# Bubbling solutions for mean field equations with variable intensities on
compact Riemann surfaces

Pablo Figueroa Instituto de Ciencias Físicas y Matemáticas, Facultad de
Ciencias, Universidad Austral de Chile, Campus Isla Teja, Valdivia, Chile.
E-mail<EMAIL_ADDRESS>Author partially supported by grant Fondecyt
Regular Nº1201884, Chile.

(August 27, 2024)

###### Abstract

For an asymmetric sinh-Poisson problem arising as a mean field equation of
equilibrium turbulence vortices with variable intensities of interest in
hydrodynamic turbulence, we address the existence of bubbling solutions on
compact Riemann surfaces. By using a Lyapunov-Schmidt reduction, we find
sufficient conditions under which there exist bubbling solutions blowing up at
$m$ different points of $S$: positively at $m_{1}$ points and negatively at
$m-m_{1}$ points with $m\geq 1$ and $m_{1}\in\\{0,1,...,m\\}$. Several
examples in different situations illustrate our results in the sphere
$\mathbb{S}^{2}$ and flat two-torus $\mathbb{T}$ including non negative
potentials with zero set non empty.

_Keywords_ : mean field equation, exponential nonlinearity, blow up solutions,
Lyapunov-Schmidt reduction

_2020 AMS Subject Classification_ : 35B44, 35J15, 35J60, 35R01

##  1 Introduction

Let $(S,g)$ be a compact Riemann surface and consider the problem

$-\Delta_{g}u=\lambda_{1}\left({V_{1}(x)e^{u}\over\int_{S}V_{1}e^{u}dv_{g}}-\frac{1}{|S|}\right)-\lambda_{2}\tau\left({V_{2}(x)e^{-\tau
u}\over\int_{S}V_{2}e^{-\tau u}dv_{g}}-\frac{1}{|S|}\right),$ (1.1)

where $\lambda_{1},\lambda_{2}\geq 0$, $\tau>0$, $V_{1}$ and $V_{2}$ are
smooth nonnegative potentials in $S$ and $|S|$ is the area of $S$. Here,
$\Delta_{g}$ is the Laplace-Beltrami operator and $dv_{g}$ is the area element
in $(S,g)$. This equation have attracted a lot of attention in recent years
due to its relevance in the statistical mechanics description of
2D-turbulence, as initiated by Onsager [49]. Precisely, in this context, under
a _deterministic_ assumption on the distribution of the vortex circulations,
Sawada, Suzuki [56] derive the following equation:

$\begin{array}[]{ll}\displaystyle-\Delta_{g}u=\lambda\int\limits_{[-1,1]}\alpha\bigg{(}{e^{\alpha
u}\over\int_{S}e^{\alpha
u}dv_{g}}-{1\over|S|}\bigg{)}d\mathcal{P}(\alpha)&\hbox{in}\ S\end{array}$
(1.2)

where $u$ is the stream function of a turbulent Euler flow, $\lambda>0$ is a
physical constant related to the inverse temperature and $\mathcal{P}$ is a
Borel probability measure in $[-1,1]$ describing the point-vortex intensities
distribution.

Equation (1.2) includes several well-known problems depending on a suitable
choice of $\mathcal{P}$. For instance, if $\mathcal{P}=\delta_{1}$ is
concentrated at $1$, then (1.2) is related to the classical mean field
equation

$-\Delta_{g}u=\lambda\left(\frac{Ve^{u}}{\int_{S}Ve^{u}\,dv_{g}}-\frac{1}{|S|}\right)\quad\text{in}\quad
S,$ (1.3)

where $V$ is a smooth nonnegative function on $S$. The latter equation has
been studied in several contexts such as conformal geometry [11, 10, 40],
statistical mechanics [6, 7, 12, 41] and the relativistic Chern-Simons-Higgs
model when $S$ is a flat two-torus [47, 57, 58]. Notice that solutions of
(1.3) are critical points of the functional

$J_{\lambda}(u)={1\over 2}\int_{S}|\nabla
u|^{2}_{g}\,dv_{g}-\lambda\log\left(\int_{S}Ve^{u}\,dv_{g}\right),\qquad
u\in\bar{H},$

where $\bar{H}=\\{u\in H^{1}(S):\int_{S}udv_{g}=0\\}$. Minimizers of
$J_{\lambda}$ for $\lambda<8\pi$ can be found by using Moser-Trudinger’s
inequality. The situation in the supercritical regime $\lambda\geq 8\pi$
becomes subtler and the existence of solutions could depend on the topology
and the geometry of the surface $S$ (or the domain). A degree argument has
been proved in [13, 14] by Chen and Lin, completing a program initiated by Li
[43], and has received a variational counterpart in [18, 46] by means of
improved forms of the Moser-Trudinger inequality.

Equation (1.1) is also related to (1.2) when
$\mathcal{P}=\sigma\delta_{1}+(1-\sigma)\delta_{-\tau}$ with $\tau\in[-1,1]$
and $\sigma\in[0,1]$. Furthermore, (1.1) is the Euler-Lagrange equation of the
functional

$J_{\lambda_{1},\lambda_{2}}(u)={1\over 2}\int_{S}|\nabla
u|_{g}^{2}\,dv_{g}-\lambda_{1}\log\left(\int_{S}V_{1}e^{u}dv_{g}\right)-\lambda_{2}\log\left(\int_{S}V_{2}e^{-\tau
u}dv_{g}\right),\>\>u\in\bar{H}.$ (1.4)

If $\tau=1$ and $V_{1}=V_{2}\equiv 1$ problem (1.1) reduces to mean field
equation of the equilibrium turbulence, see [5, 34, 37, 48, 52] or its related
sinh-Poisson version, see [3, 4, 33, 38, 39], which have received a
considerable interest in recent years. Precisely, in [48] a Trudinger-Moser
type inequality was proved: if $\lambda_{1},\lambda_{2}\in[0,8\pi)$, which can
be called the subcritical case, then solutions to (1.1) are the minimizers of
$J_{\lambda_{1},\lambda_{2}}$, since this functional is coercive; but if
$\lambda_{1},\lambda_{2}\in[0,8\pi]$ and either $\lambda_{1}=8\pi$ or
$\lambda_{2}=8\pi$ then the functional $J_{\lambda_{1},\lambda_{2}}$ still has
lower bound but it is not coercive. A minimization technique is no longer
possible if $\lambda_{i}>8\pi$ for some $i=1,2$ since
$J_{\lambda_{1},\lambda_{2}}$ becomes unbounded from below. In general, one
needs to apply variational methods to obtain the existence of critical points
(generally of saddle type) for $J_{\lambda_{1},\lambda_{2}}$. Several results
in the supercritical case can be found in [52, 59, 60]. A quantization
property was derived in [38] for a blow-up sequence $\\{u_{n}\\}_{n}$ to (1.1)
with $\tau=1$, one has

$m_{k}(p)=\lim_{r\to
0}\lim_{n\to+\infty}\frac{\lambda_{k,n}\int_{B_{r}(p)}V_{k}e^{(-1)^{k-1}u_{n}}\,dv_{g}}{\int_{S}V_{k}e^{(-1)^{k-1}u_{n}}\,dv_{g}}\in
8\pi\text{I\\!N},\quad k=1,2,$ (1.5)

extending the corresponding ones for (1.3) in [44] and for (1.1) with $\tau=1$
and $V_{1}=V_{2}\equiv 1$ in [39].

Concerning the version of problem (1.1) on bounded domains Pistoia and
Ricciardi built in [50] sequences of blowing-up solutions when $\tau>0$ and
$\lambda_{1},\lambda_{2}\tau^{2}$ are close to $8\pi$, while in [51] the same
authors built an arbitrary large number of sign-changing blowing-up solutions
when $\tau>0$ and $\lambda_{1},\lambda_{2}\tau^{2}$ are close to suitable (not
necessarily integer) multiples of $8\pi.$ Ricciardi and Takahashi in [53]
provided a complete blow-up picture for solution sequences of (1.1) and
successively in [54] Ricciardi et al. constructed min-max solutions when
$\lambda_{1}\to 8\pi^{+}$ and $\lambda_{2}\to 0$ on a multiply connected
domain (in this case the nonlinearity $e^{-\tau u}$ may be treated as a lower-
order term with respect to the main term $e^{u}$).

In a compact Riemann surface $S$, a blow-up analysis in subcritical case
$\lambda_{1}<8\pi$ and $\lambda_{2}<\frac{8\pi}{\tau^{2}}$, and supercritical
case $\lambda_{1}<16\pi$ and $\lambda_{2}<\frac{16\pi}{\tau^{2}}$,
characterizing the blow-up masses $m_{k}(p)$, $k=1,2$, defined similarly as in
(1.5), has been obtained in [36], when $0<\tau<1$. Furthermore, some existence
results are deduced. The authors in [55] obtain the minimal blow-up masses and
proved an existence result which generalize the one obtained in [52] for
$\tau=1$.

To the extend our knowledge, there are by now just few results concerning the
existence of bubbling solutions to (1.1) and its variants in different
framework. For instance bubbling solutions have been constructed for a sinh-
Poisson equation ($\tau=1$) on bounded domains in [3, 4] with Dirichlet
boundary condition and recently in [32] with Robin boundary condition.
Furthermore, recently in [24] and [29], the authors have constructed blowing-
up solutions on pierced domains with Dirichlet boundary condition for any
$\tau>0$. See also [50, 51] for generalizations to $\tau>0$ of results
obtained in [3, 33] for $\tau=1$, respectively. The construction of sign-
changing bubble tower solutions for sinh-Poisson type equations on pierced
domains have been addressed in [30].

By following some ideas presented in [3, 23], we are interested in to
construct bubbling solutions $u_{\lambda_{1},\lambda_{2}}$ to (1.1) with
$m_{1}$ positive bubbles and $m_{2}$ negative bubbles suitable centered at
$m=m_{1}+m_{2}$ different points of $S$ as both $\lambda_{1}\to 8\pi m_{1}$
and $\lambda_{2}\tau^{2}\to 8\pi m_{2}$, with $m_{1}\in\\{0,\dots,m\\}$. To
this aim, introduce the Green function $G(x,p)$ with pole at $p\in S$ as the
solution of

$\left\\{\begin{array}[]{ll}-\Delta_{g}G(\cdot,p)=\delta_{p}-\frac{1}{|S|}&\text{in
$S$}\\\ \int_{S}G(x,p)dv_{g}=0&\end{array}\right.$ (1.6)

where $\delta_{p}$ denote a Dirac mass in $p\in S$. Define for
$\xi=(\xi_{1},\dots,\xi_{m})\in\tilde{S}^{m}\setminus\Delta$ the functional

$\begin{split}\varphi_{m}^{*}(\xi)=&\ \frac{1}{4\pi}\sum_{j=1}^{m_{1}}\log
V_{1}(\xi_{j})+\frac{1}{4\pi\tau^{2}}\sum_{j=m_{1}+1}^{m}\log
V_{2}(\xi_{j})+\sum_{j=1}^{m_{1}}H(\xi_{j},\xi_{j})+{1\over\tau^{2}}\sum_{j=m_{1}+1}^{m}H(\xi_{j},\xi_{j})\\\
&+\sum_{j=1}^{m_{1}}\sum_{i=1\atop
i\not=j}^{m_{1}}G(\xi_{i},\xi_{j})-{2\over\tau}\sum_{j=1}^{m_{1}}\sum_{i=m_{1}+1}^{m}G(\xi_{i},\xi_{j})+{1\over\tau^{2}}\sum_{j=m_{1}+1}^{m}\sum_{i=m_{1}+1\atop
i\not=j}^{m}G(\xi_{i},\xi_{j}),\end{split}$ (1.7)

where $H(x,\xi)$ is the regular part of $G(x,\xi)$,
$\tilde{S}=\\{V_{1},V_{2}>0\\}$ and $\Delta=\\{\xi\in
S^{m}:\,\xi_{i}=\xi_{j}\hbox{ for }i\not=j\\}$ is the diagonal set in $S^{m}$
with $m=m_{1}+m_{2}$. Setting for $j\in\mathcal{J}_{1}:=\\{1,\dots,m_{1}\\}$

$\rho_{j}(x):=V_{1}(x)\exp\bigg{(}8\pi H(x,\xi_{j})+8\pi\sum\limits_{i=1\atop
i\neq
j}^{m_{1}}G(x,\xi_{i})-{8\pi\over\tau}\sum_{i=m_{1}+1}^{m}G(x,\xi_{i})\bigg{)},$
(1.8)

and for $j\in\mathcal{J}_{2}:=\\{m_{1}+1,\dots,m\\}$

$\rho_{j}(x):=V_{2}(x)\exp\bigg{(}8\pi
H(x,\xi_{j})-8\pi\tau\sum_{i=1}^{m_{1}}G(x,\xi_{i})+8\pi\sum_{i=m_{1}+1\atop
i\neq j}^{m}G(x,\xi_{i})\bigg{)},$ (1.9)

both for $\xi\in S^{m}\setminus\Delta$ we introduce the notation

$A_{k}^{*}(\xi)=4\pi\sum_{j\in\mathcal{J}_{k}}\left[\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j})\right],\quad
k=1,2$ (1.10)

where $K$ is the Gaussian curvature of $(S,g)$. The sign of $A_{k}^{*}$,
$k=1,2$ allows us to obtain a first existence result of bubbling solutions and
several consequences, see Theorem 2.1 and section 2. Unfortunately, there are
cases where the sign of $A_{k}^{*}(\xi)$ either $k=1$ or $k=2$ or both is not
available. For instance the case $S=\mathbb{T}$, $V_{1}=V_{2}\equiv 1$,
$m_{1}=m_{2}=1$ and $\tau=1$. See also [23] for several examples in case
$\lambda_{2}=0$, namely, $m_{2}=0$, that could be extended here. Following
ideas presented in [23], in all these situations, a more refined analysis is
necessary. To this aim, introduce the quantities for $k=1,2$

$\displaystyle B_{k}^{*}(\xi)$ $\displaystyle=$
$\displaystyle-2\pi\sum_{j\in\mathcal{J}_{k}}[\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j})]\log\rho_{j}(\xi_{j})-\frac{A_{k}^{*}(\xi)}{2}$
$\displaystyle+\lim_{r\to
0}\bigg{[}8\int_{S\setminus\cup_{j\in\mathcal{J}_{k}}B_{r}(\xi_{j})}V_{1}e^{8\pi(-\tau)^{k-1}\sum\limits_{j=1}^{m_{1}}G(x,\xi_{j})+8\pi(-\tau)^{k-2}\sum\limits_{l=m_{1}+1}^{m}G(x,\xi_{l})}dv_{g}-\frac{8\pi}{r^{2}}\sum\limits_{j\in\mathcal{J}_{k}}\rho_{j}(\xi_{j})$
$\displaystyle\qquad\qquad-A_{k}^{*}(\xi)\log\frac{1}{r}\bigg{]}$

where $B_{r}(\xi)$ denotes the pre-image of $B_{r}(0)$ through the isothermal
coordinate system at $\xi$. These types of quantities has been first used and
derived by Chang, Chen and Lin in [9] in the study of the mean field equation
on bounded domains with Dirichlet boundary condition; for the case of the
torus see [15]. Moreover, the constant $B_{k}^{*}(\xi)$ has also been used in
the construction of non-topological condensates for the relativistic abelian
Chern-Simons-Higgs model as the Chern-Simons parameter tends to zero, see [19,
23, 45]. Our main result states as follows.

######  Theorem 1.1.

Let $\mathcal{D}\subset\subset\tilde{S}^{m}\setminus\Delta$ be a stable
critical set of $\varphi_{m}^{*}$. Assume that

$\hbox{either }A_{1}^{*}(\xi)>0\>(<0\hbox{ resp.)}\qquad\hbox{or}\qquad
A_{1}^{*}(\xi)=0,\>B_{1}^{*}(\xi)>0\>(<0\hbox{ resp.)}$ (1.12)

and

$\hbox{either }A_{2}^{*}(\xi)>0\>(<0\hbox{ resp.)}\qquad\hbox{or}\qquad
A_{2}^{*}(\xi)=0,\>B_{2}^{*}(\xi)>0\>(<0\hbox{ resp.)}$ (1.13)

do hold in a closed neighborhood $U$ of $\mathcal{D}$ in
$\tilde{S}^{m}\setminus\Delta$. Then, for all $\lambda_{1}$ in a small right
(left resp.) neighborhood of $8\pi m_{1}$ and $\lambda_{2}\tau^{2}$ in a small
right (left resp.) neighborhood of $8\pi m_{2}$ there is a solution
$u_{\lambda_{1},\lambda_{2}}$ of (1.1) which concentrate (along sub-sequences)
at $m$ points, positively at $q_{1},\dots,q_{m_{1}}$ and negatively at
$q_{m_{1}+1},\dots,q_{m}$, in the sense

$\frac{\lambda_{1}V_{1}e^{u_{\lambda_{1},\lambda_{2}}}}{\int_{S}V_{1}e^{u_{\lambda_{1},\lambda_{2}}}dv_{g}}\rightharpoonup
8\pi\sum_{j=1}^{m_{1}}\delta_{q_{j}}\quad\text{ and
}\quad\frac{\lambda_{2}\tau^{2}V_{2}e^{-\tau
u_{\lambda_{1},\lambda_{2}}}}{\int_{S}V_{2}e^{-\tau
u_{\lambda_{1},\lambda_{2}}}dv_{g}}\rightharpoonup
8\pi\sum_{j=m_{1}+1}^{m}\delta_{q_{j}}$ (1.14)

as simultaneously $\lambda_{1}\to 8\pi m_{1}$ and $\lambda_{2}\tau^{2}\to 8\pi
m_{2}$ for some $q\in\mathcal{D}$.

Notice that along with (1.14) there hold
$(-\tau)^{k-1}u_{\lambda_{1},\lambda_{2}}-\log\int_{S}V_{k}e^{(-\tau)^{k-1}u_{\lambda_{1},\lambda_{2}}}\to-\infty$
in $C_{\text{loc}}(S\setminus\\{q_{1},\dots,q_{m}\\})$ and

$\sup_{\mathcal{O}_{j}}\bigg{(}(-\tau)^{k-1}u_{\lambda_{1},\lambda_{2}}-\log\int_{S}V_{k}e^{(-\tau)^{k-1}u_{\lambda_{1},\lambda_{2}}}\bigg{)}\to+\infty$

as simultaneously $\lambda_{1}\to 8\pi m_{1}$ and $\lambda_{2}\tau^{2}\to 8\pi
m_{2}$, for any neighborhood $\mathcal{O}_{j}$ of $q_{j}$ in $S$ with $k=1$
for $j=1,\dots,m_{1}$ and $k=2$ for $j=m_{1}+1,\dots,m$. Hence, we get that
$u_{\lambda_{1},\lambda_{2}}$ concentrates positively at
$q_{1},\dots,q_{m_{1}}$ and negatively at $q_{m_{1}+1},\dots,q_{m}$ as
simultaneously $\lambda_{1}\to 8\pi m_{1}$ and $\lambda_{2}\tau^{2}\to 8\pi
m_{2}$. As in [23], the notion of stability we are using here is the one
introduced in [42], see Definition 2.1 below. Conditions (1.12)-(1.13) on a
neighborhood of $\mathcal{D}$ are required to deal a with stable critical set
$\mathcal{D}$ in the sense below. Arguing as in Remark 4.5 in [23], the same
conclusion of Theorem 1.1 follows under the validity of conditions
(1.12)-(1.13) just on $\mathcal{D}=\\{\xi_{0}\\}$, where $\xi_{0}$ is a non-
degenerate local minimum/maximum point of $\varphi_{m}^{*}$. Similarly,
Theorem 1.1 is also valid in the special case
$|A_{k}^{*}(\xi)|=O(|\nabla\varphi_{m}^{*}(\xi)|_{g})$, $k=1,2$ in a
neighborhood of $\mathcal{D}$ and $B_{k}^{*}(\xi)>0$ in $\mathcal{D}$.

Now, we can address the case $S=\mathbb{T}$, $V_{1}=V_{2}\equiv 1$,
$m_{1}=m_{2}=1$ and $\tau=1$. When $\mathbb{T}$ is a rectangle, the constants
like $B_{k}^{*}(\xi)$, $k=1,2$, has been used by Chen, Lin nd Wang [15] in the
computation of the Leray-Schauder degree. Due to $H(x,x)$ is constant in
$\mathbb{T}$, we deduce that
$\varphi_{2}^{*}(\xi)=-2G(\xi_{1},\xi_{2})+\text{const.}$. Also, it is known
that the Green’s function satisfies $G(\xi_{1},\xi_{2})=G(\xi_{1}-\xi_{2},0)$
and the function $G(\cdot,0)$ has exactly three non-degenerate critical points
$q_{1}$, $q_{2}$ (saddle points) and $q_{3}$ (minimum point). According to (1)
we have that for $i,k\in\\{1,2\\}$

$B_{k}^{*}(\xi)=\lim_{r\to 0}\left[8\int_{\mathbb{T}\setminus
B_{r}(\xi_{k})}e^{8\pi G(x,\xi_{k})-8\pi G(x,\xi_{i})}-{8\pi\over
r^{2}}e^{8\pi H(\xi_{k},\xi_{k})-8\pi G(\xi_{i},\xi_{k})}\right],\ \ i\neq k.$

Assuming that $\mathbb{T}=-\mathbb{T}$ it follows that
$B_{1}^{*}(\xi)=B_{2}^{*}(\xi)$, $\xi=(\xi_{1},\xi_{2})$, since
$G(z,0)=G(-z,0)$. Furthermore, it is known that $B_{1}^{*}(\xi)>0$ when either
$\xi_{1}-\xi_{2}=q_{1}$ or $\xi_{1}-\xi_{2}=q_{2}$ and $B_{1}^{*}(\xi)<0$ when
either $\xi_{1}-\xi_{2}=q_{3}$. By Theorem 1.1 we deduce the existence of

  * •

two distinct families of solutions, for $\lambda_{1},\lambda_{2}$ in a small
right neighborhood of $8\pi$, concentrating positively at $\xi_{1}$ and
negatively at $\xi_{2}$ with either $\xi_{1}-\xi_{2}=q_{1}$ or
$\xi_{1}-\xi_{2}=q_{2}$ as $\lambda_{1}\to 8\pi$ and $\lambda_{2}\to 8\pi$;

  * •

one family of solutions, for $\lambda_{1}$, $\lambda_{2}$ in a small left
neighborhood of $8\pi$, concentrating positively at $\xi_{1}$ and negatively
at $\xi_{2}$ with $\xi_{1}-\xi_{2}=q_{3}$ as $\lambda_{1}\to 8\pi$ and
$\lambda_{2}\to 8\pi$.

