then we have

(121)
$\mathbb{P}_{g}^{4,3}(\varphi_{a_{i},\lambda},\;\varphi_{a_{j},\lambda})\,\leq\,8\pi^{2}G(a_{i},a_{j})\,+\,O(1),$

where $O(1)$ means here $O_{A,\lambda,\epsilon}(1)$ with $A=(a_{i},a_{j})$,
and for the meaning of $O_{A,\lambda,\epsilon}(1)$, see Section 2. 3) If
$\epsilon$ is positive and small, $a_{i},a_{j}\in\partial M$,
$\lambda_{i},\lambda_{j}\geq\frac{1}{\epsilon}$,
$\frac{1}{\Lambda}\leq\frac{\Lambda_{i}}{\lambda_{j}}\leq\Lambda$ and
$\lambda_{i}d_{\hat{g}}(a_{i},a_{j})\geq 4\overline{C}R$, then we have

(122)
$\mathbb{P}_{g}^{4,3}(\varphi_{a_{i},\lambda},\;\varphi_{a_{j},\lambda})\,\leq\,8\pi^{2}G(a_{i},a_{j})\,+\,O(1),$

where $O(1)$ means here $O_{A,\bar{\lambda},\epsilon}(1)$ with
$A=(a_{i},a_{j})$ and $\bar{\lambda}=(\lambda_{i},\lambda_{j})$ and for the
meaning of $O_{A,\bar{\lambda},\epsilon}(1)$, see Section 2.

######  Lemma 5.7.

Let $p\in\mathbb{N}^{*}$, $\hat{R}$ be a large positive constant, $\epsilon$
be a small positive number, $\alpha_{i}\geq 0$, $i=1,\cdots,p$,
$\sum_{i=1}^{p}\alpha_{i}=k$, $\lambda\geq\frac{1}{\epsilon}$ and
$u=\sum_{i=1}^{p}\alpha_{i}\varphi_{a_{i},\lambda}$. Assuming that there exist
two positive integer $i,j\in\\{1,\cdots,p\\}$ with $i\neq j$ such that
$\lambda d_{\hat{g}}(a_{i},a_{j})\leq\frac{\hat{R}}{4\overline{C}}$, where
$\overline{C}$ is as in (39), then we have

(123) $\mathcal{E}_{g}(u)\,\leq\mathcal{E}_{g}(v)\,+\,O(\log\hat{R}),$

with

$v:=\sum_{k\leq p,k\neq
i,j}\alpha_{k}\varphi_{a_{k},\lambda}\,+(\alpha_{i}+\alpha_{j})\varphi_{a_{i},\lambda}.$

where here $O(1)$ stand for $O_{{\bar{\alpha}},A,\lambda,\epsilon}(1)$, with
${\bar{\alpha}}=(\alpha_{1},\cdots,\alpha_{p})$ and $A=(a_{1},\cdots,a_{p})$,
and for the meaning of $O_{{\bar{\alpha}},A,\lambda,\epsilon}(1)$, we refer
the reader to Section 2.

######  Lemma 5.8.

1) If $\epsilon$ is positive and small, $a_{i},a_{j}\in\partial M$,
$\lambda\geq\frac{1}{\epsilon}$ and $\lambda d_{\hat{g}}(a_{i},a_{j})\geq
4\overline{C}R$, then

$\varphi_{a_{j},\lambda}(\cdot)=G(a_{j},\cdot)+O(1)\;\;\;\text{in}\;\;B^{a_{i}}_{a_{i}}(\frac{R}{\lambda}),$

where here $O(1)$ means here $O_{A,\lambda,\epsilon}(1)$, with
$A=(a_{i},a_{j})$, and for the meaning of $O_{A,\lambda,\epsilon}(1)$, see
Section 2.
2) If $\epsilon$ is positive and small, $a_{i},a_{j}\in\partial M$,
$\lambda_{i},\lambda_{j}\geq\frac{1}{\epsilon}$,
$\frac{1}{\Lambda}\leq\frac{\Lambda_{i}}{\lambda_{j}}\leq\Lambda$, and
$\lambda_{i}d_{\hat{g}}(a_{i},a_{j})\geq 4\overline{C}R$, then