The case $m_{2}=0$, namely, as $\lambda_{2}\tau^{2}\to 0^{+}$, can be also
addressed by this approach. Thus, we have that (1.1) can be seen as a
perturbation of (1.3). In this case the nonlinearity $e^{-\tau u}$ is treated
as a lower-order term with respect to the main term $e^{u}$. For simplicity we
denote $A(\xi)$ and $B(\xi)$ instead $A_{1}^{*}(\xi)$ and $B_{1}^{*}(\xi)$
with $m_{1}=m$ and $\mathcal{J}_{2}=\varnothing$, so that we have the
following result.

######  Theorem 1.2.

Let $\mathcal{D}\subset\subset\tilde{S}^{m}\setminus\Delta$ be a stable
critical set of $\varphi_{m}^{*}$. Assume that

$\hbox{either }A(\xi)>0\>(<0\hbox{ resp.)}\qquad\hbox{or}\qquad
A(\xi)=0,\>B(\xi)>0\>(<0\hbox{ resp.)}$ (1.15)

do hold in a closed neighborhood $U$ of $\mathcal{D}$ in
$\tilde{S}^{m}\setminus\Delta$. Then, for all $\lambda_{1}$ in a small right
(left resp.) neighborhood of $8\pi m_{1}$ and $\lambda_{2}\tau^{2}$ in a small
right neighborhood of $0$ there is a solution $u_{\lambda_{1},\lambda_{2}}$ of
(1.1) which concentrate positively (along sub-sequences) at $m$ points
$q_{1},\dots,q_{m}$

$\frac{\lambda_{1}V_{1}e^{u_{\lambda_{1},\lambda_{2}}}}{\int_{S}V_{1}e^{u_{\lambda_{1},\lambda_{2}}}dv_{g}}\rightharpoonup
8\pi\sum_{j=1}^{m}\delta_{q_{j}}\quad\text{in measure sense for some
$q\in\mathcal{D}$}$ $\text{and}\quad\frac{\lambda_{2}\tau^{2}V_{2}e^{-\tau
u_{\lambda_{1},\lambda_{2}}}}{\int_{S}V_{2}e^{-\tau
u_{\lambda_{1},\lambda_{2}}}dv_{g}}\to 0\quad\text{uniformly in $S$. }$

Notice that a similar result can be obtained in case $m_{1}=0$ and $m_{2}=m$,
namely, as $\lambda_{1}\to 0^{+}$ and $\lambda_{2}\tau^{2}\to 8\pi m$, and
$u_{\lambda_{1},\lambda_{2}}$ concentrates negatively at $m$ different points
of $S$. The same conclusion of Theorem 1.2 follows: on one hand, under the
validity of condition (1.15) just on $\mathcal{D}=\\{\xi_{0}\\}$, where
$\xi_{0}$ is a non-degenerate local minimum/maximum point of
$\varphi_{m}^{*}$; and on the other hand, in the special case
$|A(\xi)|=O(|\nabla\varphi_{m}^{*}(\xi)|_{g})$ in a neighborhood of
$\mathcal{D}$ and $B(\xi)>0$ in $\mathcal{D}$. See proof of Theorem 3.2 and
Remark 4.5 in [23] for more details. Several examples for Theorem 1.2 can be
derived from each example provided in [23] for the case $\lambda_{2}=0$.

The paper is organized as follows: Some consequences and examples are
presented in section 2. In Section 3, we construct a first approximation to a
solution to (1.1) with the required properties and we estimate the size of the
error of approximation with appropriate norms. In Section 4 we describe the
scheme of our proofs, by stating the principal results we need, and we give
the proof of our Theorem 1.1. Section 5 is devoted to the computation of the
expansion of the energy functional on the first approximation we constructed
in Section 3. The proof of Theorem 1.2 is done in Section 6. Sections 7 and 8
are devoted to prove the intermediate results we state in Section 4.

##  2 Consequences and examples

In this section we present several consequences of Theorem 1.1 and some
examples that illustrate our results in the sphere $\mathbb{S}^{2}$ and flat
two-torus $\mathbb{T}$. A special case of Theorem 1.1 is the following:

######  Theorem 2.1.

Let $\mathcal{D}\subset\subset\tilde{S}^{m}\setminus\Delta$ be a stable
critical set of $\varphi_{m}^{*}$. Assume that $A_{1}^{*}(\xi)>0$ ($<0$ resp.)
and $A_{2}^{*}(\xi)>0$ ($<0$ resp.) for all $\xi\in\mathcal{D}$. Then, for all
$\lambda_{1}$ in a small right (left resp.) neighborhood of $8\pi m_{1}$ and
$\lambda_{2}$ in a small right (left resp.) neighborhood of $\dfrac{8\pi
m_{2}}{\tau^{2}}$ there is a solution $u_{\lambda_{1},\lambda_{2}}$ of (1.1)
which concentrate (along sub-sequences) at $m$ points $q_{1},\dots,q_{m}$ in
the sense (1.14) for some $q\in\mathcal{D}$.

The notion of stability we are using here is the following:

######  Definition 2.1.

A critical set $\mathcal{D}\subset\subset\tilde{S}^{m}\setminus\Delta$ of
$\varphi_{m}$ is stable if for any closed neighborhood $U$ of $\mathcal{D}$ in
$\tilde{S}^{m}\setminus\Delta$ there exists $\delta>0$ such that, if
$\|G-\varphi_{m}\|_{C^{1}(U)}\leq\delta$, then $G$ has at least one critical
point in $U$. In particular, the minimal/maximal set of $\varphi_{m}$ is
stable (if $\varphi_{m}$ is not constant) as well as any isolated c.p. of
$\varphi_{m}$ with non-trivial local degree.

Notice that from the definition of $\rho_{j}$ in (1.8)-(1.9) and
$A_{k}^{*}(\xi)$ in (1.10), it is readily checked that

$A_{k}^{*}(\xi)=4\pi\sum_{j\in\mathcal{J}_{k}}\rho_{j}(\xi_{j})[\Delta_{g}\log
V_{1}(\xi_{j})+(-\tau)^{k-1}\frac{8\pi}{|S|}\left(m_{1}-{m_{2}\over\tau}\right)-2K(\xi_{j})],\quad
k=1,2$

for $\xi$ a c.p. of $\varphi_{m}^{*}$, in view of $\nabla\rho_{j}(\xi_{j})=0$
for all $j=1,\dots,m$. If $V_{1}\geq 0$ and $V_{2}\geq 0$ in $S$, then the
function $\varphi_{2}^{*}$ with $m_{1}=m_{2}=1$ always attains its maximum
value in $\tilde{S}^{2}\setminus\Delta$ and the maximal set is clearly stable.
Let us stress that $V_{1}$ and $V_{2}$ can vanish at some points of $S$. Thus,
we have deduced the following fact.

######  Corollary 2.1.

Assume that $V_{i}\geq 0$ in $S$ for $i=1,2$. If either
$\sup_{S}[2K-\Delta_{g}\log
V_{1}]<\frac{8\pi}{|S|}\big{(}1-{1\over\tau}\big{)}$ or
$\inf_{S}[2K-\Delta_{g}\log
V_{1}]>\frac{8\pi}{|S|}\big{(}1-{1\over\tau}\big{)}$ and either
$\sup_{S}[2K-\Delta_{g}\log V_{2}]<\frac{8\pi}{|S|}\big{(}1-\tau\big{)}$ or
$\inf_{S}[2K-\Delta_{g}\log V_{2}]>\frac{8\pi}{|S|}\big{(}1-\tau\big{)}$ then
there exist solutions $u_{\lambda_{1},\lambda_{2}}$ to (1.1) which concentrate
at two points, positively at $q_{1}$ and negatively at $q_{2}$, in the sense
(1.14) as $\lambda_{1}\to 8\pi$ and $\lambda_{2}\tau^{2}\to 8\pi$, where
$(q_{1},q_{2})$ is a maximum of $\varphi_{2}^{*}$ in
$\tilde{S}^{2}\setminus\Delta$.

When $S=\mathbb{S}^{2}$ we have that $K=\dfrac{4\pi}{|\mathbb{S}^{2}|}$, so
that, for $V_{1}=V_{2}\equiv 1$ and any $\tau>0$, Corollary 2.1 then provides
the existence of blow-up solutions $u_{\lambda_{1},\lambda_{2}}$ concentrating
at two points as $\lambda_{1}\to 8\pi$ and $\lambda_{2}\tau^{2}\to 8\pi$,
where $\lambda_{1}$ and $\lambda_{2}\tau^{2}$ belongs to a small _left_
neighborhood of $8\pi$. In case of a flat two-torus $S=\mathbb{T}$, $K=0$, so
that for $V_{1}=V_{2}\equiv 1$ and any $\tau>0$, $\tau\neq 1$, Corollary 2.1
then provides the existence of blow-up solutions $u_{\lambda_{1},\lambda_{2}}$
concentrating at two points as $\lambda_{1}\to 8\pi$ and
$\lambda_{2}\tau^{2}\to 8\pi$, where $\lambda_{1}$ belongs to a small _right_
(_left_ resp.) neighborhood of $8\pi$ if $\tau>1$ ($<1$ resp.) and
$\lambda_{2}\tau^{2}$ belongs to a small _left_ (_right_ resp.) neighborhood
of $8\pi$. However, the case $S=\mathbb{T}$, $V_{1}=V_{2}\equiv 1$,
$m_{1}=m_{2}=1$ and $\tau=1$ is an example for which $A_{1}^{*}$ and
$A_{2}^{*}$ vanishes in $\mathbb{T}^{2}\setminus\Delta$ and in particular at
c.p.’s.

Let us mention some examples where $V_{1}$ and $V_{2}$ vanish at some points
of $S$. Precisely, assume that

$V_{1}(x)=e^{-4\pi\sum\limits_{i=1}^{l_{1}}n_{1,i}G(x,p_{1,i})}\qquad\text{and}\qquad
V_{2}(x)=e^{-4\pi\sum\limits_{i=1}^{l_{2}}n_{2,i}G(x,p_{2,i})},$

with $n_{1,i},n_{2,i}>0$ and $p_{1,i},p_{2,j}\in S$, $i=1,\dots,l_{1}$ and
$j=1,\dots,l_{2}$ respectively. The zero sets are
$\\{p_{1,1},\dots,p_{1,l_{1}}\\}$ for $V_{1}$ and
$\\{p_{2,1},\dots,p_{2,l_{2}}\\}$ for $V_{2}$. So, for $m_{1}=m_{2}=1$, $m=2$
we have that

$\varphi_{2}^{*}(\xi)=-\sum\limits_{i=1}^{l_{1}}n_{1,i}G(\xi_{1},p_{1,i})-{1\over\tau^{2}}\sum\limits_{j=1}^{l_{2}}n_{2,j}G(\xi_{2},p_{2,j})-{2\over\tau}G(\xi_{1},\xi_{2}),$

and if $\xi$ is a c.p. of $\varphi_{2}^{*}$ then

$A_{k}^{*}(\xi)=4\pi\rho_{k}(\xi_{k})\left[-\frac{4\pi}{|S|}\sum\limits_{i=1}^{l_{k}}n_{k,i}+{8\pi\over|S|}\Big{(}1-\tau^{2k-3}\Big{)}-2K(\xi_{k})\right],\quad
k=1,2.$

In particular, if $S=\mathbb{S}^{2}$ then Corollary 2.1 provides the existence
of blow-up solutions $u_{\lambda_{1},\lambda_{2}}$ concentrating at two points
as $\lambda_{1}\to 8\pi$ and $\lambda_{2}\tau^{2}\to 8\pi$ when
$\sum_{i=1}^{l_{1}}n_{1,i}\neq 1-\frac{2}{\tau}$ and
$\sum_{j=1}^{l_{2}}n_{2,j}\neq 1-2\tau$. We deduce the same conclusion when
$S=\mathbb{T}$ and $\sum_{i=1}^{l_{1}}n_{1,i}\neq 2-\frac{2}{\tau}$ and
$\sum_{j=1}^{l_{2}}n_{2,j}\neq 2-2\tau$. Let us stress that there is no
restriction on $n_{1,i},n_{2,j}$’s if $\tau=1$.

Now, consider the case $m_{1}=m\geq 2$ and $m_{2}=1$, namely, $\lambda_{1}$
close to $8\pi m$ and $\lambda_{2}\tau^{2}$ close to $8\pi$. Roughly speaking,
if $u_{\lambda_{1},\lambda_{2}}$ concentrates negatively at $q$ then

$\lambda_{2}\tau\left(\frac{V_{2}e^{-\tau
u_{\lambda_{1},\lambda_{2}}}}{\int_{S}V_{2}e^{-\tau
u_{\lambda_{1},\lambda_{2}}}dv_{g}}-{1\over|S|}\right)\quad\text{ behaves like
}\quad 4\pi\cdot{2\over\tau}\left(\delta_{q}-{1\over|S|}\right)\quad\text{ as
$\lambda\tau^{2}\to 8\pi$}$

and equation (1.1) resembles the singular mean field equation

$-\Delta_{g}v=\lambda\left({he^{v}\over\int_{S}he^{v}dv_{g}}-\frac{1}{|S|}\right)-4\pi\alpha\left(\delta_{q}-{1\over|S|}\right)\qquad\text{in
$S$},$

with $\alpha=\frac{2}{\tau}$. According to a result of D’Aprile and Esposito
[17, Theorem 1.4], it follows that the functional

$\begin{split}\varphi_{m+1}^{*}(\xi)=&\ \frac{1}{4\pi}\sum_{j=1}^{m}\log
V_{1}(\xi_{j})+\frac{1}{4\pi\tau^{2}}\log
V_{2}(\xi_{m+1})+\sum_{j=1}^{m_{1}}H(\xi_{j},\xi_{j})+{1\over\tau^{2}}H(\xi_{m+1},\xi_{m+1})\\\
&+\sum_{j=1}^{m}\sum_{i=1\atop
i\not=j}^{m}G(\xi_{i},\xi_{j})-{2\over\tau}\sum_{j=1}^{m}G(\xi_{j},\xi_{m+1}),\end{split}$

has a $C^{1}$-stable critical value for $\xi_{m+1}\in S$ fixed under the
assumptions $S\neq\mathbb{S}^{2},\mathbb{RP}^{2}$ and $\frac{2}{\tau}\neq
1,\dots,m-1$. Thus, we deduce the next result.

######  Corollary 2.2.

Assume that $V_{i}>0$ in $S$ for $i=1,2$,
$S\neq\mathbb{S}^{2},\mathbb{RP}^{2}$ and $\dfrac{2}{\tau}\neq 1,\dots,m-1$.
If either $\sup_{S}[2K-\Delta_{g}\log
V_{1}]<\frac{8\pi}{|S|}\big{(}m-{1\over\tau}\big{)}$ or
$\inf_{S}[2K-\Delta_{g}\log
V_{1}]>\frac{8\pi}{|S|}\big{(}m-{1\over\tau}\big{)}$ and either
$\sup_{S}[2K-\Delta_{g}\log V_{2}]<\frac{8\pi}{|S|}\big{(}1-m\tau\big{)}$ or
$\inf_{S}[2K-\Delta_{g}\log V_{2}]>\frac{8\pi}{|S|}\big{(}1-m\tau\big{)}$ then
there exist solutions $u_{\lambda_{1},\lambda_{2}}$ to (1.1) which concentrate
at $m+1$ points, positively at $q_{1},\dots,q_{m}$ and negatively at
$q_{m+1}$, in the sense (1.14) as $\lambda_{1}\to 8\pi m$ and
$\lambda_{2}\tau^{2}\to 8\pi$, where $(q_{1},\dots,q_{m+1})$ is a max-min
critical point of $\varphi_{m+1}^{*}$ in $S^{m+1}\setminus\Delta$.

When $S=\mathbb{T}$ and $V_{1}=V_{2}\equiv 1$, for any $\tau>0$, $m\tau\neq 1$
and $\tau\notin\\{2,1,{2\over 3},\dots,{2\over m-1}\\}$, Corollary 2.1 then
provides the existence of blow-up solutions $u_{\lambda_{1},\lambda_{2}}$
concentrating at $m+1$ points as $\lambda_{1}\to 8\pi m$ and
$\lambda_{2}\tau^{2}\to 8\pi$, where $\lambda_{1}$ belongs to a small right
(left resp.) neighborhood of $8\pi m$ if $m\tau>1$ ($<1$ resp.) and
$\lambda_{2}\tau^{2}$ belongs to a small left (right resp.) neighborhood of
$8\pi$. Notice that a similar result can be obtained in case $m_{1}=1$ and
$m_{2}=m$, namely, $\lambda_{1}$ close to $8\pi$ and $\lambda_{2}\tau^{2}$
close to $8\pi m$.

Observe that on one hand, we generalize existence results of blowing-up
solutions for mean field equations (1.3) in [23] to an asymmetric problem
(1.1). And on the other hand, we perform, in a compact Riemann surface $S$, a
similar construction done for a sinh-Poisson equation in bounded domains with
Dirichlet boundary conditions by [3] and extended to an asymmetric case in
[50]. Both problems in [3, 50] do not contain any potential $V_{k}$ and the
existence of $C^{1}$-stable critical points of the corresponding
$\varphi_{m}^{*}$ implies the existence of blowing-up solutions. However, to
prove our results is not enough to assume the existence of $C^{1}$-stable
critical points of $\varphi_{m}^{*}$ in (1.7). Admissibility conditions in
terms of quantities either $A_{k}^{*}$’s or $B_{k}^{*}$’s have to be used, in
the same spirit of [23]. After completion of this work, we have learned that
in [1] the existence of $C^{1}$-stable critical points of vortex type
Hamiltonians, including $\varphi_{m}^{*}$ in (1.7), has been proved for a
surface $S$ which is not homeomorphic to the sphere nor the projective plane.

Finally, we point out that the type of arguments used to obtain our results
have been also developed in several previous works by various authors. Let us
quote a few papers from the vast literature concerning singular perturbation
problems with nonlinearities of exponential type [8, 21, 26, 27, 31].

##  3 Approximation of the solution

The main idea to construct approximating solutions of (1.1), as in [23], is to
use as “basic cells” the functions

$u_{\delta,\xi}(x)=u_{0}\Big{(}\frac{|x-\xi|}{\delta}\Big{)}-2\log\delta,\qquad\delta>0,\>\xi\in\text{I\\!R}^{2},$

where $\displaystyle u_{0}(r)=\log\frac{8}{(1+r^{2})^{2}}.$ They are all the
solutions of

$\left\\{\begin{array}[]{ll}\Delta u+e^{u}=0&\text{in $\text{I\\!R}^{2}$}\\\
\int_{\text{I\\!R}^{2}}e^{u}<\infty,&\end{array}\right.$

and do satisfy the following concentration property:
$e^{u_{\delta,\xi}}\rightharpoonup 8\pi\delta_{\xi}$ in measure sense as
$\delta\to 0$. We will use now isothermal coordinates to pull-back
$u_{\delta,\xi}$ in $S$. Let us recall that every Riemann surface $(S,g)$ is
locally conformally flat, and the local coordinates in which $g$ is conformal
to the Euclidean metric are referred to as isothermal coordinates (see for
example the simple existence proof provided by Chern [16]). For every $\xi\in
S$ it amounts to find a local chart $y_{\xi}$, with $y_{\xi}(\xi)=0$, from a
neighborhood of $\xi$ onto $B_{2r_{0}}(0)$ (the choice of $r_{0}$ is
independent of $\xi$) in which $g=e^{\hat{\varphi}_{\xi}(y_{\xi}(x))}dx$,
where $\hat{\varphi}_{\xi}\in C^{\infty}(B_{2r_{0}}(0),\mathbb{R})$. In
particular, $\hat{\varphi}_{\xi}$ relates with the Gaussian curvature $K$ of
$(S,g)$ through the relation:

$\Delta\hat{\varphi}_{\xi}(y)=-2K(y_{\xi}^{-1}(y))e^{\hat{\varphi}_{\xi}(y)}\qquad\hbox{
for }y\in B_{2r_{0}}(0).$ (3.1)

We can also assume that $y_{\xi}$, $\hat{\varphi}_{\xi}$ depends smoothly in
$\xi$ and that $\hat{\varphi}_{\xi}(0)=0$, $\nabla\hat{\varphi}_{\xi}(0)=0$.
We now pull-back $u_{\delta,0}$ in $\xi\in S$, for $\delta>0$, by simply
setting $\displaystyle
U_{\delta,\xi}(x)=u_{\delta,0}(y_{\xi}(x))=\log\frac{8\delta^{2}}{(\delta^{2}+|y_{\xi}(x)|^{2})^{2}}$
for $x\in y_{\xi}^{-1}(B_{2r_{0}}(0))$. Letting $\chi\in
C_{0}^{\infty}(B_{2r_{0}}(0))$ be a radial cut-off function so that
$0\leq\chi\leq 1$, $\chi\equiv 1$ in $B_{r_{0}}(0)$, we introduce the function
$PU_{\delta,\xi}$ as the unique solution of

$\left\\{\begin{array}[]{ll}-\Delta_{g}PU_{\delta,\xi}(x)=\chi_{\xi}(x)e^{-\varphi_{\xi}(x)}e^{U_{\delta,\xi}(x)}-\frac{1}{|S|}\int_{S}\chi_{\xi}e^{-\varphi_{\xi}}e^{U_{\delta,\xi}}dv_{g}&\text{in
}S\\\ \int_{S}PU_{\delta,\xi}dv_{g}=0,\end{array}\right.$ (3.2)

where $\chi_{\xi}(x)=\chi(|y_{\xi}(x)|)$ and
$\varphi_{\xi}(x)=\hat{\varphi}_{\xi}(y_{\xi}(x))$. Notice that the R.H.S. in
(3.2) has zero average and smoothly depends in $x$, and then (3.2) is uniquely
solvable by a smooth solution $PU_{\delta,\xi}$.

Let us recall the transformation law for $\Delta_{g}$ under conformal changes:
if $\tilde{g}=e^{\varphi}g$, then

$\Delta_{\tilde{g}}=e^{-\varphi}\Delta_{g}.$ (3.3)

Decompose now the Green function $G(x,\xi)$, $\xi\in S$, as $\displaystyle
G(x,\xi)=-\frac{1}{2\pi}\chi_{\xi}(x)\log|y_{\xi}(x)|+H(x,\xi),$ and by (1.6)
then deduce that

$\left\\{\begin{array}[]{ll}-\Delta_{g}H=-\frac{1}{2\pi}\Delta_{g}\chi_{\xi}\,\log|y_{\xi}(x)|-\frac{1}{\pi}\langle\nabla\chi_{\xi},\nabla\log|y_{\xi}(x)|\rangle_{g}-\frac{1}{|S|}&\text{in
$S$}\\\
\int_{S}H(\cdot,\xi)\,dv_{g}=\frac{1}{2\pi}\int_{S}\chi_{\xi}\log|y_{\xi}(\cdot)|dv_{g}.&\end{array}\right.$

We have used that
$\displaystyle\Delta_{g}\log|y_{\xi}(x)|=e^{-\hat{\varphi}_{\xi}(y)}\Delta\log|y|\Big{|}_{y=y_{\xi}(x)}=2\pi\delta_{\xi}$
in view of (3.3). For $r\leq 2r_{0}$ define
$B_{r}(\xi)=y_{\xi}^{-1}(B_{r}(0))$, $A_{r}(\xi)=B_{r}(\xi)\setminus
B_{r/2}(\xi)$, and set

$f_{\xi}={\Delta_{g}\chi_{\xi}\over|y_{\xi}(x)|^{2}}+2\Big{\langle}\nabla\chi_{\xi},\nabla|y_{\xi}(x)|^{-2}\Big{\rangle}_{g}+{2\over|S|}\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\over|y|^{3}}\,dy.$

Setting
$\Psi_{\delta,\xi}(x)=PU_{\delta,\xi}(x)-\chi_{\xi}[U_{\delta,\xi}-\log(8\delta^{2})]-8\pi
H(x,\xi),$ by the definition of $f_{\xi}$ we then have that
$-\Delta_{g}\Psi_{\delta,\xi}=-2\delta^{2}f_{\xi}+O(\delta^{4})$ in $S$ so
that

$\int_{S}f_{\xi}dv_{g}=\frac{1}{2\delta^{2}}\int_{S}\Delta_{g}\Psi_{\delta,\xi}dv_{g}+O(\delta^{2})=O(\delta^{2})$

for all $\delta>0$, and hence $\int_{S}f_{\xi}dv_{g}=0$. Therefore, $F_{\xi}$
is well defined as the unique solution of

$\left\\{\begin{array}[]{ll}-\Delta_{g}F_{\xi}=f_{\xi}&\text{in }S\\\
\int_{S}F_{\xi}dv_{g}=0.&\end{array}\right.$ (3.4)

We have the following asymptotic expansion of $PU_{\delta,\xi}$ as $\delta\to
0$, as shown in [23]:

######  Lemma 3.1.

The function $PU_{\delta,\xi}$ satisfies

$PU_{\delta,\xi}=\chi_{\xi}\left[U_{\delta,\xi}-\log(8\delta^{2})\right]+8\pi
H(x,\xi)+\alpha_{\delta,\xi}-2\delta^{2}F_{\xi}+O(\delta^{4}|\log\delta|)$

uniformly in $S$, where $F_{\xi}$ is given in (3.4) and

$\alpha_{\delta,\xi}=-{4\pi\over|S|}\delta^{2}\log\delta+2{\delta^{2}\over|S|}\left(\int_{\mathbb{R}^{2}}\chi(|y|)\frac{e^{\hat{\varphi}_{\xi}(y)}-1}{|y|^{2}}dy+\pi-\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\log|y|\over|y|}dy\right).$

In particular, there holds

$PU_{\delta,\xi}=8\pi
G(x,\xi)-2{\delta^{2}\chi_{\xi}\over|y_{\xi}(x)|^{2}}+\alpha_{\delta,\xi}-2\delta^{2}F_{\xi}+O(\delta^{4}|\log\delta|)$

locally uniformly in $S\setminus\\{\xi\\}$.

The ansatz will be constructed as follows. Given $m\in\mathbb{N}$, let us
consider distinct points $\xi_{j}\in S$ and $\delta_{j}>0$, $j=1,\dots,m$. In
order to have a good approximation, we will assume that $\exists\,C_{0}>1\,:$

$\delta_{j}^{2}=\begin{cases}\mu_{1}^{2}\delta^{2}\rho_{j}(\xi_{j})&\text{ for
$j\in\\{1,\dots,m_{1}\\}$}\\\ \mu_{2}^{2}\delta^{2}\rho_{j}(\xi_{j})&\text{
for $j\in\\{m_{1}+1,\dots,m\\}$}\end{cases},\ \ \ \text{with }\ 0<\mu_{i}\leq
C_{0},\quad i=1,2$ (3.5) $|\lambda_{1}-8\pi m_{1}|\leq
C_{0}\delta^{2}|\log\delta|\quad\text{and}\quad|\lambda_{2}\tau^{2}-8\pi
m_{2}|\leq C_{0}\delta^{2}|\log\delta|,$ (3.6)

where $\delta>0$, $m_{1}\in\\{1,\dots,m-1\\}$, $m_{2}=m-m_{1}$ and $\rho_{j}$
is given by (1.8)-(1.9). Up to take $r_{0}$ smaller, we assume that the points
$\xi_{j}$’s are well separated and $V_{1}(\xi_{j})$, $V_{2}(\xi_{j})$ are
uniformly away from zero, namely, we choose
$\xi=(\xi_{1},\dots,\xi_{m})\in\Xi$, where

$\Xi=\\{(\xi_{1},\dots,\xi_{m})\in S^{m}\mid d_{g}(\xi_{i},\xi_{j})\geq
4r_{0}\>\text{ and }\>V_{1}(\xi_{j}),\>V_{2}(\xi_{j})\geq
r_{0}\>\>\forall\>i,j=1,\dots,m,\>i\not=j\\}.$

Denote $U_{j}:=U_{\delta_{j},\xi_{j}}$ and $W_{j}=PU_{j}$, $j=1,\dots,m$,
where $P$ is the projection operator defined by (3.2). Thus, our approximating
solution is $\displaystyle
W(x)=\sum_{j=1}^{m_{1}}W_{j}(x)-{1\over\tau}\sum_{j=m_{1}+1}^{m}W_{j}(x)$,
parametrized by $(\mu,\xi)\in\mathcal{M}\times\Xi$, with
$\mu=(\mu_{1},\mu_{2})$ and $\mathcal{M}=(0,C_{0}]\times(0,C_{0}]$. Notice
that for $r_{0}$ small enough we have that
$\mathcal{D}\subset\Xi\subset\tilde{S}^{m}\setminus\Delta$. We will look for a
solution $u$ of (1.1) in the form $u=W+\phi$, for some small remainder term
$\phi$. In terms of $\phi$, the problem (1.1) is equivalent to find
$\phi\in\bar{H}$ so that

$L(\phi)=-[R+N(\phi)]\qquad\text{ in $S$},$ (3.7)

where the linear operator $L$ is defined as

$L(\phi)=\Delta_{g}\phi+\sum_{i=1}^{2}\lambda_{i}\tau^{2(i-1)}{V_{i}(x)e^{(-\tau)^{i-1}W}\over\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}\left(\phi-{\int_{S}V_{i}e^{(-\tau)^{i-1}W}\phi
dv_{g}\over\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}\right),$ (3.8)

the nonlinear part $N$ is given by

$N(\phi)=N_{1}(\phi)-N_{2}(\phi)$ (3.9)

with

$\begin{split}N_{i}(\phi)=&\,\lambda_{i}\tau^{i-1}\bigg{(}{V_{i}e^{(-\tau)^{i-1}(W+\phi)}\over\int_{S}V_{i}e^{(-\tau)^{i-1}(W+\phi)}dv_{g}}-{(-\tau)^{i-1}V_{i}e^{(-\tau)^{i-1}W}\over\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}\left[\phi-\frac{\int_{S}V_{i}e^{(-\tau)^{i-1}W}\phi
dv_{g}}{\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}\right]\\\
&\,-{V_{i}e^{(-\tau)^{i-1}W}\over\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}\bigg{)}\end{split}$
(3.10)

for $i=1,2$ and the approximation rate of $W$ is encoded in

$R=\Delta_{g}W+\lambda_{1}\left({V_{1}(x)e^{W}\over\int_{S}V_{1}e^{W}dv_{g}}-{1\over|S|}\right)-\lambda_{2}\tau\left({V_{2}(x)e^{-\tau
W}\over\int_{S}V_{2}e^{-\tau W}dv_{g}}-{1\over|S|}\right).$ (3.11)

Notice that for all $\phi\in\bar{H}$

$\int_{S}L(\phi)dv_{g}=\int_{S}N(\phi)dv_{g}=\int_{S}Rdv_{g}=0.$

In order to get the invertibility of $L$, let us introduce the weighted norm
for any $h\in L^{\infty}(S)$

$\|h\|_{*}=\sup_{x\in
S}\left[\sum_{j=1}^{m}\frac{\delta_{j}^{\sigma}}{(\delta_{j}^{2}+\chi_{B_{r_{0}}(\xi_{j})}(x)|y_{\xi_{j}}(x)|^{2}+r_{0}^{2}\chi_{S\setminus
B_{r_{0}}(\xi_{j})}(x))^{1+\sigma/2}}\right]^{-1}|h(x)|,$

where $0<\sigma<1$ is a small fixed constant and $\chi_{A}$ denotes the
characteristic function of the set $A$. Let us evaluate the approximation rate
of $W$ in $\|\cdot\|_{*}$ and recall that $m=m_{1}+m_{2}$:

######  Lemma 3.2.

Assume (3.5)-(3.6). There exists a constant $C>0$, independent of $\delta>0$
small, such that

$\|R\|_{*}\leq
C\left(\delta\,|\nabla\varphi_{m}^{*}(\xi)|_{g}+\delta^{2-\sigma}|\log\delta|\right)$
(3.12)

for all $\xi\in\Xi$, where $|\nabla\varphi_{m}^{*}(\xi)|_{g}^{2}$ stands for
$\displaystyle\sum_{j=1}^{m}|\nabla_{\xi_{j}}\varphi_{m}^{*}(\xi)|_{g}^{2}$.

###### Proof:.

We shall argue in the same way as in [23, Lemma 2.1]. First, from Lemma 3.1 we
note that for any $j\in\\{1,\dots,m\\}$,
$W_{j}(x)=U_{j}(x)-\log(8\delta_{j}^{2})+8\pi
H(x,\xi_{j})+O(\delta^{2}|\log\delta|)$ uniformly for $x\in
B_{r_{0}}(\xi_{j})$ and $W_{j}(x)=8\pi G(x,\xi_{j})+O(\delta^{2}|\log\delta|)$
uniformly for $x$ on compact subsets of $S\setminus\\{\xi_{j}\\}$. Since by
symmetry and $\hat{\varphi}_{\xi_{j}}(0)=0$ we have that

$\int_{B_{r_{0}}(\xi_{j})}\rho_{j}(x)e^{U_{j}}dv_{g}=8\pi\rho_{j}(\xi_{j})+O(\delta^{2}|\log\delta|),$

we then get that for $j\in\\{1,\dots,m_{1}\\}$

$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{W}dv_{g}$ $\displaystyle=$
$\displaystyle\frac{1}{8\delta_{j}^{2}}\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{U_{j}+8\pi
H(x,\xi_{j})+8\pi\sum\limits_{l=1,l\neq
j}^{m_{1}}G(x,\xi_{l})-{8\pi\over\tau}\sum\limits_{l=m_{1}+1}^{m}G(x,\xi_{l})+O(\delta^{2}|\log\delta|)}dv_{g}$
(3.13) $\displaystyle=$
$\displaystyle{1\over\delta_{j}^{2}}[\pi\rho_{j}(\xi_{j})+O(\delta^{2}|\log\delta|)]={\pi\over\mu_{1}^{2}\delta^{2}}+O(|\log\delta|)$

and for $j\in\\{m_{1}+1,\dots,m\\}$

$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{W}dv_{g}$ $\displaystyle=$
$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{-{1\over\tau}[U_{j}-\log(8\delta_{j}^{2})+8\pi
H(x,\xi_{j})]+8\pi\sum\limits_{l=1}^{m_{1}}G(x,\xi_{l})-{8\pi\over\tau}\sum\limits_{l=m_{1}+1,l\neq
j}^{m}G(x,\xi_{l})+O(\delta^{2}|\log\delta|)}dv_{g}$ (3.14) $\displaystyle=$
$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{1}(x)\Big{[}{\rho_{j}(x)\over
V_{2}(x)}\Big{]}^{-1/\tau}(\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2})^{2/\tau}(1+O(\delta^{2}|\log\delta|))dv_{g}$
$\displaystyle=$ $\displaystyle O(1).$

So, by using (3.13)-(3.14) we have that

$\displaystyle\int_{S}V_{1}e^{W}dv_{g}=\sum_{j=1}^{m_{1}}\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{W}dv_{g}+O(1)={\pi
m_{1}\over\mu_{1}^{2}\delta^{2}}+O(|\log\delta|).$ (3.15)

Similarly, for $j\in\\{1,\dots,m_{1}\\}$ we get that

$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{2}e^{-\tau W}dv_{g}$
$\displaystyle=$ $\displaystyle O(1)$ (3.16)

and for $j\in\\{m_{1}+1,\dots,m\\}$

$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{2}e^{-\tau W}dv_{g}$
$\displaystyle=$
$\displaystyle{1\over\delta_{j}^{2}}[\pi\rho_{j}(\xi_{j})+O(\delta^{2}|\log\delta|)]={\pi\over\mu_{2}^{2}\delta^{2}}+O(|\log\delta|).$
(3.17)

So, by using (3.16)-(3.17) we have that

$\displaystyle\int_{S}V_{2}e^{-\tau W}dv_{g}$ $\displaystyle=$
$\displaystyle\sum_{j=m_{1}+1}^{m}\int_{B_{r_{0}}(\xi_{j})}V_{2}e^{W}dv_{g}+O(1)={\pi
m_{2}\over\mu_{2}^{2}\delta^{2}}+O(|\log\delta|).$ (3.18)

By Lemma 3.1 and (3.5), (3.15), (3.18) we have that

  * •

in $S\setminus\cup_{j=1}^{m}B_{r_{0}}(\xi_{j})$ there holds
$\lambda_{1}\frac{V_{1}e^{W}}{\int_{S}V_{1}e^{W}dv_{g}}=O(\delta^{2})$ in view
of $W(x)=O(1)$;

  * •

in $B_{r_{0}}(\xi_{j})$, $j\in\\{1,\dots,m_{1}\\}$, there holds

$\displaystyle\frac{V_{1}e^{W}}{\int_{S}V_{1}e^{W}dv_{g}}$ $\displaystyle=$
$\displaystyle\frac{V_{1}e^{-\log(8\delta_{j}^{2})+8\pi
H(x,\xi_{j})+8\pi\sum\limits_{l=1,l\neq
j}^{m_{1}}G(x,\xi_{l})-{8\pi\over\tau}\sum\limits_{l=m_{1}+1}^{m}G(x,\xi_{l})+O(\delta^{2}|\log\delta|)}}{\pi
m_{1}\mu_{1}^{-2}\delta^{-2}+O(|\log\delta|)}e^{U_{j}}$ $\displaystyle=$
$\displaystyle\frac{1}{8\pi
m_{1}}\bigg{[}1+\Big{\langle}\frac{\nabla(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)}{\rho_{j}(\xi_{j})},y_{\xi_{j}}(x)\Big{\rangle}+O(|y_{\xi_{j}}(x)|^{2}+\delta^{2}|\log\delta|)\bigg{]}e^{U_{j}};$

  * •

in $B_{r_{0}}(\xi_{j})$, $j\in\\{m_{1}+1,\dots,m\\}$, there holds

$\frac{V_{1}e^{W}}{\int_{S}V_{1}e^{W}dv_{g}}=\frac{V_{1}(x)[\rho_{j}(x)/V_{2}(x)]^{-1/\tau}+O(\delta^{2}|\log\delta|)}{\pi
m_{1}\mu_{1}^{-2}\delta^{-2}+O(|\log\delta|)}(\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2})^{2/\tau}=O(\delta^{2}).$

Similarly as above, we have that

  * •

in $S\setminus\cup_{j=1}^{m}B_{r_{0}}(\xi_{j})$ there holds
$\lambda_{2}\tau\frac{V_{2}e^{-\tau W}}{\int_{S}V_{2}e^{-\tau
W}dv_{g}}=O(\delta^{2})$ in view of $W(x)=O(1)$;

  * •

in $B_{r_{0}}(\xi_{j})$, $j\in\\{1,\dots,m_{1}\\}$, there holds

$\frac{V_{2}e^{-\tau W}}{\int_{S}V_{2}e^{-\tau
W}dv_{g}}=\frac{V_{2}(x)[\rho_{j}(x)/V_{1}(x)]^{-\tau}+O(\delta^{2}|\log\delta|)}{\pi
m_{2}\mu_{2}^{-2}\delta^{-2}+O(|\log\delta|)}(\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2})^{2\tau}=O(\delta^{2}),$

  * •

in $B_{r_{0}}(\xi_{j})$, $j\in\\{m_{1}+1,\dots,m\\}$, there holds

$\frac{V_{2}e^{-\tau W}}{\int_{S}V_{2}e^{-\tau W}dv_{g}}=\frac{1}{8\pi
m_{2}}\bigg{[}1+\Big{\langle}\frac{\nabla(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)}{\rho_{j}(\xi_{j})},y_{\xi_{j}}(x)\Big{\rangle}+O(|y_{\xi_{j}}(x)|^{2}+\delta^{2}|\log\delta|)\bigg{]}e^{U_{j}}.$

Since as before

$\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}dv_{g}=\int_{B_{r_{0}}(0)}{8\delta_{j}^{2}\over(\delta_{j}^{2}+|y|^{2})^{2}}dy+O(\delta^{2})=8\pi+O(\delta^{2})$

with $\varphi_{j}=\varphi_{\xi_{j}}$, for $R$ given by (3.11) we then have
that

$\displaystyle R$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{m_{1}}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}+{\lambda_{1}V_{1}e^{W}\over\int_{S}V_{1}e^{W}dv_{g}}+{8\pi
m_{1}-\lambda_{1}\over|S|}+O(\delta^{2})$
$\displaystyle+{1\over\tau}\sum_{j=m_{1}+1}^{m}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}-{\lambda_{2}\tau
V_{2}e^{-\tau W}\over\int_{S}V_{2}e^{-\tau W}dv_{g}}+{\lambda_{2}\tau^{2}-8\pi
m_{2}\over|S|\tau}+O(\delta^{2}),$

where $\chi_{j}=\chi_{\xi_{j}}$. By previous computations we now deduce that
$R(x)=O(\delta^{2})$ in $S\setminus\cup_{j=1}^{m}B_{r_{0}}(\xi_{j})$,

$\displaystyle R$ $\displaystyle=$
$\displaystyle\left[-e^{-\varphi_{j}}+{\lambda_{1}\over 8\pi
m_{1}}+O\big{(}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)||y_{\xi_{j}}(x)|+|y_{\xi_{j}}(x)|^{2}+\delta^{2}|\log\delta|\big{)}\right]e^{U_{j}}$
$\displaystyle+O(|\lambda_{1}-8\pi m_{1}|+|\lambda_{2}\tau^{2}-8\pi
m_{2}|+\delta^{2})$ $\displaystyle=$ $\displaystyle
e^{U_{j}}O\left(|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)||y_{\xi_{j}}(x)|+|y_{\xi_{j}}(x)|^{2}+|\lambda_{1}-8\pi
m_{1}|+\delta^{2}|\log\delta|\right)$ $\displaystyle+O(|\lambda_{1}-8\pi
m_{1}|+|\lambda_{2}\tau^{2}-8\pi m_{2}|+\delta^{2})$

in $B_{r_{0}}(\xi_{j})$, $j\in\\{1,\dots,m_{1}\\}$ and similarly,

$\displaystyle R$ $\displaystyle=$ $\displaystyle
e^{U_{j}}O\left(|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)||y_{\xi_{j}}(x)|+|y_{\xi_{j}}(x)|^{2}+|\lambda_{2}\tau^{2}-8\pi
m_{2}|+\delta^{2}|\log\delta|\right)$ $\displaystyle+O(|\lambda_{1}-8\pi
m_{1}|+|\lambda_{2}\tau^{2}-8\pi m_{2}|+\delta^{2})$

in $B_{r_{0}}(\xi_{j})$, $j\in\\{m_{1}+1,\dots,m\\}$, in view of
$\varphi_{j}(\xi_{j})=0$ and $\nabla\varphi_{j}(\xi_{j})=0$. From the
definition of $\|\cdot\|_{*}$ and (3.6) we deduce the validity of (3.12). This
finishes the proof. ∎

##  4 Variational reduction and proof of main results

The solvability theory for the linear operator $L$ given in (3.8), obtained as
the linearization of (1.1) at the approximating solution $W$, is a key step in
the so-called nonlinear Lyapunov-Schimdt reduction. Notice that formally the
operator $L$ approaches $\hat{L}$ defined in $\mathbb{R}^{2}$ as

$\hat{L}(\phi)=\Delta\phi+{8\over(1+|y|^{2})^{2}}\left(\phi-{1\over\pi}\int_{\text{I\\!R}^{2}}{\phi(z)\over(1+|z|^{2})^{2}}\,dz\right),$

by setting $y=y_{\xi_{j}}(x)/\delta_{j}$ as $\delta\to 0$. Due to the
intrinsic invariances, the kernel of $\hat{L}$ in
$L^{\infty}(\text{I\\!R}^{2})$ is non-empty and is spanned by $1$ and $Y_{j}$,
$j=0,1,2$, where $\displaystyle Y_{i}(y)={4y_{i}\over 1+|y|^{2}}$, $i=1,2$,
and $\displaystyle Y_{0}(y)=2\,{1-|y|^{2}\over 1+|y|^{2}}$. Since [20, 23, 25]
it is by now rather standard to show the invertibility of $L$ in a suitable
“orthogonal” space, and a sketched proof of it will be given in Appendix A.
However, as it was observed in [23], for Dirichlet Liouville-type equations on
bounded domains as in [20, 25], the corresponding limiting operator
$\tilde{L}$ takes the form
$\tilde{L}(\phi)=\Delta\phi+{8\over(1+|y|^{2})^{2}}\phi$ and the function $1$
does not belong to its kernel, making possible to disregard the “dilation
parameters” $\delta_{i}$ in the reduction. As we will see, two additional
parameters $\mu_{1}$ and $\mu_{2}$ are needed in the reduction (one associated
to all “positive bubbles” and the other one to all “negative bubbles”) and in
this respect our problem displays a new feature w.r.t. Dirichlet Liouville-
type equations, making our situation very similar to the one arising in the
study of critical problems in higher dimension. Roughly speaking, $L$ resemble
a “direct sum” of linear operators for mean field type equations.