$\varphi_{a_{j},\lambda_{j}}(\cdot)=G(a_{j},\cdot)+O(1)\;\;\;\text{in}\;\;B^{a_{i}}_{a_{i}}(\frac{R}{\lambda_{i}}),$

where here $O(1)$ means here $O_{A,\bar{\lambda},\epsilon}(1)$, with
$A=(a_{i},a_{j})$, $\bar{\lambda}=(\lambda_{i},\lambda_{j})$ and for the
meaning of $O_{A,\lambda,\epsilon}(1)$, see Section 2.

######  Lemma 5.9.

There exists $\Gamma_{0}$ and $\tilde{\Lambda}_{0}$ two large positive
constant such that for every $a\in\partial M$,
$\lambda\geq\tilde{\Lambda}_{0}$, and $w\in
F_{a,\lambda}:=\\{w\in\mathcal{H}_{\frac{\partial}{\partial
n}},\overline{w}_{(Q,T)}=\left\langle\varphi_{a,\lambda},w\right\rangle_{\mathbb{P}^{4,3}}=\left\langle
v_{r},w\right\rangle_{\mathbb{P}^{4,3}}=0,\;r=1\cdots,\bar{k}\\}$, we have

(124) $\oint_{\partial
M}e^{3\hat{\delta}_{a,\lambda}}w^{2}dV_{g_{a}}\leq\Gamma_{0}||w||^{2}_{\mathbb{P}^{4,3}}.$

######  Lemma 5.10.

Assuming that $\eta$ is a small positive real number with $0<2\eta<\varrho$
where $\varrho$ is as in (41), then there exists a small positive constant
$c_{0}:=c_{0}(\eta)$ and $\Lambda_{0}:=\Lambda_{0}(\eta)$ such that for every
$a_{i}\in\partial M$ concentrations points with $d_{\hat{g}}(a_{i},a_{j})\geq
4\overline{C}\eta$ where $\bar{C}$ is as in (39), for every $\lambda_{i}>0$
concentrations parameters satisfying $\lambda_{i}\geq\Lambda_{0}$, with
$i=1,\cdots,k$, and for every $w\in
E_{A,\bar{\lambda}}^{*}=\cap_{i=1}^{k}E^{*}_{a_{i},\lambda_{i}}$ with
$A:=(a_{1},\cdots,a_{k}$), $\bar{\lambda}:=(\lambda_{1},\cdots,\lambda_{k})$
and $E^{*}_{a_{i},\lambda_{i}}=\\{w\in\mathcal{H}_{\frac{\partial}{\partial
n}}:\;\;\left\langle\varphi_{a_{i},\lambda_{i}},w\right\rangle_{\mathbb{P}^{4,3}}=\left\langle\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial\lambda_{i}},w\right\rangle_{\mathbb{P}^{4,3}}=\left\langle\frac{\partial\varphi_{a_{i},\lambda_{i}}}{\partial
a_{i}},w\right\rangle_{\mathbb{P}^{4,3}}=\overline{w}_{(Q,T)}=\left\langle
v_{r},w\right\rangle_{\mathbb{P}^{4,3}}=0,\;r=1\cdots,\bar{k}=0\\}$, there
holds

(125) $||w||^{2}_{\mathbb{P}^{4,3}}-6\sum_{i=1}^{k}\oint_{\partial
M}e^{3\hat{\delta}_{a_{i},\lambda_{i}}}w^{2}dS_{g_{a_{i}}}\geq
c_{0}||w||^{2}_{\mathbb{P}^{4,3}}.$