To be more precise, for $i=0,1,2$ and $j=1,\dots,m$ introduce the functions

$Z_{ij}(x)=Y_{i}\left({y_{\xi_{j}}(x)\over\delta_{j}}\right)=\left\\{\begin{array}[]{ll}\displaystyle
2{\delta_{j}^{2}-|y_{\xi_{j}}(x)|^{2}\over\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2}}&\hbox{for
}i=0\\\\[14.22636pt]
\displaystyle{4\delta_{j}(y_{\xi_{j}}(x))_{i}\over\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2}}&\hbox{for
}i=1,2,\end{array}\right.$

and set $Z_{1}=\displaystyle\sum_{l=1}^{m_{1}}Z_{0l}$ and
$Z_{2}=\displaystyle\sum_{l=m_{1}+1}^{m}Z_{0l}$. For $i=1,2$ and
$j=1,\dots,m$, let $PZ_{i}$, $PZ_{ij}$ be the projections of $Z_{i}$, $Z_{ij}$
as the solutions in $\bar{H}$ of

$\begin{array}[]{rl}\Delta_{g}PZ_{i}&\displaystyle=\chi_{j}\Delta_{g}Z_{i}-\frac{1}{|S|}\int_{S}\chi_{j}\Delta_{g}Z_{i}dv_{g}\\\\[14.22636pt]
\Delta_{g}PZ_{ij}&\displaystyle=\chi_{j}\Delta_{g}Z_{ij}-\frac{1}{|S|}\int_{S}\chi_{j}\Delta_{g}Z_{ij}dv_{g}.\end{array}$
(4.1)

In Appendix A we prove the following result:

######  Proposition 4.1.

There exists $\delta_{0}>0$ so that for all $0<\delta\leq\delta_{0}$, $h\in
C(S)$ with $\int_{S}h\,dv_{g}=0$, $\mu\in\mathcal{M}$, $\xi\in\Xi$ there is a
unique solution $\phi\in\bar{H}\cap W^{2,2}(S)$ and
$c_{0i},c_{ij}\in\mathbb{R}$ of

$\left\\{\begin{array}[]{ll}L(\phi)=h+\displaystyle\sum_{i=1}^{2}\Big{[}c_{0i}\Delta_{g}PZ_{i}+\sum_{j=1}^{m}c_{ij}\Delta_{g}PZ_{ij}\Big{]}&\text{in
}S\\\
\displaystyle\int_{S}\phi\Delta_{g}PZ_{i}dv_{g}=\int_{S}\phi\Delta_{g}PZ_{ij}dv_{g}=0&\forall\>i=1,2,\,j=1,\dots,m.\end{array}\right.$
(4.2)

Moreover, the map $(\mu,\xi)\mapsto(\phi,c_{0i},c_{ij})$ is twice-
differentiable in $\mu$ and one-differentiable in $\xi$ with

$\displaystyle\|\phi\|_{\infty}\leq
C|\log\delta|\|h\|_{*}\>,\qquad\displaystyle\sum_{i=1}^{2}\Big{[}|c_{0i}|+\sum_{j=1}^{m}|c_{ij}|\Big{]}\leq
C\|h\|_{*}$ (4.3)
$\displaystyle\sum_{i=1}^{2}\bigg{[}\|\partial_{\mu_{i}}\phi\|_{\infty}+\sum_{k=1}^{2}{1\over|\log\delta|}\|\partial_{\mu_{i}\mu_{k}}\phi\|_{\infty}+\sum_{j=1}^{m}\delta\|\partial_{(\xi_{j})_{i}}\phi\|_{\infty}\bigg{]}\leq
C|\log\delta|^{2}\|h\|_{*}$ (4.4)

for some $C>0$.

Let us recall that $u=W+\phi$ solves (1.1) if $\phi\in\bar{H}$ does satisfy
(3.7). Since the operator $L$ is not fully invertible, in view of Proposition
4.1 one can solve the nonlinear problem (3.7) just up to a linear combination
of $\Delta_{g}PZ_{1}$, $\Delta_{g}PZ_{2}$ and $\Delta_{g}PZ_{ij}$, as
explained in the following (see Appendix B the proof):

######  Proposition 4.2.

There exists $\delta_{0}>0$ so that for all $0<\delta\leq\delta_{0}$,
$\mu\in\mathcal{M}$, $\xi\in\Xi$ problem

$\left\\{\begin{array}[]{ll}L(\phi)=-[R+N(\phi)]+\displaystyle\sum_{i=1}^{2}\Big{[}c_{0i}\Delta_{g}PZ_{i}+\sum_{j=1}^{m}c_{ij}\Delta_{g}PZ_{ij}\Big{]}&\text{in
}S\\\
\displaystyle\int_{S}\phi\Delta_{g}PZ_{i}dv_{g}=\int_{S}\phi\Delta_{g}PZ_{ij}dv_{g}=0&\forall\>i=1,2,\,j=1,\dots,m\end{array}\right.$
(4.5)

admits a unique solution $\phi(\mu,\xi)\in\bar{H}\cap W^{2,2}(S)$ and
$c_{0i}(\mu,\xi),\,c_{ij}(\mu,\xi)\in\text{I\\!R}$, $i=1,2$ and $j=1,\dots,m$,
where $\delta_{j}>0$ are as in (3.5) and $N$, $R$ are given by (3.9), (3.11),
respectively. Moreover, the map
$(\mu,\xi)\mapsto(\phi(\mu,\xi),c_{0i}(\mu,\xi),c_{ij}(\mu,\xi))$ is twice-
differentiable in $\mu$ and one-differentiable in $\xi$ with

$\displaystyle\|\phi\|_{\infty}$ $\displaystyle\leq$ $\displaystyle
C\left(\delta|\log\delta|\,|\nabla\varphi_{m}(\xi)|_{g}+\delta^{2-\sigma}|\log\delta|^{2}\right)$
(4.6)
$\displaystyle\sum_{i=1}^{2}\Big{[}\|\partial_{\mu_{i}}\phi\|_{\infty}+\sum_{j=1}^{m}\delta\|\partial_{(\xi_{j})_{i}}\phi\|_{\infty}+\sum_{k=1}^{2}{\|\partial_{\mu_{i}\mu_{k}}\phi\|_{\infty}\over|\log\delta|}\Big{]}$
$\displaystyle\leq$ $\displaystyle
C\left(\delta|\log\delta|^{2}|\nabla\varphi_{m}(\xi)|_{g}+\delta^{2-\sigma}|\log\delta|^{3}\right)$
(4.7)

The function $[W+\phi](\mu,\xi)$ will be a true solution of (3.7) if
$\mu\in\mathcal{M}$ and $\xi\in\Xi$ are such that
$c_{0i}(\mu,\xi)=c_{ij}(\mu,\xi)=0$ for all $i=1,2,$ and $j=1,\dots,m$. This
problem is equivalent to finding critical points of the reduced energy
$E_{\lambda_{1},\lambda_{2}}(\mu,\xi)=J_{\lambda_{1},\lambda_{2}}\big{(}[W+\phi](\mu,\xi)\big{)}$,
where $J_{\lambda_{1},\lambda_{2}}$ is given by (1.4), as stated in (we omit
its proof):

######  Lemma 4.1.

There exists $\delta_{0}$ such that, if $(\mu,\xi)\in\mathcal{M}\times\Xi$ is
a critical point of $E_{\lambda_{1},\lambda_{2}}$ for
$0<\delta\leq\delta_{0}$, then $u=W(\mu,\xi)+\phi(\mu,\xi)$ is a solution to
(1.1), where $\delta_{i}$ are given by (3.5).

Once equation (1.1) has been reduced to the search of c.p.’s for
$E_{\lambda_{1},\lambda_{2}}$, it becomes crucial to show that the main
asymptotic term of $E_{\lambda_{1},\lambda_{2}}$ is given by
$J_{\lambda_{1},\lambda_{2}}(W)$, for which an expansion has been given in
Theorem 5.1. More precisely, by estimates in Appendix B we have that

######  Theorem 4.1.

Assume (3.5) - (3.6). The following expansion does hold

$\displaystyle E_{\lambda_{1},\lambda_{2}}(\mu,\xi)$ $\displaystyle=$
$\displaystyle-8\pi\Big{(}m_{1}+{m_{2}\over\tau^{2}}\Big{)}-\lambda_{1}\log(\pi
m_{1})-\lambda_{2}\log(\pi m_{2})+2\big{(}\lambda_{1}-8\pi
m_{1}\big{)}\log\delta$
$\displaystyle\,\,+{2\over\tau^{2}}\big{(}\lambda_{2}\tau^{2}-8\pi
m_{2}\big{)}\log\delta-32\pi^{2}\varphi_{m}^{*}(\xi)+2\big{(}\lambda_{1}-8\pi
m_{1}\big{)}\log\mu_{1}$
$\displaystyle\,\,+A_{1}^{*}(\xi)\mu_{1}^{2}\delta^{2}\log\delta+\left[A_{1}^{*}(\xi)\mu_{1}^{2}\log\mu_{1}-B_{1}^{*}(\xi)\mu_{1}^{2}\right]\delta^{2}+{1\over\tau^{2}}\Big{\\{}2\big{(}\lambda_{2}\tau^{2}-8\pi
m_{2}\big{)}\log\mu_{2}$
$\displaystyle\,\,+A_{2}^{*}(\xi)\mu_{2}^{2}\delta^{2}\log\delta+\left[A_{2}^{*}(\xi)\mu_{2}^{2}\log\mu_{2}-B_{2}^{*}(\xi)\mu_{2}^{2}\right]\delta^{2}\Big{\\}}+o(\delta^{2})+r_{\lambda_{1},\lambda_{2}}(\mu,\xi)$

in $C^{2}(\mathbb{R}^{2})$ and $C^{1}(\Xi)$ as $\delta\to 0^{+}$, where
$\varphi_{m}^{*}(\xi)$, $A_{k}^{*}(\xi)$ and $B_{k}^{*}(\xi)$, $k=1,2$ are
given by (1.7), (1.10) and (1), $k=1,2$, respectively. The term
$r_{\lambda_{1},\lambda_{2}}(\mu,\xi)$ satisfies

$\displaystyle|r_{\lambda_{1},\lambda_{2}}(\mu,\xi)|$ $\displaystyle+$
$\displaystyle\frac{\delta|\nabla_{\xi}r_{\lambda_{1},\lambda_{2}}(\mu,\xi)|}{|\log\delta|}+\frac{|\nabla_{\mu}r_{\lambda_{1},\lambda_{2}}(\mu,\xi)|}{|\log\delta|}$
(4.9) $\displaystyle+$
$\displaystyle\frac{|D^{2}_{\mu}r_{\lambda_{1},\lambda_{2}}(\mu,\xi)|}{|\log\delta|^{2}}\leq
C\left(\delta^{2}|\log\delta|\,|\nabla\varphi_{m}^{*}(\xi)|_{g}^{2}+\delta^{3-\sigma}|\log\delta|^{2}\right)$

for some $C>0$ independent of $(\mu,\xi)\in\mathcal{M}\times\Xi$.

We are now in position to establish the main result stated in the
Introduction. We shall argue similarly to [23, Theorem 1.5].

###### Proof (of Theorem 1.1):.

According to Lemma 4.1, we just need to find a critical point of
$E=E_{\lambda_{1},\lambda_{2}}(\mu,\xi)$ with $\mu=(\mu_{1},\mu_{2})$. Recall
that $\tau>0$ is fixed. Assumptions (1.12) and (1.13) allows us to choose
$\mu_{k}=\mu_{k}(\lambda_{k},\xi)$ for $\lambda_{k}\tau^{2(k-1)}$ close to
$8\pi m_{k}$, $k=1,2$, respectively. Precisely, fixing $k\in\\{1,2\\}$ we
choose $\lambda_{k}\tau^{2(k-1)}-8\pi m_{k}=\delta^{2}$ ($-\delta^{2}$ resp.)
if either $A_{k}^{*}(\xi)>0$ ($<0$ resp.) or $A^{*}(\xi)=0$,
$B_{k}^{*}(\xi)>0$ ($<0$ resp.) in $U$. Thus, we deduce the expansions

$\begin{split}\frac{\tau^{2(k-1)}\partial_{\mu_{k}}E(\mu,\xi)}{\lambda_{k}\tau^{2(k-1)}-8\pi
m_{k}}=&\,\frac{2}{\mu_{k}}+2A_{k}^{*}(\xi)\mu_{k}\log\delta+A_{k}^{*}(\xi)(2\mu_{k}\log\mu_{k}+\mu_{k})-2B_{k}^{*}(\xi)\mu_{k}\\\
&\,+o(1)+O\left(|\log\delta|^{2}\,|\nabla\varphi_{m}^{*}(\xi)|^{2}_{g}\right)\end{split}$

and

$\begin{split}\frac{\tau^{2(k-1)}\partial_{\mu_{k}\mu_{k}}E(\mu,\xi)}{\lambda_{k}\tau^{2(k-1)}-8\pi
m_{k}}=&\,-\frac{2}{\mu_{k}^{2}}+2A_{k}^{*}(\xi)\log\delta+A_{k}^{*}(\xi)(2\log\mu_{k}+3)-2B_{k}^{*}(\xi)\\\
&\,+o(1)+O\left(|\log\delta|^{3}\,|\nabla\varphi_{m}^{*}(\xi)|^{2}_{g}\right),\end{split}$

as $\delta\to 0^{+}$. Arguing in the same way as in the proof of Theorem 3.2
in [23], we conclude existence of a $C^{1}$ map
$\mu_{k}=\mu_{k}(\lambda_{k},\xi)$ satisfying
$\partial_{\mu_{k}}E(\mu(\lambda,\xi),\xi)=0$, with
$\lambda=(\lambda_{1},\lambda_{2})$ and $\mu=(\mu_{1},\mu_{2})$ for all
$\xi\in U$. Now, considering
$\tilde{E}(\xi)=E_{\lambda}\big{(}\mu_{1}(\lambda_{1},\xi),\mu_{2}(\lambda_{2},\xi),\xi\big{)}$
and again arguing in the same way as in the proof of Theorem 3.2 in [23] it
follows that
$\tilde{E}(\xi)=-32\pi^{2}\varphi_{m}^{*}(\xi)+O(\delta^{2}|\log\delta|)$,

$\begin{split}\nabla_{\xi}\tilde{E}(\xi)&=\nabla_{\xi}E\big{(}\mu_{1}(\lambda_{1},\xi),\mu_{2}(\lambda_{2},\xi),\xi\big{)}+\nabla_{\mu}E\big{(}\mu_{1}(\lambda_{1},\xi),\mu_{2}(\lambda_{2},\xi),\xi\big{)}\nabla_{\xi}\mu(\lambda,\xi)\\\
&=-32\pi^{2}\nabla\varphi_{m}^{*}(\xi)+O(\delta|\log\delta|^{2})\end{split}$

uniformly in $\xi\in U$ and there exists a critical point
$\xi_{\lambda_{1},\lambda_{2}}=\xi_{\delta}\in U$ of $\tilde{E}(\xi)$, since
$\mathcal{D}$ is a stable critical set of $\varphi_{m}^{*}$ (see Definition
2.1). Up to take $U$ smaller so that $\nabla\varphi_{m}^{*}(\xi)\neq 0$ for
all $\xi\in U\setminus\mathcal{D}$, it can be deduced that the pair
$\big{(}\mu(\lambda_{1},\lambda_{2},\xi_{\delta}),\xi_{\delta}\big{)}$ is a
c.p. of $E(\mu,\xi)$ and, along a sub-sequence, $\xi_{\delta}\to
q\in\mathcal{D}$ as $\delta\to 0$, namely, as $\lambda_{1}\to 8\pi m_{1}$ and
$\lambda_{2}\tau^{2}\to 8\pi m_{2}$. By construction, the corresponding
solution has the required asymptotic properties (1.14). See proof of Theorem
1.5 in [23] for more details. This completes the proof. ∎

##  5 The reduced energy

The purpose of this section is to give an asymptotic expansion of the “reduced
energy” $J_{\lambda_{1},\lambda_{2}}(W)$, where $J_{\lambda_{1},\lambda_{2}}$
is the energy functional given by (1.4). For technical reasons, we will be
concerned with establishing it in a $C^{2}$-sense in $\mu$ and just in a
$C^{1}$-sense in $\xi$. To this aim, the following result will be very useful,
see [23, Lemma 3.1] for a proof.

######  Lemma 5.1.

Letting $f\in C^{2,\gamma}(S)$ (possibly depending in $\xi$), $0<\gamma<1$,
denote as $P_{2}(f)$ the second-order Taylor expansion of $f(x)$ at $\xi$:

$P_{2}f(x)=f(\xi)+\langle\nabla(f\circ
y_{\xi}^{-1})(0),y_{\xi}(x)\rangle+{1\over 2}\langle D^{2}(f\circ
y_{\xi}^{-1})(0)y_{\xi}(x),y_{\xi}(x)\rangle.$

The following expansions do hold as $\delta\to 0$:

$\displaystyle\int_{S}\chi_{\xi}e^{-\varphi_{\xi}}f(x)e^{U_{\delta,\xi}}dv_{g}$
$\displaystyle=$ $\displaystyle 8\pi
f(\xi)-2\delta^{2}\Delta_{g}f(\xi)\left[2\pi\log\delta+\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\log|y|\over|y|}dy+\pi\right]$
$\displaystyle+8\delta^{2}\int_{S}\chi_{\xi}e^{-\varphi_{\xi}}{f(x)-P_{2}(f)(x)\over|y_{\xi}(x)|^{4}}\,dv_{g}+4\delta^{2}f(\xi)\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\over|y|^{3}}dy+o(\delta^{2}),$
$\displaystyle\int_{S}\chi_{\xi}e^{-\varphi_{\xi}}f(x)e^{U_{\delta,\xi}}\frac{dv_{g}}{\delta^{2}+|y_{\xi}(x)|^{2}}=\frac{4\pi}{\delta^{2}}f(\xi)+\pi\Delta_{g}f(\xi)+O(\delta^{\gamma})$

and

$\displaystyle\int_{S}\chi_{\xi}e^{-\varphi_{\xi}}f(x)e^{U_{\delta,\xi}}\frac{a\delta^{2}-|y_{\xi}(x)|^{2}}{(\delta^{2}+|y_{\xi}(x)|^{2})^{2}}dv_{g}=\frac{4\pi}{3\delta^{2}}(2a-1)f(\xi)+(a-2)\frac{\pi}{3}\Delta_{g}f(\xi)+O(\delta^{\gamma})$

for $a\in\mathbb{R}$.

We are now ready to establish the expansion of
$J_{\lambda_{1},\lambda_{2}}(W)$:

######  Theorem 5.1.

Assume (3.5)-(3.6). The following expansion does hold

$\displaystyle J_{\lambda_{1},\lambda_{2}}(W)$ $\displaystyle=$
$\displaystyle\,\,-8\pi\Big{(}m_{1}+{m_{2}\over\tau^{2}}\Big{)}-\lambda_{1}\log(\pi
m_{1})-\lambda_{2}\log(\pi m_{2})+2\big{(}\lambda_{1}-8\pi
m_{1}\big{)}\log\delta$
$\displaystyle\,\,+{2\over\tau^{2}}\big{(}\lambda_{2}\tau^{2}-8\pi
m_{2}\big{)}\log\delta-32\pi^{2}\varphi_{m}^{*}(\xi)+2\big{(}\lambda_{1}-8\pi
m_{1}\big{)}\log\mu_{1}$
$\displaystyle\,\,+A_{1}^{*}(\xi)\mu_{1}^{2}\delta^{2}\log\delta+\left[A_{1}^{*}(\xi)\mu_{1}^{2}\log\mu_{1}-B_{1}^{*}(\xi)\mu_{1}^{2}\right]\delta^{2}+{1\over\tau^{2}}\Big{\\{}2\big{(}\lambda_{2}\tau^{2}-8\pi
m_{2}\big{)}\log\mu_{2}$
$\displaystyle\,\,+A_{2}^{*}(\xi)\mu_{2}^{2}\delta^{2}\log\delta+\left[A_{2}^{*}(\xi)\mu_{2}^{2}\log\mu_{2}-B_{2}^{*}(\xi)\mu_{2}^{2}\right]\delta^{2}\Big{\\}}+o(\delta^{2})$

in $C^{2}(\mathbb{R}^{2})$ and $C^{1}(\Xi)$ as $\delta\to 0^{+}$, where
$\varphi_{m}^{*}(\xi)$, $A_{1}^{*}(\xi)$, $A_{2}^{*}(\xi)$, $B_{1}^{*}(\xi)$
and $B_{2}^{*}(\xi)$ are given by (1.7), (1.10) and (1), $k=1,2$,
respectively.

As in [23, Theorem 3.2], the proof will be divided into several steps.

###### Proof (of (5.1) in $C(\mathbb{R}^{2}\times\Xi)$):.

First, let us consider the term. Integrating by parts we have that

$\begin{split}\int_{S}|\nabla
W|_{g}^{2}dv_{g}&=\sum_{j,l=1}^{m_{1}}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}-{1\over\tau}\sum_{j=1}^{m_{1}}\sum_{l=m_{1}+1}^{m}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}\\\
&\ \ \
-{1\over\tau}\sum_{j=m_{1}+1}^{m}\sum_{l=1}^{m_{1}}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}+{1\over\tau^{2}}\sum_{j,l=m_{1}+1}^{m}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}\end{split}$

in view of $\int_{S}Wdv_{g}=0$. Since by (1.6) and (3.2)

$\displaystyle\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}G(x,\xi_{l})dv_{g}=\int_{S}(-\Delta_{g}PU_{j})G(x,\xi_{l})dv_{g}=PU_{j}(\xi_{l})$
(5.2)

for all $j,l=1,\dots,m$, by Lemmata 3.1, 5.1, (5.2) and computations done in
the proof of [23, Theorem 3.2], we have that for $l=j$

$\displaystyle\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{j}dv_{g}=-16\pi-32\pi\log\delta_{j}+64\pi^{2}H(\xi_{j},\xi_{j})+16\pi\alpha_{\delta_{j},\xi_{j}}-32\pi\delta_{j}^{2}F_{\xi_{j}}(\xi_{j})+O(\delta^{4}|\log\delta|^{2}).$

Similarly, by Lemmata 3.1, 5.1 and (5.2) we have that for $l\not=j$

$\displaystyle\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}$
$\displaystyle=$ $\displaystyle
64\pi^{2}G(\xi_{l},\xi_{j})+8\pi(\alpha_{\delta_{j},\xi_{j}}+\alpha_{\delta_{l},\xi_{l}})-16\pi(\delta_{j}^{2}F_{\xi_{j}}(\xi_{l})+\delta_{l}^{2}F_{\xi_{l}}(\xi_{j})$
$\displaystyle+O(\delta^{4}|\log\delta|^{2}).$

Setting
$\displaystyle\alpha_{1,\delta,\xi}=\sum_{j=1}^{m_{1}}\alpha_{\delta_{j},\xi_{j}}$,
$\displaystyle\alpha_{2,\delta,\xi}=\sum_{j=m_{1}+1}^{m}\alpha_{\delta_{j},\xi_{j}},$
$\displaystyle
F_{1,\delta,\xi}(x)=\sum_{j=1}^{m_{1}}\delta_{j}^{2}F_{\xi_{j}}(x)$ and
$\displaystyle
F_{2,\delta,\xi}(x)=\sum_{j=m_{1}+1}^{m}\delta_{j}^{2}F_{\xi_{j}}(x)$, we find
that

$\displaystyle\sum_{j,l=1}^{m_{1}}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}=-16\pi
m_{1}+\sum_{j=1}^{m_{1}}\Big{[}-32\pi\log(\mu_{1}\delta)-16\pi\log
V_{1}(\xi_{j})-64\pi^{2}H(\xi_{j},\xi_{j})$
$\displaystyle-64\pi^{2}\sum_{i=1\atop i\neq
j}^{m_{1}}G(\xi_{j},\xi_{i})+{128\pi^{2}\over\tau}\sum_{i=m_{1}+1}^{m}G(\xi_{j},\xi_{i})\Big{]}+16\pi
m_{1}\alpha_{1,\delta,\xi}-32\pi\sum_{j=1}^{m_{1}}F_{1,\delta,\xi}(\xi_{j})+O(\delta^{4}|\log\delta|^{2}),$
$\displaystyle\sum_{j=1}^{m_{1}}\sum_{l=m_{1}+1}^{m}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}$
$\displaystyle=$ $\displaystyle
64\pi^{2}\sum_{j=1}^{m_{1}}\sum_{l=m_{1}+1}^{m}G(\xi_{j},\xi_{l})+8\pi
m_{2}\alpha_{1,\delta,\xi}+8\pi m_{1}\alpha_{2,\delta,\xi}$
$\displaystyle-16\pi\sum_{j=m_{1}+1}^{m}F_{1,\delta,\xi}(\xi_{j})-16\pi\sum_{j=1}^{m_{1}}F_{2,\delta,\xi}(\xi_{j})+O(\delta^{4}|\log\delta|^{2}),$
$\displaystyle\sum_{j=m_{1}+1}^{m}\sum_{l=1}^{m_{1}}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}$
$\displaystyle=$ $\displaystyle
64\pi^{2}\sum_{j=m_{1}+1}^{m}\sum_{l=1}^{m_{1}}G(\xi_{j},\xi_{l})+8\pi
m_{2}\alpha_{1,\delta,\xi}+8\pi m_{1}\alpha_{2,\delta,\xi}$
$\displaystyle-16\pi\sum_{j=m_{1}+1}^{m}F_{1,\delta,\xi}(\xi_{j})-16\pi\sum_{j=1}^{m_{1}}F_{2,\delta,\xi}(\xi_{j})+O(\delta^{4}|\log\delta|^{2}),$

and

$\displaystyle\sum_{j,l=m_{1}+1}^{m}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}W_{l}dv_{g}=-16\pi
m_{2}+\sum_{j=m_{1}+1}^{m}\Big{[}-32\pi\log(\mu_{2}\delta)-16\pi\log
V_{2}(\xi_{j})-64\pi^{2}H(\xi_{j},\xi_{j})$
$\displaystyle-128\pi^{2}\tau\sum_{i=1}^{m_{1}}G(\xi_{j},\xi_{i})-64\pi^{2}\sum_{i=m_{1}+1\atop
i\neq j}^{m}G(\xi_{j},\xi_{i})\Big{]}+16\pi
m_{2}\alpha_{2,\delta,\xi}-32\pi\sum_{j=m_{1}+1}^{m}F_{2,\delta,\xi}(\xi_{j})+O(\delta^{4}|\log\delta|^{2})$

in view of (3.5). Now, setting
$\displaystyle\alpha_{\delta,\xi}=\alpha_{1,\delta_{j},\xi_{j}}-{1\over\tau}\alpha_{2,\delta,\xi}$
and $\displaystyle
F_{\delta,\xi}(x)=F_{1,\delta,\xi}(x)-{1\over\tau}F_{2,\delta,\xi}(x),$
summing up the four previous expansions, for the gradient term we get that

$\displaystyle{1\over 2}\int_{S}|\nabla W|_{g}^{2}dv_{g}$ $\displaystyle=$
$\displaystyle-\,8\pi\left(m_{1}+{m_{2}\over\tau^{2}}\right)-16\pi\left(m_{1}\log(\mu_{1}\delta)+{m_{2}\over\tau^{2}}\log(\mu_{2}\delta)\right)-32\pi^{2}\varphi_{m}^{*}(\xi)$
$\displaystyle+\,8\pi\left(m_{1}-{m_{2}\over\tau}\right)\alpha_{\delta,\xi}-16\pi\sum_{j=1}^{m_{1}}F_{\delta,\xi}(\xi_{j})+{16\pi\over\tau}\sum_{j=m_{1}+1}^{m}F_{\delta,\xi}(\xi_{j})+o(\delta^{2})$

in view of (1.7).

Let us now expand the potential terms in $J_{\lambda_{1},\lambda_{2}}(W)$,
similarly to the proof of [23, Theorem 3.2]. By Lemma 3.1 for any
$j=1,\dots,m_{1}$ we find that

$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{W}dv_{g}$ $\displaystyle=$
$\displaystyle{e^{\alpha_{\delta,\xi}}\over
8\delta_{j}^{2}}\left[\int_{S}\chi_{j}e^{U_{j}}\rho_{j}e^{-2F_{\delta,\xi}}dv_{g}-8\delta_{j}^{2}\int_{A_{2r_{0}}(\xi_{j})}\frac{\chi_{j}\rho_{j}}{|y_{\xi_{j}}(x)|^{4}}dv_{g}+O(\delta^{4}|\log\delta|)\right].$

By Lemma 5.1 (with
$f(x)=e^{\varphi_{j}}\rho_{j}e^{\alpha_{\delta,\xi}-2F_{\delta,\xi}}$) we can
now deduce that

$\displaystyle
8\delta_{j}^{2}e^{-\alpha_{\delta,\xi}}\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{W}dv_{g}=8\pi\rho_{j}(\xi_{j})e^{-2F_{\delta,\xi}(\xi_{j})}-4\pi\left(\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j})\right)\delta_{j}^{2}\log\delta_{j}$
$\displaystyle-2\left(\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j})\right)\left(\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\log|y|\over|y|}dy+\pi\right)\delta_{j}^{2}+4\delta_{j}^{2}\rho_{j}(\xi_{j})\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\over|y|^{3}}dy$
$\displaystyle+8\delta_{j}^{2}\int_{B_{r_{0}}(\xi_{j})}\left[V_{1}e^{8\pi\sum\limits_{j=1}^{m}G(x,\xi_{j})-{8\pi\over\tau}\sum\limits_{i=m_{1}+1}^{m}G(x,\xi_{i})}-e^{-\varphi_{j}}\frac{P_{2}(e^{\varphi_{j}}\rho_{j})}{|y_{\xi_{j}}(x)|^{4}}\right]dv_{g}$
$\displaystyle-8\delta_{j}^{2}\int_{A_{2r_{0}}(\xi_{j})}\chi_{j}e^{-\varphi_{j}}\frac{P_{2}(e^{\varphi_{j}}\rho_{j})}{|y_{\xi_{j}}(x)|^{4}}\,dv_{g}+o(\delta^{2})$

in view of
$\frac{\rho_{j}(x)}{|y_{\xi_{j}}(x)|^{4}}=V_{1}e^{8\pi\sum\limits_{j=1}^{m_{1}}G(x,\xi_{j})-{8\pi\over\tau}\sum\limits_{i=m_{1}+1}^{m}G(x,\xi_{i})}$
in $B_{r_{0}}(\xi_{j})$ and by (3.1)

$\displaystyle\Delta_{g}\left[e^{\varphi_{j}}\rho_{j}\right](\xi_{j})=\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j}).$
(5.4)

Now, by Lemma 3.1 for any $j=m_{1}+1,\dots,m$ we find that

$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{W}dv_{g}$ $\displaystyle=$
$\displaystyle\int_{B_{r_{0}}(\xi_{j})}V_{1}\Big{[}{\rho_{j}\over
V_{2}}\Big{]}^{-1/\tau}e^{-{1\over\tau}[U_{j}-\log(8\delta_{j}^{2})]+\alpha_{\delta,\xi}+O(\delta^{2})}dv_{g}$
$\displaystyle=$ $\displaystyle
e^{\alpha_{\delta,\xi}}\bigg{[}\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{8\pi\sum\limits_{j=1}^{m_{1}}G(x,\xi_{j})-{8\pi\over\tau}\sum\limits_{i=m_{1}+1}^{m}G(x,\xi_{i})}dv_{g}+O(\delta^{2})\bigg{]}.$

On the other hand, we have that

$\int_{S\setminus\cup_{j=1}^{m}B_{r_{0}}(\xi_{j})}V_{1}e^{W}dv_{g}=e^{\alpha_{\delta,\xi}}\bigg{[}\int_{S\setminus\cup_{j=1}^{m}B_{r_{0}}(\xi_{j})}V_{1}e^{8\pi\sum\limits_{j=1}^{m_{1}}G(x,\xi_{j})-{8\pi\over\tau}\sum\limits_{i=m_{1}+1}^{m}G(x,\xi_{i})}dv_{g}+O(\delta^{2})\bigg{]}.$

Since

$\sum_{j=1}^{m_{1}}e^{-2F_{\delta,\xi}(\xi_{j})}=m_{1}-2\sum_{j=1}^{m_{1}}F_{\delta,\xi}(\xi_{j})+O(\delta^{4})$
(5.5)

and by (3.5) there holds
$\displaystyle\delta_{j}^{2}\log\delta_{j}=\rho_{j}(\xi_{j})\mu_{i}^{2}\delta^{2}\log\delta+\rho_{j}(\xi_{j})\mu_{i}^{2}\delta^{2}\log\mu_{i}+\frac{1}{2}\rho_{j}(\xi_{j})\log\rho_{j}(\xi_{j})\mu_{i}^{2}\delta^{2},$
we then obtain that

$\displaystyle\frac{1}{\pi}e^{-\alpha_{\delta,\xi}}\mu_{1}^{2}\delta^{2}\int_{S}V_{1}e^{W}dv_{g}=m_{1}-\frac{A_{1}^{*}(\xi)}{8\pi}\mu_{1}^{2}\delta^{2}\log(\mu_{1}\delta)+\frac{B_{1,\chi}(\xi)}{8\pi}\mu_{1}^{2}\delta^{2}-2\sum_{j=1}^{m_{1}}F_{\delta,\xi}(\xi_{j})+o(\delta^{2}),$
(5.6)

where

$\displaystyle B_{1,\chi}(\xi)$ $\displaystyle=$
$\displaystyle-2\pi\sum_{j=1}^{m_{1}}[\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j})]\log\rho_{j}(\xi_{j})$
$\displaystyle-\frac{A_{1}^{*}(\xi)}{2\pi}\bigg{(}\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\log|y|\over|y|}dy+\pi\bigg{)}+4\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\over|y|^{3}}dy\sum_{j=1}^{m_{1}}\rho_{j}(\xi_{j})$
$\displaystyle+8\int_{S}\left[V_{1}e^{8\pi\sum\limits_{j=1}^{m_{1}}G(x,\xi_{j})-{8\pi\over\tau}\sum\limits_{l=m_{1}+1}^{m}G(x,\xi_{l})}-\sum_{j=1}^{m_{1}}\chi_{j}e^{-\varphi_{j}}\frac{P_{2}(e^{\varphi_{j}}\rho_{j})}{|y_{\xi_{j}}(x)|^{4}}\right]dv_{g},$

By integration by parts on integrals involving $\chi$ and the splitting of $S$
as the union of $\cup_{j=1}^{m_{1}}B_{r}(\xi_{j})$ and
$S\setminus\cup_{j=1}^{m_{1}}B_{r}(\xi_{j})$, $r\leq r_{0}$, we easily deduce
that

$\displaystyle B_{1,\chi}(\xi)$ $\displaystyle=$
$\displaystyle-2\pi\sum_{j=1}^{m_{1}}[\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j})]\log\rho_{j}(\xi_{j})-\frac{A_{1}^{*}(\xi)}{2}$
$\displaystyle+8\int_{S\setminus\cup_{j=1}^{m_{1}}B_{r}(\xi_{j})}V_{1}e^{8\pi\sum\limits_{j=1}^{m}G(x,\xi_{j})-{8\pi\over\tau}\sum\limits_{l=m_{1}+1}^{m}G(x,\xi_{l})}dv_{g}-\frac{8\pi}{r^{2}}\sum_{j=1}^{m_{1}}\rho_{j}(\xi_{j})-A_{1}^{*}(\xi)\log\frac{1}{r}$
$\displaystyle+8\sum_{j=1}^{m_{1}}\int_{B_{r}(\xi_{j})}\frac{e^{\varphi_{j}(x)}\rho_{j}(x)-P_{2}(e^{\varphi_{j}}\rho_{j})(x)}{|y_{\xi_{j}}(x)|^{4}}\,e^{-\varphi_{j}(x)}dv_{g}$

in view of (5.4) and the definitions of $A_{1}^{*}(\xi)$,
$P_{2}(e^{\varphi_{j}}\rho_{j})$. As a by-product we have that
$B_{1,\chi}(\xi)$ does not depend on $\chi$ and $r\leq r_{0}$. Since

$\lim_{r\to
0}\int_{B_{r}(\xi_{j})}\frac{e^{\varphi_{j}(x)}\rho_{j}(x)-P_{2}(e^{\varphi_{j}}\rho_{j})(x)}{|y_{\xi_{j}}(x)|^{4}}\,e^{-\varphi_{j}(x)}dv_{g}=0$

in view of
$e^{\varphi_{j}(x)}\rho_{j}(x)-P_{2}(e^{\varphi_{j}}\rho_{j})(x)=o(|y_{\xi_{j}}(x)|^{2})$
as $x\to\xi_{j}$, we have that $B_{1,\chi}(\xi)$ coincides with
$B_{1}^{*}(\xi)$ as defined in (1) with $k=1$.

Similar as above, by Lemmata 3.1, 5.1 (with
$f(x)=e^{\varphi_{j}}\rho_{j}e^{-\tau\alpha_{\delta,\xi}+2\tau
F_{\delta,\xi}}$), (5.4),

$\sum_{j=m_{1}+1}^{m}e^{2\tau
F_{\delta,\xi}(\xi_{j})}=m_{2}+2\tau\sum_{j=m_{1}+1}^{m}F_{\delta,\xi}(\xi_{j})+O(\delta^{4})$
(5.7)

and by (3.5), we then obtain that

$\displaystyle\frac{1}{\pi}e^{\tau\alpha_{\delta,\xi}}\mu_{2}^{2}\delta^{2}\int_{S}V_{2}e^{-\tau
W}dv_{g}$ $\displaystyle=$ $\displaystyle
m_{2}-\frac{A_{2}^{*}(\xi)}{8\pi}\mu_{2}^{2}\delta^{2}\log(\mu_{2}\delta)+\frac{B_{2,\chi}(\xi)}{8\pi}\mu_{2}^{2}\delta^{2}$
$\displaystyle+2\tau\sum_{j=m_{1}+1}^{m}F_{\delta,\xi}(\xi_{j})+o(\delta^{2}),$

where

$\displaystyle B_{2,\chi}(\xi)$ $\displaystyle=$
$\displaystyle-2\pi\sum_{j=m_{1}+1}^{m}[\Delta_{g}\rho_{j}(\xi_{j})-2K(\xi_{j})\rho_{j}(\xi_{j})]\log\rho_{j}(\xi_{j})-\frac{A_{2}^{*}(\xi)}{2}$
$\displaystyle+8\int_{S\setminus\cup_{j=m_{1}+1}^{m}B_{r}(\xi_{j})}V_{2}e^{-8\pi\tau\sum\limits_{j=1}^{m}G(x,\xi_{j})+8\pi\sum\limits_{l=m_{1}+1}^{m}G(x,\xi_{l})}dv_{g}-\frac{8\pi}{r^{2}}\sum_{j=m_{1}+1}^{m}\rho_{j}(\xi_{j})$
$\displaystyle-
A_{2}^{*}(\xi)\log\frac{1}{r}+8\sum_{j=m_{1}+1}^{m}\int_{B_{r}(\xi_{j})}\frac{e^{\varphi_{j}(x)}\rho_{j}(x)-P_{2}(e^{\varphi_{j}}\rho_{j})(x)}{|y_{\xi_{j}}(x)|^{4}}\,e^{-\varphi_{j}(x)}dv_{g},$

$B_{2,\chi}(\xi)$ does not depend on $\chi$ and $r\leq r_{0}$, and coincides
with $B_{2}^{*}(\xi)$ as defined in (1) with $k=2$.

Finally, from (3.6), expansions (5), (5.6) and (5) and Taylor’s expansion for
$a\geq 1$, $\log(a+t)=\log a+{t\over a}+O(t^{2})$ as $t\to 0$, we get the
expansion (5.1) as $\delta\to 0$ and the proof is complete. ∎

We establish now expansion (5.1) in a $C^{1}$-sense in $\xi$, where the
derivatives in $\xi$ are with respect to a given coordinate system. Recall we
use ideas in [23, Theorem 3.2].

###### Proof (of (5.1) in $C^{1}(\Xi)$):.

We just need to expand the derivatives of $J_{\lambda_{1},\lambda_{2}}(W)$ in
$\xi$. Let us fix $i\in\\{1,2\\}$ and $j\in\\{1,\dots,m\\}$. We have that

$\partial_{(\xi_{j})_{i}}[J_{\lambda_{1},\lambda_{2}}(W)]=-\int_{S}\left[\Delta_{g}W+{\lambda_{1}V_{1}e^{W}\over\int_{S}V_{1}e^{W}dv_{g}}-{\lambda_{1}\tau
V_{2}e^{-\tau W}\over\int_{S}V_{2}e^{-\tau
W}dv_{g}}\right]\partial_{(\xi_{j})_{i}}Wdv_{g}.$

Notice that as in Lemma 3.1, it follows that

$\displaystyle\partial_{(\xi_{j})_{i}}W_{q}$ $\displaystyle=$
$\displaystyle-2{\chi_{q}\over\delta_{q}^{2}+|y_{\xi_{q}}(x)|^{2}}\left[\partial_{(\xi_{j})_{i}}|y_{\xi_{q}}(x)|^{2}+\delta_{q}^{2}\partial_{(\xi_{j})_{i}}(\log\rho_{q}(\xi_{q}))\right]$
$\displaystyle-4\log|y_{\xi_{q}}(x)|\partial_{(\xi_{j})_{i}}\chi_{q}+8\pi\partial_{(\xi_{j})_{i}}H(x,\xi_{q})+O(\delta^{2}|\log\delta|)$

does hold uniformly in $S$. Hence, by using (5) and expansions in the proof of
(35) in $C^{1}(\Xi)$ in [23, Theorem 3.2], we deduce that

$\displaystyle-\int_{S}\Delta_{g}W\partial_{(\xi_{j})_{i}}Wdv_{g}$
$\displaystyle=$
$\displaystyle\sum_{l=1}^{m_{1}}\int_{S}\chi_{l}e^{-\varphi_{l}}e^{U_{l}}\partial_{(\xi_{j})_{i}}Wdv_{g}-{1\over\tau}\sum_{l=m_{1}+1}^{m}\int_{S}\chi_{l}e^{-\varphi_{l}}e^{U_{l}}\partial_{(\xi_{j})_{i}}Wdv_{g}$
(5.10) $\displaystyle=$
$\displaystyle-32\pi^{2}\partial_{(\xi_{j})_{i}}\varphi^{*}_{m}(\xi)+O(\delta^{2}|\log\delta|)$

for $j\in\\{1,\dots,m_{1}\\}$. Similarly, for $j\in\\{m_{1}+1,\dots,m\\}$ we
compute

$-\int_{S}\Delta_{g}W\partial_{(\xi_{j})_{i}}Wdv_{g}=-32\pi^{2}\partial_{(\xi_{j})_{i}}\varphi_{m}(\xi)+O(\delta^{2}|\log\delta|).$

In order to give an expansion of the second term in
$\partial_{(\xi_{j})_{i}}[J_{\lambda}(W)]$, first observe that by Lemma 3.1
there hold

$V_{1}e^{W}={e^{\alpha_{\delta,\xi}-2F_{\delta,\xi}(x)}\over
8\delta_{j}^{2}}\rho_{j}e^{U_{j}}[1+O(\delta^{4}|\log\delta|)]\quad\text{uniformly
in $B_{r_{0}}(\xi_{j})$, $j=1\dots,m_{1}$}$ (5.11) $V_{1}e^{W}=O(1)$ uniformly
in $S\setminus\cup_{j=1}^{m_{1}}B_{r_{0}}(\xi_{j})$, (5.12) $V_{2}e^{-\tau
W}={e^{-\tau\alpha_{\delta,\xi}+2\tau F_{\delta,\xi}(x)}\over
8\delta_{j}^{2}}\rho_{j}e^{U_{j}}[1+O(\delta^{4}|\log\delta|)]\quad\text{uniformly
in $B_{r_{0}}(\xi_{j})$, $j=m_{1}+1\dots,m$}$ (5.13) $\text{ and
$V_{2}e^{-\tau W}=O(1)$ uniformly in
$S\setminus\cup_{j=m_{1}+1}^{m}B_{r_{0}}(\xi_{j})$}.$ (5.14)

So, arguing in the same way as in the proof of (35) in $C^{1}(\Xi)$ in [23,
Theorem 3.2] and taking into account that for $k=1,2$

$\int_{S}V_{k}e^{(-\tau)^{k-1}W}\partial_{(\xi_{j})_{i}}Wdv_{g}=\sum_{l=1}^{m_{1}}\int_{S}V_{k}e^{(-\tau)^{k-1}W}\partial_{(\xi_{j})_{i}}W_{l}-{1\over\tau}\sum_{l=m_{1}+1}^{m}\int_{S}V_{k}e^{(-\tau)^{k-1}W}\partial_{(\xi_{j})_{i}}W_{l}$

we have that

$\int_{S}{V_{k}e^{(-\tau)^{k-1}W}\over\int_{S}V_{k}e^{(-\tau)^{k-1}W}dv_{g}}\partial_{(\xi_{j})_{i}}Wdv_{g}=O(\delta^{2}|\log\delta|),\quad
k=1,2.$ (5.15)

In conclusion, by (5.10)-(5.15) we can write

$\displaystyle\partial_{(\xi_{j})_{i}}[J_{\lambda_{1},\lambda_{2}}(W)]=-32\pi^{2}\partial_{(\xi_{j})_{i}}\varphi^{*}_{m}(\xi)+O(\delta^{2}|\log\delta|).$

and the proof is complete. ∎

Finally, we address the expansions for the derivatives of
$J_{\lambda_{1},\lambda_{2}}(W)$ in $\mu$. Recall that we argue similar to the
proof of (35) in $C^{2}(\mathbb{R})$ in [23, Theorem 3.2].