## References

  * [1] Ahmedou M., Ndiaye C. B Morse theory and the resonant $Q$-curvature problem, Preprint: arxiv:.1409.7919.
  * [2] Ahmedou, M., Kallel S., Ndiaye, C. B., The resonant boundary Q-curvature problem and boundary-weighted barycenters. Preprint: arXiv:1604.03745.
  * [3] Aubin T., Some nonlinear problems in Riemannian geometry, Springer Monographs in Mathematics, Springer-Verlag, Berlin 1998.
  * [4] Aubin T, Bahri A., Methods of algebraic topology for the problem of prescribed scalar curvature. J. Math. Pures Appl. (9) 76 (1997), no. 6, 525-549.
  * [5] Aubin T., Bahri A., A topological hypothesis for the problem of prescribed scalar curvature. J. Math. Pures Appl. (9) 76 (1997), no. 10, 843-850.
  * [6] Bahri A., Critical points at infinity in some variational problems, Research Notes in Mathematics, 182, Longman-Pitman, London, 1989.
  * [7] Bahri A., Un problème variationel sans compacité dans la géométrie de contact, Comptes Rendus Mathématique Académie des Sciences, Paris, Série I 299 (1984) 754-760.
  * [8] Bahri A., Pseudo-orbits of Contact Forms, Pitman Research Notes in Mathematics Series, 173. Longman Scientific & Technical, Harlow, 1988.
  * [9] Bahri A., An invariant for Yamabe-type flows with applications to scalar-curvature problems in high dimension. A celebration of John F. Nash, Jr. Duke Math. J. 81 (1996), no. 2, 323-466.
  * [10] Bahri A., Brezis H., Equations elliptiques non linéaires sur des variétés avec exposant de Sobolev critique. C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), no. 11, 573–576.
  * [11] Bahri A., Coron J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41-3 (1988), 253-294.
  * [12] Bahri A., and Rabinowitz P., Periodic solutions of Hamiltonian systems of $3$-body, Ann. Inst. Poincaré, Anal. non linéaire 8(1991), 561-649.
  * [13] Bahri, A., Coron, J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. Pure Appl. Math. 41-3 (1988),-294.
  * [14] Branson T.P., Oersted., Explicit functional determinants in four dimensions, Proc. Amer. Math. Soc 113-3(1991), 669-682.
  * [15] Bredon G.E., Topology and geometry, Graduate Texts in Mathematics, 139, 1997. cand., 57-2 (1995), 293-345.
  * [16] Brendle S., Curvature flows on surfaces with boundary, Math. Ann. 324 (2002), no. 3, 491-519.
  * [17] Brendle S., A family of curvature flows on surfaces with boundary, Math. Z. 241, 829-869 (2002).
  * [18] Brendle S., A generalization of the Yamabe flow for manifolds with boundary. Asian J. Math. 6 (2002), no. 4, 625–644.
  * [19] Brendle S., Convergence of the Yamabe flow for arbitrary initial energy, J. Diff. Geom. 69 (2005),217-278.
  * [20] Brendle S., Chen S. Z., An existence theorem for the Yamabe problem on manifolds with boundary. J. Eur. Math. Soc. (JEMS) 16 (2014), no. 5, 991-1016.
  * [21] Catino G., Ndiaye C. B., Integral pinching results for manifolds with boundary, Ann. Sc. Norm. Super. Pisa Cl. Sci. 9 (2010), 785-813.
  * [22] Chang S.Y.A., Qing J.,., Yang P.C.,Compactification of a class of conformally flat $4$-manifold, Invent. Math. 142-1(2000), 65-93.
  * [23] Chang S.Y.A., Qing J.,The Zeta Functional Determinants on manifolds with boundary 1. The Formula, Journal of Functional Analysis 147, 327-362 (1997)
  * [24] Chang S.Y.A., Qing J.,The Zeta Functional Determinants on manifolds with boundary II. Extremal Metrics and Compactness of Isospectral Set, Journal of Functional Analysis 147, 363-399 (1997)
  * [25] Chang S.Y.A., Qing J.,., Yang P.C.,On the Chern-Gauss-Bonnet integral for conformal metrics on ${\bf R}^{4}$, Duke Math. J.03-3(2000),523-544.