###### Proof (of (5.1) in $C^{2}(\mathbb{R}^{2})$):.

We just focus on the first and second derivative of
$J_{\lambda_{1},\lambda_{2}}(W)$ in $\mu_{i}$, $i=1,2$. Since
$\partial_{\mu_{i}}=\delta\rho_{l}^{\frac{1}{2}}(\xi_{l})\partial_{\delta_{l}}$,
$i=1$ for $l\in\\{1,\dots,m_{1}\\}$ and $i=2$ for $l\in\\{m_{1}+1,\dots,m\\}$,
in view of (3.5), arguing as in Lemma 3.1, it is easy to show that

$\displaystyle\delta^{-1}\rho_{l}^{-\frac{1}{2}}(\xi_{l})\partial_{\mu_{i}}W_{l}=-\chi_{l}\frac{4\delta_{l}}{\delta_{l}^{2}+|y_{\xi_{l}}(x)|^{2}}+\beta_{\delta_{l},\xi_{l}}-4\delta_{l}F_{\xi_{l}}+O(\delta^{3}|\log\delta|)$
(5.16)
$\displaystyle\delta^{-2}\rho_{l}^{-1}(\xi_{l})\partial_{\mu_{i}\mu_{i}}W_{l}=4\chi_{l}\frac{\delta_{l}^{2}-|y_{\xi_{l}}(x)|^{2}}{(\delta_{l}^{2}+|y_{\xi_{l}}(x)|^{2})^{2}}+\gamma_{\delta_{l},\xi_{l}}-4F_{\xi_{l}}+O(\delta^{2}|\log\delta|)$
(5.17)

do hold uniformly in $S$, where

$\beta_{\delta_{l},\xi_{l}}=-{8\pi\over|S|}\delta_{l}\log\delta_{l}+{4\delta_{l}\over|S|}\left(\int_{\mathbb{R}^{2}}\chi(|y|)\frac{e^{\hat{\varphi}_{\xi}(y)}-1}{|y|^{2}}dy-\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\log|y|\over|y|}dy\right)$

and

$\gamma_{\delta_{l},\xi_{l}}=-{8\pi\over|S|}\log\delta_{l}+{4\over|S|}\left(\int_{\mathbb{R}^{2}}\chi(|y|)\frac{e^{\hat{\varphi}_{\xi}(y)}-1}{|y|^{2}}dy-2\pi-\int_{\mathbb{R}^{2}}{\chi^{\prime}(|y|)\log|y|\over|y|}dy\right).$

Note that $\partial_{\mu_{i}}W_{l}=0$ either if $i=1$ and
$l\in\\{m_{1}+1,\dots,m\\}$ or $i=2$ and $l\in\\{1,\dots,m\\}$. Let us stress
that $\partial_{\mu_{i}\mu_{k}}W_{l}=0$ for all $l=1,\dots,m$ and $i\neq k$,
so that $\partial_{\mu_{i}\mu_{k}}W=0$ for $i\neq k$. By Lemma 5.1 we then
have that either for $i=1$, $l\in\\{1,\dots,m_{1}\\}$ or $i=2$,
$l\in\\{m_{1}+1,\dots,m\\}$

$\delta^{-1}\rho_{l}^{-\frac{1}{2}}(\xi_{l})\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\partial_{\mu_{i}}W_{l}dv_{g}=-\frac{16\pi}{\delta_{j}}\delta_{jl}+8\pi\beta_{\delta_{l},\xi_{l}}-32\pi\delta_{l}F_{\xi_{l}}(\xi_{j})+O(\delta^{3}|\log\delta|^{2}),$
$\displaystyle\delta^{-2}\rho_{l}^{-1}(\xi_{l})\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\partial_{\mu_{i}\mu_{i}}W_{l}dv_{g}=\frac{16\pi}{3\delta_{j}^{2}}\delta_{jl}+8\pi\gamma_{\delta_{l},\xi_{l}}-32\pi
F_{\xi_{l}}(\xi_{j})+O(\delta^{2}|\log\delta|^{2})$ (5.18)

and either for $k=1$, $j\in\\{1,\dots,m_{1}\\}$ or $k=2$,
$l\in\\{m_{1}+1,\dots,m\\}$

$\displaystyle\delta^{-1}\rho_{l}^{-\frac{1}{2}}(\xi_{l})\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\partial_{\mu_{k}}U_{j}\partial_{\mu_{i}}W_{l}dv_{g}=\frac{2}{\mu_{k}}\delta^{-1}\rho_{l}^{-\frac{1}{2}}(\xi_{l})\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\frac{|y_{\xi_{j}}(x)|^{2}-\delta_{j}^{2}}{\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2}}\partial_{\mu_{i}}W_{l}dv_{g}$
$\displaystyle=\frac{32\pi}{3\delta_{j}^{2}}\delta\rho_{j}(\xi_{j})^{\frac{1}{2}}\delta_{jl}+O(\delta^{\gamma})$
(5.19)

in view of $\int_{\mathbb{R}^{2}}\frac{|y|^{2}-1}{(1+|y|^{2})^{3}}dy=0$, where
$\delta_{jl}$ denotes the Kronecker’s symbol. Note that
$\partial_{\mu_{k}}U_{j}=0$ either for $k=1$ and $j\in\\{m_{1}+1,\dots,m\\}$
or $k=2$ and $j\in\\{1,\dots,m_{1}\\}$. Since
$\int_{S}\partial_{\mu_{i}}Wdv_{g}=\int_{S}\partial_{\mu_{i}\mu_{k}}Wdv_{g}=0$,
we then deduce the following expansions:

$\displaystyle\int_{S}(-\Delta_{g}W)\partial_{\mu_{1}}Wdv_{g}$
$\displaystyle=$
$\displaystyle\sum_{j,l=1}^{m_{1}}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\partial_{\mu_{1}}W_{l}dv_{g}-{1\over\tau}\sum_{j=m_{1}+1}^{m}\sum_{l=1}^{m_{1}}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\partial_{\mu_{1}}W_{l}dv_{g}$
$\displaystyle=$ $\displaystyle-\frac{16\pi m_{1}}{\mu_{1}}+8\pi
m_{1}\delta\sum_{l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}-32\pi\mu_{1}\delta^{2}\sum_{j,l=1}^{m_{1}}\rho_{l}(\xi_{l})F_{\xi_{l}}(\xi_{j})$
$\displaystyle-{8\pi
m_{2}\over\tau}\delta\sum_{l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}+{32\pi\over\tau}\mu_{1}\delta^{2}\sum_{j=m_{1}+1}^{m}\sum_{l=1}^{m_{1}}\rho_{l}(\xi_{l})F_{\xi_{l}}(\xi_{j})+O(\delta^{4}|\log\delta|^{2}),$

and

$\displaystyle\int_{S}(-\Delta_{g}W)\partial_{\mu_{2}}Wdv_{g}$
$\displaystyle=$
$\displaystyle-{1\over\tau}\sum_{j=1}^{m_{1}}\sum_{l=m_{1}+1}^{m}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\partial_{\mu_{2}}W_{l}dv_{g}+{1\over\tau^{2}}\sum_{j,l=m_{1}+1}^{m}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}\partial_{\mu_{2}}W_{l}dv_{g}$
$\displaystyle=$ $\displaystyle-\frac{16\pi m_{2}}{\mu_{2}\tau^{2}}-{8\pi
m_{1}\over\tau}\delta\sum_{l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}+{32\pi\over\tau}\mu_{2}\delta^{2}\sum_{j=1}^{m_{1}}\sum_{l=m_{1}+1}^{m}\rho_{l}(\xi_{l})F_{\xi_{l}}(\xi_{j})$
$\displaystyle+{8\pi
m_{2}\over\tau^{2}}\delta\sum_{l=m_{1}+1}^{m}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}-{32\pi\over\tau^{2}}\mu_{2}\delta^{2}\sum_{j,l=m_{1}+1}^{m}\rho_{l}(\xi_{l})F_{\xi_{l}}(\xi_{j})+O(\delta^{4}|\log\delta|^{2}),$

as $\delta\to 0$. Since by Lemma 3.1 there hold (5.11), (5.12) and
$\partial_{\mu_{1}}W=O(\delta^{2}|\log\delta|)$ uniformly in
$S\setminus\cup_{j=1}^{m_{1}}B_{r_{0}}(\xi_{j})$, by Lemma 5.1 we can write
that

$\displaystyle\displaystyle\int_{S}$ $\displaystyle
V_{1}e^{W}\partial_{\mu_{1}}Wdv_{g}=\sum_{j,l=1}^{m_{1}}\int_{B_{r_{0}}(\xi_{j})}V_{1}e^{W}\partial_{\mu_{1}}W_{l}dv_{g}+O(\delta^{2}|\log\delta|)$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{m_{1}}{e^{\alpha_{\delta,\xi}}\over
2\mu_{1}}\int_{B_{r_{0}}(\xi_{j})}e^{-2F_{\delta,\xi}(x)}\frac{\rho_{j}e^{U_{j}}}{\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2}}dv_{g}$
$\displaystyle+\;\pi{e^{\alpha_{\delta,\xi}}\over\mu_{1}^{2}\delta}\left(m_{1}\sum_{l=1}^{m}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}-4\sum_{j,l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\delta_{l}F_{\xi_{l}}(\xi_{j})\right)+\,O(\delta|\log\delta|)$
$\displaystyle=$
$\displaystyle\pi{e^{\alpha_{\delta,\xi}}\over\mu_{1}^{2}\delta^{2}}\left(-\frac{2m_{1}}{\mu_{1}}+m_{1}\delta\sum_{l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}-{\mu_{1}\delta^{2}\over
8\pi}A_{1}^{*}(\xi)-{4\over\mu_{1}\tau}\sum_{j=1}^{m_{1}}F_{2,\delta,\xi}(\xi_{j})+O(\delta^{2+\gamma})\right)$

in view of (5.4) and from (5.5)

$\displaystyle\sum_{j=1}^{m_{1}}e^{-2F_{\delta,\xi}(\xi_{j})}=m_{1}-2\sum_{j,l=1}^{m_{1}}\delta_{l}^{2}F_{\xi_{l}}(\xi_{j})+{2\over\tau}\sum_{j=1}^{m_{1}}F_{2,\delta,\xi}(\xi_{j})+O(\delta^{4}).$

Combining with (5.6) we then get that

$\displaystyle\frac{\int_{S}V_{1}e^{W}\partial_{\mu_{1}}Wdv_{g}}{\int_{S}V_{1}e^{W}dv_{g}}$
$\displaystyle=$
$\displaystyle-\frac{2}{\mu_{1}}+\delta\sum_{l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}-{\delta^{2}A_{1}^{*}(\xi)\over
8\pi m_{1}}[\mu_{1}+2\mu_{1}\log\mu_{1}]$
$\displaystyle-\frac{A_{1}^{*}(\xi)}{4\pi
m_{1}}\mu_{1}\delta^{2}\log\delta+\frac{B_{1}^{*}(\xi)}{4\pi
m_{1}}\mu_{1}\delta^{2}-\frac{4}{m_{1}\mu_{1}}\sum_{j=1}^{m_{1}}F_{1,\delta,\xi}(\xi_{j})+o(\delta^{2}).$

Similarly as above, there hold (5.13), (5.14) and and
$\partial_{\mu_{1}}W=O(\delta^{2}|\log\delta|)$ uniformly in
$S\setminus\cup_{j=1}^{m_{1}}B_{r_{0}}(\xi_{j})$, so that

$\displaystyle\int_{S}V_{2}e^{-\tau
W}\partial_{\mu_{1}}Wdv_{g}=\sum_{j,l=1}^{m_{1}}\int_{B_{r_{0}}(\xi_{j})}V_{2}e^{-\tau
W}\partial_{\mu_{1}}W_{l}dv_{g}+\sum_{j=m_{1}+1}^{m}\sum_{l=1}^{m_{1}}\int_{B_{r_{0}}(\xi_{j})}V_{2}e^{-\tau
W}\partial_{\mu_{1}}W_{l}dv_{g}$ $\displaystyle\ \
+\>O(\delta^{2}|\log\delta|)$
$\displaystyle=\pi{e^{-\tau\alpha_{\delta,\xi}}\over\mu_{2}^{2}\delta^{2}}\left(m_{2}\delta\sum_{l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}-{4\over\mu_{1}}\sum_{j=m_{1}+1}^{m}F_{1,\delta,\xi}(\xi_{j})+O(\delta^{4}|\log\delta|)\right)$

in view of $\tau>0$, (5.7) and

$\displaystyle\int_{B_{r_{0}}(\xi_{j})}\frac{V_{2}e^{-\tau
W}}{\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2}}dv_{g}=O\left(\int_{B_{r_{0}}(\xi_{j})}(\delta_{j}^{2}+|y_{\xi_{j}}(x)|^{2})^{\tau-1}dv_{g}\right)=O(1).$

Combining with (5) we then get that

$\displaystyle\frac{\int_{S}V_{2}e^{-\tau
W}\partial_{\mu_{1}}Wdv_{g}}{\int_{S}V_{2}e^{-\tau W}dv_{g}}$ $\displaystyle=$
$\displaystyle\delta\sum_{l=1}^{m_{1}}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}-\frac{4}{m_{2}\mu_{1}}\sum_{j=1}^{m_{1}}F_{1,\delta,\xi}(\xi_{j})+o(\delta^{2}),$
(5.23)

which yields to

$\displaystyle\partial_{\mu_{1}}[J_{\lambda_{1},\lambda_{2}}(W)]$
$\displaystyle=$
$\displaystyle\int_{S}(-\Delta_{g}W)\partial_{\mu_{1}}Wdv_{g}-\lambda_{1}\frac{\int_{S}V_{1}e^{W}\partial_{\mu_{1}}Wdv_{g}}{\int_{S}V_{1}e^{W}dv_{g}}+\lambda_{2}\tau\frac{\int_{S}V_{2}e^{-\tau
W}\partial_{\mu_{1}}Wdv_{g}}{\int_{S}V_{2}e^{-\tau W}dv_{g}}$ (5.24)
$\displaystyle=$ $\displaystyle\frac{2(\lambda_{1}-8\pi
m_{1})}{\mu_{1}}+2A_{1}^{*}(\xi)\mu_{1}\delta^{2}\log\delta+[A_{1}^{*}(\xi)\\{\mu_{1}+2\mu_{1}\log\mu_{1}\\}-2B_{1}^{*}(\xi)\mu_{1}]\delta^{2}$
$\displaystyle+o(\delta^{2})$

in view of (5), so that we deduce the validity of (5.1) for the first
derivative in $\mu_{1}$. Now, for the first derivative in $\mu_{2}$, similarly
as above we have that

$\displaystyle\frac{\int_{S}V_{1}e^{W}\partial_{\mu_{2}}Wdv_{g}}{\int_{S}V_{1}e^{W}dv_{g}}$
$\displaystyle=$
$\displaystyle-{\delta\over\tau}\sum_{l=m_{1}+1}^{m}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}+\frac{4}{m_{1}\mu_{2}\tau}\sum_{j=1}^{m_{1}}F_{2,\delta,\xi}(\xi_{j})+o(\delta^{2}).$
(5.25)

in view of (5.6), and

$\displaystyle\frac{\int_{S}V_{2}e^{-\tau
W}\partial_{\mu_{2}}Wdv_{g}}{\int_{S}V_{2}e^{-\tau W}dv_{g}}$ $\displaystyle=$
$\displaystyle\frac{2}{\mu_{2}\tau}-{\delta\over\tau}\sum_{l=m_{1}+1}^{m}\rho_{l}^{\frac{1}{2}}(\xi_{l})\beta_{\delta_{l},\xi_{l}}+{\delta^{2}A_{2}^{*}(\xi)\over
8\pi m_{2}\tau}[\mu_{2}+2\mu_{2}\log\mu_{2}]$
$\displaystyle+\frac{A_{2}^{*}(\xi)}{4\pi
m_{2}\tau}\mu_{2}\delta^{2}\log\delta-\frac{B_{2}^{*}(\xi)}{4\pi
m_{2}\tau}\mu_{2}\delta^{2}+\frac{4}{m_{2}\mu_{2}\tau}\sum_{j=m_{1}+1}^{m}F_{2,\delta,\xi}(\xi_{j})+o(\delta^{2}).$

by using (5.7) and combining with (5). Thus, by using (5) we conclude the
validity of (5.1) for the first derivative in $\mu_{2}$:

$\displaystyle\partial_{\mu_{2}}[J_{\lambda_{1},\lambda_{2}}(W)]$
$\displaystyle=$ $\displaystyle\frac{2(\lambda_{2}\tau^{2}-8\pi
m_{2})}{\mu_{2}\tau^{2}}+\frac{2A_{2}^{*}(\xi)}{\tau^{2}}\mu_{2}\delta^{2}\log\delta+[A_{2}^{*}(\xi)\\{\mu_{2}+2\mu_{2}\log\mu_{2}\\}-2B_{2}^{*}(\xi)\mu_{2}]\frac{\delta^{2}}{\tau^{2}}$
(5.27) $\displaystyle+o(\delta^{2})$

Towards the expansion of the second derivatives in $\mu$, we proceed in a
similar way to obtain (5.24) and (5.27) with the aid of the expansions (5.16)
for $\partial_{\mu_{i}}W$ and (5.17) for $\partial_{\mu_{i}\mu_{i}}W_{l}$,
(5.18) and (5) (see also validity of expansion (35) in $C^{2}(\mathbb{R})$ in
[23, Theorem 3.2]). We omit the details, thus, we conclude the validity of
(5.1) also for the second derivatives in $\mu$ and the proof is complete. ∎

##  6 Proof of Theorem 1.2

In this section, we shall study the existence of blowing-up solutions as
$\lambda_{1}\to 8\pi m_{1}$ and $\lambda_{2}\tau^{2}\to 0$, which resembles
the equation (1.3). For simplicity, we shall denote $m_{1}=m$ so that our
approximating solution is $\displaystyle W(x)=\sum_{j=1}^{m}W_{j}(x)$, and we
look for solutions to (1.1) in the form $u=W+\phi$. Assumptions (3.5)-(3.6)
are replaced by

$\begin{split}\delta_{j}^{2}=\mu^{2}\delta^{2}\rho_{j}(\xi_{j}),\ \
j=1,\dots,m\qquad&\text{with}\qquad 0<\mu\leq C_{0}\\\\[2.84544pt]
|\lambda_{1}-8\pi m|\leq C\delta^{2}|\log\delta|\qquad&\text{and}\qquad
0<\lambda_{2}\tau^{2}\leq C\delta^{2}|\log\delta|.\end{split}$ (6.1)

Notice from similar computations above to obtain (5), we have that

$\int_{S}V_{2}e^{-\tau
W}dv_{g}=e^{-\tau\alpha_{\delta,\xi}}\left[\int_{S}V_{2}e^{-8\pi\tau\sum\limits_{j=1}^{m}G(x,\xi_{j})}dv_{g}+O(\delta^{2})\right]\geq\eta_{0}>0$

for some $\eta_{0}>0$. By conditions (6.1) we get that

${\lambda_{2}\tau V_{2}e^{-\tau W}\over\int_{S}V_{2}e^{-\tau
W}}=O(\delta^{2}|\log\delta|)\quad\text{uniformly in $S$}.$ (6.2)

Hence, it follows estimate (3.12). Now, denote $\displaystyle
Z=\sum_{l=1}^{m}Z_{0l}$ and $PZ$ its projection according to (4.1). By using
(6.2) and similar arguments used in the proofs of [23, Proposition 4.1] and
Proposition 4.1, it follows the invertibility of $L$ in (3.8) in this case (as
$\lambda_{1}\to 8\pi m$ and $\lambda_{2}\tau^{2}\to 0$), and we deduced the
following fact.

######  Proposition 6.1.

There exists $\delta_{0}>0$ so that for all $0<\delta\leq\delta_{0}$,
$\mu\in(0,C_{0}]$, $\xi\in\Xi$ problem

$\left\\{\begin{array}[]{ll}L(\phi)=-[R+N(\phi)]+c_{0}\Delta_{g}PZ+\displaystyle\sum_{i=1}^{2}\sum_{j=1}^{m}c_{ij}\Delta_{g}PZ_{ij}&\text{in
}S\\\
\int_{S}\phi\Delta_{g}PZdv_{g}=\int_{S}\phi\Delta_{g}PZ_{ij}dv_{g}=0&\forall\>i=1,2,\,j=1,\dots,m\end{array}\right.$

admits a unique solution $\phi(\mu,\xi)\in\bar{H}\cap W^{2,2}(S)$ and
$c_{0}(\mu,\xi),\,c_{ij}(\mu,\xi)\in\text{I\\!R}$, $i=1,2$ and $j=1,\dots,m$,
where $\delta_{j}>0$ are as in (6.1) and $N$, $R$ are given by (3.9), (3.11),
respectively. Moreover, the map
$(\mu,\xi)\mapsto(\phi(\mu,\xi),c_{0}(\mu,\xi),c_{ij}(\mu,\xi))$ is twice-
differentiable in $\mu$ and one-differentiable in $\xi$ with

$\|\phi\|_{\infty}+{\|\partial_{\mu}\phi\|_{\infty}\over|\log\delta|}+\sum_{i=1}^{2}\sum_{j=1}^{m}{\delta\|\partial_{(\xi_{j})_{i}}\phi\|_{\infty}\over|\log\delta|}+{\|\partial_{\mu\mu}\phi\|_{\infty}\over|\log\delta|^{2}}\leq
C\left(\delta|\log\delta||\nabla\varphi_{m}^{*}(\xi)|_{g}+\delta^{2-\sigma}|\log\delta|^{2}\right)$

As in the case $m_{2}\geq 1$, the function $[W+\phi](\mu,\xi)$ will be a true
solution of (3.7) if $\mu\in[C_{0}^{-1},C_{0}]$ and $\xi\in\Xi$ are such that
$c_{0}(\mu,\xi)=c_{ij}(\mu,\xi)=0$ for all $i=1,2,$ and $j=1,\dots,m$.
Similarly to Lemma 4.1, this problem is equivalent to finding critical points
of the reduced energy
$E_{\lambda_{1},\lambda_{2}}(\mu,\xi)=J_{\lambda_{1},\lambda_{2}}\big{(}[W+\phi](\mu,\xi)\big{)}$,
where $J_{\lambda_{1},\lambda_{2}}$ is given by (1.4). Notice that

$\lambda_{2}\log\left(\int_{S}V_{2}e^{-\tau
W}dv_{g}\right)=-\lambda_{2}\tau\alpha_{\delta,\xi}+\lambda_{2}\log\left(\int_{S}V_{2}e^{-8\pi\tau\sum\limits_{j=1}^{m}G(x,\xi_{j})}dv_{g}\right)+O(\delta^{4}|\log\delta|).$

Let us stress that
$\displaystyle\lambda_{2}\log\bigg{(}\int_{S}V_{2}e^{-8\pi\tau\sum\limits_{j=1}^{m}G(x,\xi_{j})}dv_{g}\bigg{)}$
is independent of $\mu$. Taking into account computations in the proof of [23,
Theorem 3.2] and similar ones in the proof of Theorem 5.1 we have that

$\begin{split}J_{\lambda_{1},\lambda_{2}}(W)=&\,-8\pi m-\lambda_{1}\log(\pi
m)+2(\lambda_{1}-8\pi
m)\log(\mu\delta)-32\pi^{2}\varphi_{m}^{*}(\xi)+A(\xi)\mu^{2}\delta^{2}\log\delta\\\
&+[A(\xi)\mu^{2}\log\mu-B(\xi)\mu^{2}]\delta^{2}-\lambda_{2}\log\left(\int_{S}V_{2}e^{-8\pi\tau\sum\limits_{j=1}^{m}G(x,\xi_{j})}dv_{g}\right)+o(\delta^{2}).\end{split}$

Consequently, from estimates in Appendix B we obtain that

######  Theorem 6.1.

Assume (6.1). The following expansion does hold

$\displaystyle E_{\lambda_{1},\lambda_{2}}(\mu,\xi)$ $\displaystyle=$
$\displaystyle-8\pi m-\lambda_{1}\log(\pi m)-2\big{(}\lambda_{1}-8\pi
m\big{)}\log\delta-32\pi^{2}\varphi_{m}^{*}(\xi)$
$\displaystyle\,\,+2\big{(}\lambda_{1}-8\pi
m\big{)}\log\mu+A(\xi)\mu^{2}\delta^{2}\log\delta+\left[A(\xi)\mu^{2}\log\mu-B(\xi)\mu^{2}\right]\delta^{2}$
$\displaystyle\,\,-\lambda_{2}\log\left(\int_{S}V_{2}e^{-8\pi\tau\sum\limits_{j=1}^{m}G(x,\xi_{j})}dv_{g}\right)+o(\delta^{2})+r_{\lambda_{1},\lambda_{2}}(\mu,\xi)$

in $C^{2}(\mathbb{R})$ and $C^{1}(\Xi)$ as $\delta\to 0^{+}$, where
$\varphi_{m}^{*}(\xi)$, $A(\xi)$ and $B(\xi)$ are given by (1.7), (1.10) and
(1) with $k=1$, respectively. The term $r_{\lambda_{1},\lambda_{2}}(\mu,\xi)$
satisfies (4.9) for some $C>0$ independent of
$(\mu,\xi)\in(0,C_{0}]\times\Xi$.

###### Proof (of Theorem 1.2):.

We argue in the same way as in the proof of Theorem 1.1 with $k=1$. ∎

##  7 Appendix A

We shall argue in the same way as in Appendix A in [23]. We first address
a-priori estimates for the operator $L$ when all the $c_{ij}$’s vanish:

######  Proposition 7.1.

There exists $\delta_{0}>0$ and $C>0$ so that, for all
$0<\delta\leq\delta_{0}$, $h\in C(S)$ with $\int_{S}hdv_{g}=0$, $\xi\in\Xi$
and $\phi\in H_{0}^{1}(S)\cap W^{2,2}(S)$ a solution of (4.2) with
$c_{0i}=c_{ij}=0$, $i=1,2$ and $j=1,\dots,m$, one has

$\|\phi\|_{\infty}\leq C|\log\delta|\|h\|_{*}.$

###### Proof:.

By contradiction, assume the existence of sequences $\delta\to 0$,
$\mu=(\mu_{1},\mu_{2})$ with $\mu\to\mu^{*}$, points $\xi\in\Xi$ with
$\xi\to\xi^{*}$, functions $h$ with $|\log\delta|\|h\|_{*}=o(1)$ and solutions
$\phi$ with $\|\phi\|_{\infty}=1$. Recall that
$\delta_{j}^{2}=\mu_{i}\delta^{2}\rho_{j}(\xi_{j})$. Setting
$\mathcal{K}_{i}=\frac{\lambda_{i}\tau^{2(i-1)}V_{i}e^{(-\tau)^{i-1}W}}{\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}$,
$\psi_{i}=\phi+\tilde{c}_{i}(\phi)$,
$\tilde{c}_{i}(\phi)=-\frac{\int_{S}V_{i}e^{(-\tau)^{i-1}W}\phi
dv_{g}}{\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}$ for $i=1,2$, we have that
$\psi_{1}-\tilde{c}_{1}(\phi)=\psi_{2}-\tilde{c}_{2}(\phi)$,
$\Delta_{g}\psi_{1}+\mathcal{K}_{1}\psi_{1}+\mathcal{K}_{2}[\psi_{1}-\tilde{c}_{1}(\phi)+\tilde{c}_{2}(\phi)]=h$
and
$\Delta_{g}\psi_{2}+\mathcal{K}_{1}[\psi_{2}-\tilde{c}_{2}(\phi)+\tilde{c}_{1}(\phi)]+\mathcal{K}_{2}\psi_{2}=h$
in $S$ and $\psi_{i}$, $i=1,2$ does satisfy the same orthogonality conditions
as $\phi$.

Since $\|\psi_{i,n}\|_{\infty}\leq 2\|\phi_{n}\|_{\infty}\leq 2$ and
$\Delta_{g}\psi_{i}=o(1)$ in
$C_{\hbox{loc}}(S\setminus\\{\xi_{1}^{*},\dots,\xi_{m}^{*}\\})$, we can assume
that $\psi_{i,n}\to\psi_{i,\infty}$ in
$C^{1}_{\hbox{loc}}(S\setminus\\{\xi_{1}^{*},\dots,\xi_{m}^{*}\\})$. Since
$\psi_{i,\infty}$ is bounded, it extends to an harmonic function in $S$, and
then
$\psi_{i,\infty}=\tilde{c}_{i,0}:=-\lim\frac{\int_{S}V_{i}e^{(-\tau)^{i-1}W}\phi
dv_{g}}{\int_{S}V_{i}e^{(-\tau)^{i-1}W}dv_{g}}$ in view of
${1\over|S|}\int_{S}\psi_{i,n}dv_{g}=\tilde{c}_{i,n}(\phi)$.

The function $\Psi_{i,j}=\psi_{i}(y_{\xi_{j}}^{-1}(\delta_{j}y))$ $i=1$, for
$j=1,\dots,m_{1}$ and $i=2$ for $j=m_{1}+1,\dots,m$ satisfy
$\Delta\Psi_{1,j}+\mathcal{\tilde{K}}_{1,j}\Psi_{1,j}+\mathcal{\tilde{K}}_{2,j}[\Psi_{1,j}-\tilde{c}_{1}+\tilde{c}_{2}]=\tilde{h}_{j}$
and
$\Delta\Psi_{2,j}+\mathcal{\tilde{K}}_{1,j}[\Psi_{2,j}-\tilde{c}_{2}+\tilde{c}_{1}]+\mathcal{\tilde{K}}_{2,j}\Psi_{2,j}=\tilde{h}_{j}$
in $B_{2r_{0}\over\delta_{j}}(0)$, where
$\mathcal{\tilde{K}}_{i,j}=\delta_{j}^{2}\mathcal{K}_{i}(y_{\xi_{j}}^{-1}(\delta_{j}y))$
and $\tilde{h}_{j}(y)=\delta_{j}^{2}h(y_{\xi_{j}}^{-1}(\delta_{j}y))$. Since
$|\tilde{h}_{j}|\leq C\|h\|_{*}$,

$\mathcal{\tilde{K}}_{1,j}=\begin{cases}{8\over(1+|y|^{2})^{2}}(1+O(\delta^{2}|\log\delta|))&\text{
for $j=1,\dots,m_{1}$}\\\ O(\delta_{j}^{2})&\text{ for $j=m_{1}+1,\dots,m$
}\end{cases}$

and

$\mathcal{\tilde{K}}_{2,j}=\begin{cases}O(\delta_{j}^{2})&\text{ for
$j=1,\dots,m_{1}$}\\\
{8\over(1+|y|^{2})^{2}}(1+O(\delta^{2}|\log\delta|))&\text{ for
$j=m_{1}+1,\dots,m$ }\end{cases}$

uniformly in $B_{\frac{2r_{0}}{\delta}}(0)$, in view of Lemma 3.1, (5.6) and
(5), up to a sub-sequence, by elliptic estimates
$\Psi_{i,j}\to\Psi_{j,\infty}$ with $i=1$ if $j=1,\dots,m_{1}$ and $i=2$ if
$j=m_{1}+1,\dots,m$ in $C^{1}_{\hbox{loc}}(\mathbb{R}^{2})$, where
$\Psi_{j,\infty}$ is a bounded solution of
$\Delta\Psi_{j,\infty}+{8\over(1+|y|^{2})^{2}}\Psi_{j,\infty}=0$ of the form
$\Psi_{j,\infty}=\displaystyle\sum_{i=0}^{2}a_{ij}Y_{i}$ (see for example
[2]). Since
$-\Delta_{g}PZ_{ij}=\chi_{j}e^{-\varphi_{j}}e^{U_{j}}Z_{ij}-\frac{1}{|S|}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}Z_{ij}dv_{g}$
in view of (4.1) and $\Delta_{g}=e^{-\varphi_{j}}\Delta$ in
$B_{2r_{0}}(\xi_{j})$ through $y_{\xi_{j}}$, we have that

$0=-\int_{S}\psi_{l}\Delta_{g}PZ_{ij}=32\int_{\mathbb{R}^{2}}\Psi_{l,j}\frac{y_{i}}{(1+|y|^{2})^{3}}dy-\frac{32}{|S|}\int_{\mathbb{R}^{2}}\frac{y_{i}}{(1+|y|^{2})^{3}}dy\int_{S}\psi_{l,n}+O(\delta^{3}),$

with $l=1$ if $j=1,\dots,m_{1}$ and $l=2$ if $j=m_{1}+1,\dots,m$. Since then
$\int_{\mathbb{R}^{2}}\Psi_{j,\infty}\frac{y_{i}}{(1+|y|^{2})^{3}}dy=0$, we
deduce that $a_{1j}=a_{2j}=0$. By the orthogonality condition
$\int_{S}\phi\Delta_{g}PZ_{1}=0$, similarly we deduce that

$\displaystyle 0$ $\displaystyle=$
$\displaystyle-\sum_{j=1}^{m_{1}}\int_{S}\psi_{1}\Delta_{g}PZ_{0j}dv_{g}$
$\displaystyle=$ $\displaystyle
16\sum_{j=1}^{m_{1}}\int_{\mathbb{R}^{2}}\Psi_{j}\frac{1-|y|^{2}}{(1+|y|^{2})^{3}}dy-\frac{16}{|S|}m_{1}\int_{\mathbb{R}^{2}}\frac{1-|y|^{2}}{(1+|y|^{2})^{3}}dy\int_{S}\psi_{1,n}+O(\delta^{2}),$

which implies $\displaystyle\sum_{j=1}^{m_{1}}a_{0j}=0$ in view of
$\int_{\mathbb{R}^{2}}\frac{1-|y|^{2}}{(1+|y|^{2})^{3}}dy=0$. By using the
same argument, the orthogonality condition $\int_{S}\phi\Delta_{g}PZ_{2}=0$
implies that $\displaystyle\sum_{j=m_{1}+1}^{m}a_{0j}=0$. By dominated
convergence we have that

$\displaystyle\int_{S}G(y,\xi_{j})\mathcal{K}_{1}\psi_{1}dv_{g}=-{1\over
2\pi}\log\delta\int_{B_{r_{0}}(\xi_{j})}\mathcal{K}_{1}\psi_{1}dv_{g}+\int_{\mathbb{R}^{2}}\Big{[}-{1\over
2\pi}\log|y|+H(\xi_{j},\xi_{j})\Big{]}\frac{8}{(1+|y|^{2})^{2}}\Psi_{j,\infty}dy$
$\displaystyle+\sum_{i=1\atop
i\not=j}^{m_{1}}G(\xi_{i},\xi_{j})\int_{\mathbb{R}^{2}}\frac{8}{(1+|y|^{2})^{2}}\Psi_{i,\infty}dy+o(1)=-{1\over
2\pi}\log\delta\int_{B_{r_{0}}(\xi_{j})}\mathcal{K}_{1}\psi_{1}dv_{g}+4a_{0j}+o(1)$

in view of
$\int_{\mathbb{R}^{2}}\log|y|\frac{1-|y|^{2}}{(1+|y|^{2})^{3}}dy=-\frac{\pi}{2}$
and

$\begin{split}\int_{S}G(y,\xi_{j})\mathcal{K}_{2}\psi_{2}dv_{g}=&\,\sum_{i=m_{1}+1}^{m}G(\xi_{i},\xi_{j})\int_{\mathbb{R}^{2}}\frac{8}{(1+|y|^{2})^{2}}\Psi_{i,\infty}(y)\,dy\\\
&+O\bigg{(}\delta^{2}\int_{B_{r_{0}}(\xi_{j})}|G(y,\xi_{j})|dv_{g}\bigg{)}+o(1)=o(1)\end{split}$

for $j=1,\dots,m_{1}$. In view of $\int_{S}\mathcal{K}_{l}\psi_{l}=0$, $l=1,2$
and

$\bigg{|}\int_{S}G(y,\xi_{j})hdv_{g}\bigg{|}\leq
C|\log\delta|\int_{S}|h|dv_{g}+\frac{\|h\|_{*}}{\delta^{2}}\bigg{|}\int_{B_{\delta}(\xi_{j})}G(y,\xi_{j})dv_{g}\bigg{|}\leq
C^{\prime}|\log\delta|\|h\|_{*}=o(1),$

by the Green’s representation formula

$\displaystyle\sum_{j=1}^{m_{1}}\Psi_{j}(0)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{1}}\psi_{1}(\xi_{j})={m_{1}\over|S|}\int_{S}\psi_{1}dv_{g}+\sum_{j=1}^{m_{1}}\int_{S}G(y,\xi_{j})[\mathcal{K}_{1}\psi_{1}+\mathcal{K}_{2}\psi_{2}-h]dv_{g}$
$\displaystyle=$ $\displaystyle
m_{1}\tilde{c}_{1,0}+4\sum_{j=1}^{m_{1}}a_{0j}+o(1)$

which gives $\displaystyle
2\sum_{j=1}^{m_{1}}a_{0j}=m_{1}\tilde{c}_{1,0}+4\displaystyle\sum_{j=1}^{m_{1}}a_{0j}$
as $n\to+\infty$. Since $\displaystyle\sum_{j=1}^{m_{1}}a_{0j}=0$, we get that
$\tilde{c}_{1,0}=0$. By using a similar argument, we obtain that

$\int_{S}G(y,\xi_{j})\mathcal{K}_{1}\psi_{1}dv_{g}=o(1)\quad\text{for
$j=1,\dots,m_{1}$ and}$
$\int_{S}G(y,\xi_{j})\mathcal{K}_{2}\psi_{2}dv_{g}=-{1\over
2\pi}\log\delta\int_{B_{r_{0}}(\xi_{j})}\mathcal{K}_{2}\psi_{2}dv_{g}+4a_{0j}+o(1)$

for $j=m_{1}+1,\dots,m$, so that, from the Green’s representation formula for
$\Psi_{j}(0)$ for $j=m_{1}+1,\dots,m$ we get that $\tilde{c}_{2,0}=0$.
Following [25], let $P\hat{Z}_{j}\in H_{0}^{1}(S)$ be s.t.
$\Delta_{g}P\hat{Z}_{j}=\chi_{j}\Delta_{g}\hat{Z}_{j}-\frac{1}{|S|}\int_{S}\chi_{j}\Delta_{g}\hat{Z}_{j}dv_{g}$
in $S$, where

$\hat{Z}_{j}(x)=\beta_{j}\Big{(}\frac{y_{\xi_{j}}(x)}{\delta_{j}}\Big{)}\,,\qquad\beta_{j}(y)={4\over
3}[2\log\delta_{j}+\log(1+|y|^{2})]\frac{1-|y|^{2}}{1+|y|^{2}}+{8\over
3}\frac{1}{1+|y|^{2}}$

satisfies
$e^{\varphi_{j}}\Delta_{g}\hat{Z}_{j}+e^{U_{j}}\hat{Z}_{j}=e^{U_{j}}Z_{0j}$ in
$B_{2r_{0}}(\xi_{j})$. Since it is easily seen that
$P\hat{Z}_{j}=\chi_{j}\hat{Z}_{j}+{16\pi\over
3}H(\cdot,\xi_{j})+O(\delta^{2}|\log\delta|^{2})$ uniformly in $S$, we test
the equation of $\psi_{1}$ against $P\hat{Z}_{j}$, $j=1,\dots,m_{1}$ to get:

$\displaystyle
o(1)=\int_{S}hP\hat{Z}_{j}=\int_{S}\psi_{1}\bigg{[}\chi_{j}\Delta_{g}\hat{Z}_{j}-\frac{1}{|S|}\int_{S}\chi_{j}\Delta_{g}\hat{Z}_{j}dv_{g}\bigg{]}dv_{g}+\int_{S}\big{[}\mathcal{K}_{1}\psi_{1}+\mathcal{K}_{2}(\psi_{1}-\tilde{c}_{1}+\tilde{c}_{2})\big{]}P\hat{Z}_{j}dv_{g}$
$\displaystyle=\int_{S}\chi_{j}\psi_{1}[\Delta_{g}\hat{Z}_{j}+\mathcal{K}_{1}\hat{Z}_{j}]dv_{g}+o(1)=\int_{S}\chi_{j}\psi
e^{U_{j}}Z_{0j}dv_{g}+o(1)=16\int_{\mathbb{R}^{2}}\Psi_{j}\frac{1-|y|^{2}}{(1+|y|^{2})^{3}}dy+o(1)$

in view of , $\int_{S}\mathcal{K}_{1}\psi_{1}dv_{g}=0$,
$\int_{S}\mathcal{K}_{2}[\psi_{1}-\tilde{c}_{1}+\tilde{c}_{2}]P\hat{Z}_{j}dv_{g}=o(1)$,
$\int_{S}\psi_{1}dv_{g}=o(1)$,
$\int_{S}\chi_{j}\Delta_{g}\hat{Z}_{j}dv_{g}=O(1)$,
$\int_{S}\chi_{j}\psi_{1}[\mathcal{K}_{1}-e^{U_{j}}]\hat{Z}_{j}dv_{g}=O(\delta^{2}|\log\delta|^{2})$
and $\int_{S}hP\hat{Z}_{j}=O(|\log\delta|\|h\|_{*})=o(1)$, $j=1,\dots,m_{1}$.
Since $\int_{\mathbb{R}^{2}}\Psi_{j}\frac{1-|y|^{2}}{(1+|y|^{2})^{3}}dy=0$ we
have that $a_{0j}=0$, $j=1,\dots,m_{1}$. Now, testing the equation of
$\psi_{2}$ against $P\hat{Z}_{j}$, $j=m_{1}+1,\dots,m$, lead us to deduce that
$a_{0j}=0$,$j=m_{1}+1,\dots,m$. So far, we have shown that $\psi_{i}\to 0$ in
$C_{\hbox{loc}}(S\setminus\\{\xi_{1}^{*},\dots,\xi_{m}^{*}\\})$ and uniformly
in $\cup_{j=1}^{m}B_{R\delta_{j}}(\xi_{j})$, for all $R>0$ for both $i=1,2$,
in view of $\psi_{1}-\tilde{c}_{1}=\psi_{2}-\tilde{c}_{2}$.