  * [26] Chen C. C., Lin C. S, Topological degree for a mean field equation on Riemann surfaces, Comm. Pure, Appl. Math, 56-12 (2003), 1667-1727.
  * [27] Chen, S.S., Conformal deformation on manifolds with boundary, Geom. Funct. Anal. 19 (2009), no. 4, 1029–1064.
  * [28] Dold A., Algebraic topology, Lectures on algebraic topology. Reprint of the 1972 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995. xii+377 pp. ISBN: 3-540-58660-1 55-02 (01A75).
  * [29] Djadli Z., Malchiodi A., Existence of conformal metrics with constant $Q$-curvature, Ann. of Math. (2) 168 (2008), no. 3, 813–858.
  * [30] Druet O., Robert F., Bubbling phenomena for fourth-order four-dimensional PDEs with exponential growth, Proc. Amer. Math. Soc 134(2006) no. 3, 897-908.
  * [31] Escobar J.F., The Yamabe problem on manifolds with boundary, Journal Differential Geometry 35 (1992) no.1, 21-84.
  * [32] Gilbar D., Trudinger N., Elliptic Partial Differential Equations of Second Order, 2nd edition, Springr-Verlag, 1983.
  * [33] Günther M., Conformal normal coordinates, Ann. Global. Anal. Geom. 11 (1993), 173-184.
  * [34] Kallel S., Karoui R., Symmetric joins and weighted barycenters, Advanced Nonlinear Studies, 11(2011), 117–143.
  * [35] Lee J., Parker T., The Yamabe problem, Bull. A.M.S. 17 (1987),37-81.
  * [36] Lin C. S., Wei J., Sharp estimates for bubbling solutions of a fourth order mean field equation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 4, 599-630.
  * [37] Lucia M., A deformation lemma with an application to a mean field equation. Topol. Methods Nonlinear Anal. 30 (2007), no. 1, 113–138.
  * [38] Lucia M., Orák J., A minimax theorem in the presence of unbounded Palais-Smale sequences, Israel J. Math. 172 (2009).
  * [39] Malchiodi A., Morse theory and a scalar field equation on compact surfaces, Adv. Diff. Eq., 13 (2008), 1109-1129.
  * [40] Malchiodi A., Compactness of solutions to some geometric fourth-order equations, J. Reine Angew. Math. 594 (2006), 137–174.
  * [41] Marques F. C., Existence results for the Yamabe problem on manifolds with boundary, Indiana Univ. Math. J. (2005), 1599-1620.
  * [42] Ndiaye C. B., Constant $Q$-curvature metrics in arbitrary dimension, J. Funct. Anal. 251 (2007), no. 1, 1–58. 53.
  * [43] Ndiaye C.B., Conformal metrics with constant $Q$-curvature for manifolds with boundary, Comm. Anal. Geom. 16 (2008), no. 5, 1049–1124.
  * [44] Ndiaye C.B., Constant $T$-curvature conformal metric on 4-manifolds with boundary, Pacific J. Math. 240 (2009), no. 1, 151-184.
  * [45] Ndiaye C. B., $Q$-curvature flow on manifolds with boundary, Math. Z. 269 (2011), 83-114. .
  * [46] Ndiaye C. B., Algebraic topological methods for the supercritical Q-curvature problem. Adv. Math, 277, 277, pp 56–99, 2015.
  * [47] Ndiaye C. B., Topological methods for the resonant Q-curvature problem in arbitrary even dimension, Journal of Geometry and Physics, Volume 140, June 2019, Pages 178-213.
  * [48] Ndiaye C. B., Xiao J, An upper bound of the total $Q$-curvature and its isoperimetric deficit for higher-dimensional conformal Euclidean metrics, Calc. Var. Partial Differential Equations, 38, no. 1-2, 1–27, 2010.
  * [49] Paneitz S., A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, preprint, 1983.
  * [50] Osgood B., Phillips R., Sarnak P., Extremal of determinants of Laplacians. J. Funct. Anal, 80, 148-211 (1988).
  * [51] Weinstein G., Zhang L., The profile of bubbling solutions of a class of fourth order geometric equations on 4-manifolds, J. Funct. Anal. 257 (2009), no. 12, 3895–3929