Setting $\hat{\psi}_{i,j}(y)=\psi_{i}(y_{\xi_{j}}^{-1}(y))$,
$\mathcal{\hat{K}}_{j}(y)=[\mathcal{K}_{1}+\mathcal{K}_{2}](y_{\xi_{j}}^{-1}(y))$
and $\hat{h}_{j}(y)=h(y_{\xi_{j}}^{-1}(y))$ for $y\in B_{2r_{0}}(0)$, we have
that
$e^{\hat{\varphi}_{j}}\Delta\hat{\psi}_{1,j}+\mathcal{\hat{K}}_{j}\hat{\psi}_{1,j}=\hat{h}_{j}+\mathcal{K}_{2}(y_{\xi_{j}}^{-1}(y))[\tilde{c}_{1}-\tilde{c}_{2}]$.
By now it is rather standard to show that the operator
$\hat{L}_{j}=e^{\varphi_{j}}\Delta+\mathcal{\hat{K}}_{j}$ satisfies the
maximum principle in $B_{r}(0)\setminus B_{R\delta_{j}}(0)$ for $R$ large and
$r>0$ small enough, see for example [20]. As a consequence, we get that
$\psi_{1}\to 0$ in $L^{\infty}(S)$. Similarly, we also get that $\psi_{2}\to
0$ in $L^{\infty}(S)$. Since $\psi_{i}=\phi+\tilde{c}_{i}(\phi)$ and
$\tilde{c}_{i}(\phi)\to\tilde{c}_{i,0}=0$ along a sub-sequence,
$\|\psi_{i}\|_{\infty}\to 0$ implies $\phi\to 0$ in $L^{\infty}(S)$, in
contradiction with $\|\phi\|_{\infty}=1$. This completes the proof. ∎

We are now ready for

###### Proof (of Proposition 4.1):.

Since $\|\Delta_{g}PZ_{ij}\|_{*}\leq C$ for all $i=0,1,2$, $j=1,\dots,m$, by
Proposition 7.1 any solution of (4.2) satisfies

$\|\phi\|_{\infty}\leq
C|\log\delta|\bigg{[}\|h\|_{*}+\sum_{i=1}^{2}\Big{(}|c_{0i}|+\sum_{j=1}^{m}|c_{ij}|\Big{)}\bigg{]}.$

To estimate the values of the $c_{ij}$’s, test equation (4.2) against
$PZ_{ij}$, $i=1,2$ and $j=1,\dots,m$:

$\int_{S}\phi
L(PZ_{ij})dv_{g}=\int_{S}hPZ_{ij}dv_{g}+\sum_{k=1}^{2}\Big{[}c_{0k}\sum_{l=0}^{m}\int_{S}\Delta_{g}PZ_{0l}PZ_{ij}dv_{g}+\sum_{l=1}^{m}c_{kl}\int_{S}\Delta_{g}PZ_{kl}PZ_{ij}dv_{g}\Big{]}.$

Since for $j=1,\dots,m$ we have the following estimates in $C(S)$

$PZ_{ij}=\chi_{j}Z_{ij}+O(\delta)\,,\>\>\>i=1,2\,,\qquad
PZ_{0j}=\chi_{j}(Z_{0j}+2)+O(\delta^{2}|\log\delta|),$ (7.1)

it readily follows that $\int_{S}\Delta_{g}PZ_{kl}PZ_{ij}dv_{g}=-{32\pi\over
3}\delta_{ki}\delta_{lj}+O(\delta)$, where the $\delta_{ij}$’s are the
Kronecker’s symbols. By Lemma 3.1, (3.5), (5.6), (5) and (7.1) we have that
for $i=1,2$

$L(PZ_{ij})=\chi_{j}\Delta_{g}Z_{ij}+e^{U_{j}}PZ_{ij}+O\Big{(}\delta^{2}+\delta\sum_{k=1}^{m}e^{U_{k}}\Big{)}=e^{U_{j}}[PZ_{ij}-e^{-\varphi_{j}}\chi_{j}Z_{ij}]+O\Big{(}\delta^{2}+\delta\sum_{k=1}^{m}e^{U_{k}}\Big{)}$

in view of
$\frac{\int_{S}V_{1}e^{W}PZ_{ij}dv_{g}}{\int_{S}V_{1}e^{W}dv_{g}}=O(\delta)$
and $\frac{\int_{S}V_{2}e^{-\tau W}PZ_{ij}dv_{g}}{\int_{S}V_{2}e^{-\tau
W}dv_{g}}=O(\delta)$ for all $j=1,\dots,m$, leading to
$\|L(PZ_{ij})\|_{*}=O(\delta)$. Similarly, we have that

$\displaystyle L(PZ_{1})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{m_{1}}e^{U_{j}}[PZ_{0j}-\chi_{j}e^{-\varphi_{j}}Z_{0j}-2\chi_{j}]+O(\delta^{2})+O\bigg{(}\delta\sum_{k=1}^{m}e^{U_{k}}\bigg{)}$

in view of
$\frac{\int_{S}V_{1}e^{W}PZ_{0j}dv_{g}}{\int_{S}V_{1}e^{W}dv_{g}}=\frac{2}{m_{1}}+O(\delta^{2}|\log\delta|)$
and $\frac{\int_{S}V_{2}e^{-\tau W}PZ_{0j}dv_{g}}{\int_{S}V_{2}e^{-\tau
W}dv_{g}}=O(\delta^{2}|\log\delta|)$ for $j=1,\dots,m_{1}$, leading to
$\|L(PZ_{1})\|_{*}=O(\delta)$. Also, by using a similar argument for
$j=m_{1}+1,\dots,m$, we find that $\|L(PZ_{2})\|_{*}=O(\delta)$. Hence, we get
that

$\begin{split}\sum_{i=1}^{2}\Big{[}|c_{0i}|+\sum_{j=1}^{m}|c_{ij}|\Big{]}&\leq
C^{\prime}\|h\|_{*}+\delta|\log\delta|O\Big{(}\sum_{i=1}^{2}\Big{[}|c_{0i}|+\sum_{j=1}^{m}|c_{ij}|\Big{]}\bigg{)},\end{split}$

yielding to the desired estimates $\|\phi\|_{\infty}=O(|\log\delta|\|h\|_{*})$
and
$\displaystyle\sum_{i=1}^{2}\Big{[}|c_{0i}|+\sum_{j=1}^{m}|c_{ij}|\Big{]}=O(\|h\|_{*})$.
To prove the solvability assertion, problem (4.2) is equivalent to finding
$\phi\in H$ such that

$\begin{split}\int_{S}\langle\nabla\phi,\nabla\psi\rangle_{g}dv_{g}=\int_{S}&\left[{\lambda_{1}V_{1}e^{W}\over\int_{S}V_{1}e^{W}dv_{g}}\left(\phi-{\int_{S}V_{1}e^{W}\phi
dv_{g}\over\int_{S}V_{1}e^{W}dv_{g}}\right)\right.\\\
&\left.\;+\,{\lambda_{2}\tau^{2}V_{2}e^{-\tau W}\over\int_{S}V_{2}e^{-\tau
W}dv_{g}}\left(\phi-{\int_{S}V_{2}e^{-\tau W}\phi
dv_{g}\over\int_{S}V_{2}e^{-\tau W}dv_{g}}\right)-h\right]\psi
dv_{g}\qquad\forall\,\psi\in\mathcal{H},\end{split}$

where $\mathcal{H}=\\{\phi\in
H_{0}^{1}(S)\,:\>\int_{S}\phi\Delta_{g}PZ_{ij}dv_{g}=\int_{S}\phi\Delta_{g}PZ_{i}dv_{g}=0,\,i=1,2,\,j=1,\dots,m\\}$.
With the aid of Riesz representation theorem, the Fredholm’s alternative
guarantees unique solvability for any $h$ provided that the homogeneous
equation has only the trivial solution: for (4.2) with $h=0$, the a-priori
estimate (4.3) gives that $\phi=0$.

So far, we have seen that, if $T(h)$ denotes the unique solution $\phi$ of
(4.2), the operator $T$ is a continuous linear map from $\\{h\in
L^{\infty}(S):\,\int_{S}hdv_{g}=0\\}$, endowed with the $\|\cdot\|_{*}$-norm,
into $\\{\phi\in L^{\infty}(S):\,\int_{S}\phi dv_{g}=0\\}$, endowed with
$\|\cdot\|_{\infty}$-norm. The argument below is heuristic but can be made
completely rigourous. The operator $T$ and the coefficients $c_{0i},\,c_{ij}$
are differentiable w.r.t. $\xi_{l}$, $l=1,\dots,m$, or $\mu_{k}$, $k=1,2$. We
shall argue in the same way to obtain (57) in [23, Appendix A],
differentiating equation (4.2), we formally get that $X=\partial_{\beta}\phi$,
where $\beta=\xi_{l}$ with $l=1,\dots,m$ or $\beta=\mu_{k}$, $k=1,2$,
satisfies
$L(X)=\tilde{h}(\phi)+\sum_{i}d_{0i}\Delta_{g}PZ+\sum_{i,j}d_{ij}\Delta_{g}PZ_{ij}$,
for a suitable choice of $\tilde{h}(\phi)$, $d_{0i}=\partial_{\beta}c_{0i}$,
$d_{ij}=\partial_{\beta}c_{ij}$, and the orthogonality conditions become

$\int_{S}X\Delta_{g}PZ_{ij}dv_{g}=-\int_{S}\phi\partial_{\beta}(\Delta_{g}PZ_{ij})dv_{g}\,,\qquad\int_{S}X\Delta_{g}PZ_{i}dv_{g}=-\int_{S}\phi\partial_{\beta}(\Delta_{g}PZ_{i})dv_{g}.$

Now, finding and estimating suitable coefficients $b_{0i}$, $b_{ij}$ so that
$Y=X+\sum_{k}b_{0k}PZ_{k}+\sum_{k,l}b_{kl}PZ_{kl}$ satisfies the orthogonality
conditions
$\int_{S}Y\Delta_{g}PZ_{i}dv_{g}=\int_{S}Y\Delta_{g}PZ_{ij}dv_{g}=0$, the
function $X$ can be uniquely expressed as
$X=T(f)-\sum_{i}b_{0}PZ_{i}-\sum_{i,j}b_{ij}PZ_{ij}$, where
$f=\tilde{h}(\phi)+\sum_{i}b_{0i}L(PZ_{i})+\sum_{i,j}b_{ij}L(PZ_{ij})$.
Moreover, we find that $\|f\|_{*}\leq C{|\log\delta|\over\delta}\|h\|_{*},$
for $\beta=\xi_{l}$ and $\|f\|_{*}\leq C|\log\delta|\|h\|_{*}$ for
$\beta=\mu_{k}$. By (4.3) we deduce that for any first derivative

$\|\partial_{\xi_{l}}\phi\|_{\infty}\leq
C\Big{[}|\log\delta|\|f\|_{*}+{\|\phi\|_{\infty}\over\delta}\Big{]}\leq
C^{\prime}{|\log\delta|^{2}\over\delta}\|h\|_{*}.$

and $\|\partial_{\mu_{k}}\phi\|_{\infty}\leq C|\log\delta|^{2}\|h\|_{*}$.
Differentiating once more in $\mu_{j}$ the equation satisfied by
$\partial_{\mu_{i}}\phi$ and arguing as above, we finally obtain that
$\|\partial_{\mu_{i}\mu_{j}}\phi\|_{\infty}\leq C|\log\delta|^{3}\|h\|_{*}$,
and the proof is complete. ∎

##  8 Appendix B

We shall argue in the same way to [23, Proposition 4.2], so that by
Proposition 4.1 we now deduce the following.

###### Proof (of Proposition 6.1):.

In terms of the operator $T$, problem (4.5) takes the form
$\mathcal{A}(\phi)=\phi$, where $\mathcal{A}(\phi):=-T(R+N(\phi))$. After [20,
23, 24, 25, 28], a standard fixed point argument can be used to obtain that
$\mathcal{A}$ is a contraction mapping of $\mathcal{F}_{\nu}$ into itself,
where

$\mathcal{F}_{\nu}=\left\\{\phi\in
C(S)\,:\>\|\phi\|_{\infty}\leq\nu\bigg{[}\delta|\log\delta|\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|+\delta^{2-\sigma}|\log\delta|^{2}\bigg{]}\right\\}.$

Therefore has a unique fixed point $\phi\in\mathcal{F}_{\nu}$.

By the Implicit Function Theorem it follows that the map
$(\mu,\xi)\to(\phi(\mu,\xi),c_{0i}(\mu,\xi),c_{ij}(\mu,\xi))$ is (at least)
twice-differentiable in $\mu$ and one differentiable in $\xi$. Differentiating
$\phi=-T(R+N(\phi))$ w.r.t. $\beta=\xi_{l}$, $l=1,\dots,m$, or $\beta=\mu$, we
get that
$\partial_{\beta}\phi=-\partial_{\beta}T(R+N(\phi))-T(\partial_{\beta}R+\partial_{\beta}N(\phi))$.
By Lemma 3.2 and (4.4) we have that

$\|\partial_{\xi_{l}}T(R+N(\phi))\|_{\infty}\leq
C{|\log\delta|^{2}\over\delta}(\|R\|_{*}+\|N(\phi)\|_{*})=O\bigg{(}|\log\delta|^{2}\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|+\delta^{1-\sigma}|\log\delta|^{3}\bigg{)},$

for $l=1,\dots,m$, in view of
$\|\partial_{\xi_{l}}W\|_{\infty}\leq\frac{C}{\delta}$ and

$\|\partial_{\mu}T(R+N(\phi))\|_{\infty}\leq
C|\log\delta|^{2}(\|R\|_{*}+\|N(\phi)\|_{*})=O\bigg{(}\delta|\log\delta|^{2}\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|+\delta^{2-\sigma}|\log\delta|^{3}\bigg{)},$

in view of $\|\partial_{\mu}W\|_{\infty}\leq C$. So, differentiating
$\partial_{\beta}N_{i}(\phi)$ as in [23, Appendix A] with $N_{i}(\phi)$ in
(3.10), we find that

$\|\partial_{\beta}N(\phi)\|_{*}\leq
C\left[\|\partial_{\beta}W\|_{\infty}\|\phi\|_{\infty}^{2}+\|\phi\|_{\infty}\|\partial_{\beta}\phi\|_{\infty}\right]$
(8.1)

and

$\displaystyle\|\partial_{\xi_{l}}N(\phi)\|_{*}$ $\displaystyle=$
$\displaystyle
O\bigg{(}\delta|\log\delta|^{2}\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|^{2}+\delta^{3-2\sigma}|\log\delta|^{4}\bigg{)}+o\left(\frac{\|\partial_{\xi_{l}}\phi\|_{\infty}}{|\log\delta|}\right)$
$\displaystyle\|\partial_{\mu}N(\phi)\|_{*}$ $\displaystyle=$ $\displaystyle
O\bigg{(}\delta^{2}|\log\delta|^{2}\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|^{2}+\delta^{4-2\sigma}|\log\delta|^{4}\bigg{)}+o\left(\frac{\|\partial_{\mu}\phi\|_{\infty}}{|\log\delta|}\right).$

Since
$\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}dv_{g}=\int_{\mathbb{R}^{2}}\chi(|y|)\frac{8\mu_{k}^{2}\delta^{2}\rho_{j}(\xi_{j})}{(\mu_{k}^{2}\delta^{2}\rho_{j}(\xi_{j})+|y|^{2})^{2}}dy$,
either $k=1$ for $j=1,\dots,m_{1}$ or $k=2$ for $j=m_{1}+1,\dots,m$, we have
that

$\partial_{\xi_{l}}\bigg{(}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}dv_{g}\bigg{)}=8\partial_{\xi_{l}}\log\rho_{j}(\xi_{j})\int_{\mathbb{R}^{2}}\frac{1-|y|^{2}}{(1+|y|^{2})^{3}}+O(\delta^{2})=O(\delta^{2})$

and similarly,

$\begin{split}\partial_{\mu_{k}}\bigg{(}\int_{S}\chi_{j}e^{-\varphi_{j}}e^{U_{j}}dv_{g}\bigg{)}&=\int_{\mathbb{R}^{2}}\chi(|y|)\frac{16\mu_{k}\delta^{2}\rho_{j}(\xi_{j})(|y|^{2}-\mu_{k}^{2}\delta^{2}\rho_{j}(\xi_{j}))}{(\mu_{k}^{2}\delta^{2}\rho_{j}(\xi_{j})+|y|^{2})^{3}}dy=O(\delta^{2}).\end{split}$

Since $\varphi_{j}(\xi_{j})=0$ and $\nabla\varphi_{j}(\xi_{j})=0$, we have
that $e^{-\varphi_{j}}=1+O(|y_{\xi_{j}}(x)|^{2})$ and
$\partial_{\beta}(\chi_{j}e^{-\varphi_{j}}(x))=O(|y_{\xi_{j}}(x)|)$, and then
$\displaystyle\Delta_{g}\partial_{\xi_{l}}W=-\sum_{j=1}^{m_{1}}\chi_{j}e^{U_{j}}\partial_{\xi_{l}}U_{j}+{1\over\tau}\sum_{j=m_{1}+1}^{m}\chi_{j}e^{U_{j}}\partial_{\xi_{l}}U_{j}+O(\delta^{1-\sigma})$,
in view of $|\partial_{\xi_{l}}U_{j}|=O(\frac{1}{\delta})$, $l=1,\dots,m$ and
$\displaystyle\Delta_{g}\partial_{\mu}W=-\sum_{j=1}^{m}\chi_{j}e^{U_{j}}\partial_{\mu}U_{j}+{1\over\tau}\sum_{j=m_{1}+1}^{m}\chi_{j}e^{U_{j}}\partial_{\mu}U_{j}+O(\delta^{2-\sigma})$,
in view of $|\partial_{\mu}U_{j}|=O(1)$, where the big $O$ is estimated in
$\|\cdot\|_{*}$-norm. Note that in $B_{r_{0}}(\xi_{j})$

$\partial_{\xi_{l}}W=\begin{cases}\partial_{\xi_{l}}U_{j}+O(\delta^{2}|\log\delta|+|y_{\xi_{j}}(x)|+|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|),&\text{ for }j\in\\{1,\dots,m_{1}\\},\\\\[11.38092pt]
-\dfrac{1}{\tau}\big{[}\partial_{\xi_{l}}U_{j}+O(\delta^{2}|\log\delta|+|y_{\xi_{j}}(x)|+|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|)\big{]},&\text{ for
}j\in\\{m_{1}+1,\dots,m\\},\end{cases}$

and

$\displaystyle\partial_{\mu_{k}}W=\begin{cases}\partial_{\mu_{k}}U_{j}-\dfrac{2}{\mu_{k}}+O(\delta^{2}|\log\delta|),&\text{
for }j\in\\{1,\dots,m_{1}\\}\\\\[11.38092pt]
-\dfrac{1}{\tau}\Big{[}\partial_{\mu_{k}}U_{j}-\dfrac{2}{\mu_{k}}+O(\delta^{2}|\log\delta|)\Big{]},&\text{
for }j\in\\{m_{1}+1,\dots,m\\}\end{cases}.$

Furthermore, $\partial_{\xi_{l}}W=O(1)$ and
$\partial_{\mu_{k}}W=O(\delta^{2}|\log\delta|)$ in
$S\setminus\cup_{j=1}^{m}B_{r_{0}}(\xi_{j})$. From computations in the proof
of Lemma 3.1 we find that

$\displaystyle\dfrac{\lambda_{1}V_{1}e^{W}}{\int_{S}V_{1}e^{W}dv_{g}}=\frac{\lambda_{1}}{8\pi
m_{1}}\sum_{j=1}^{m_{1}}\chi_{j}\bigg{[}1+\Big{\langle}\frac{\nabla(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)}{\rho_{j}(\xi_{j})},y_{\xi_{j}}(x)\Big{\rangle}+O(|y_{\xi_{j}}(x)|^{2}+\delta^{2}|\log\delta|)\bigg{]}e^{U_{j}}$
(8.2) $\displaystyle\hskip 39.83368pt+\
O(\delta^{2})\chi_{S\setminus\cup_{j=1}^{m_{1}}B_{r_{0}}(\xi_{j})},$

and

$\displaystyle\dfrac{\lambda_{2}\tau V_{2}e^{-\tau W}}{\int_{S}V_{2}e^{-\tau
W}dv_{g}}=\frac{\lambda_{2}\tau}{8\pi
m_{2}}\sum_{j=m_{1}+1}^{m}\chi_{j}\bigg{[}1+\Big{\langle}\frac{\nabla(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)}{\rho_{j}(\xi_{j})},y_{\xi_{j}}(x)\Big{\rangle}+O(|y_{\xi_{j}}(x)|^{2}+\delta^{2}|\log\delta|)\bigg{]}e^{U_{j}}$
(8.3) $\displaystyle\hskip 36.98866pt+\
O(\delta^{2})\chi_{S\setminus\cup_{j=m_{1}+1}^{m}B_{r_{0}}(\xi_{j})}.$

By (5.15), (5), (5.23), (5.25), (5), (8.2) and (8.3) we deduce for
$\partial_{\beta}R$ the estimate

$\|\partial_{\xi_{l}}R\|_{*}+{1\over\delta}\|\partial_{\mu_{k}}R\|_{*}=O\bigg{(}\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|+\delta^{1-\sigma}|\log\delta|\bigg{)},\quad
l=1,\dots,m,\ k=1,2$

Combining all the estimates, we then get that

$\|\partial_{\xi_{l}}\phi\|_{\infty}=O\bigg{(}|\log\delta|^{2}\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|+\delta^{1-\sigma}|\log\delta|^{3}\bigg{)}+o\big{(}\|\partial_{\xi_{l}}\phi\|_{\infty}\big{)}$

and

$\|\partial_{\mu_{k}}\phi\|_{\infty}=O\bigg{(}\delta|\log\delta|^{2}\sum_{j=1}^{m}|\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)|+\delta^{2-\sigma}|\log\delta|^{3}\bigg{)}+o\big{(}\|\partial_{\mu_{k}}\phi\|_{\infty}\big{)},$

which in turn provides the validity of (4.7). We proceed in the same way to
obtain the estimate (4.7) on $\partial_{\mu_{i}\mu_{j}}\phi$, and the proof is
complete. ∎

Lemma 4.1 is rather standard and we will omit its proof. Since the problem has
been reduced to find c.p.’s of the reduced energy
$E_{\lambda_{1},\lambda_{2}}(\mu,\xi)=J_{\lambda_{1},\lambda_{2}}(W+\phi(\mu,\xi))$,
where $J_{\lambda_{1},\lambda_{2}}$ is given by (1.4), the last key step is
show that the main asymptotic term of $E_{\lambda_{1},\lambda_{2}}$ is given
by $J_{\lambda_{1},\lambda_{2}}(W)$.

###### Proof (of Theorem 4.1):.

We argue in the same way as in the proof of [23, Theorem 4.4]. For simplicity
we write $J$ instead of $J_{\lambda_{1},\lambda_{2}}$. Thus, we get that

$\displaystyle J(W+\phi)-J(W)=-{1\over
2}\int_{S}\left[R\phi-N(\phi)\phi\right]dv_{g}+\int_{0}^{1}\\!\\!\\!\int_{0}^{1}[D^{2}J(W+ts\phi)-D^{2}J(W)][\phi,\phi]\,t\,dsdt,$

so that, it is straighforward to deduce that

$|J(W+\phi)-J(W)|=O(\|R\|_{*}\|\phi\|_{\infty}+\|\phi\|_{\infty}^{3})=O\left(\delta^{2}|\log\delta|\,|\nabla\varphi_{m}^{*}(\xi)|^{2}+\delta^{3-\sigma}|\log\delta|^{2}\right)$

in view of (4.6),
$4\pi\nabla_{\xi_{j}}\varphi_{m}^{*}(\xi)=\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)$, for $j=1,\dots,m_{1}$ and
$4\pi\tau^{2}\nabla_{\xi_{j}}\varphi_{m}^{*}(\xi)=\nabla\log(\rho_{j}\circ
y_{\xi_{j}}^{-1})(0)$, for $j=m_{1}+1,\dots,m$. Now, differentiating w.r.t.
$\beta=\xi_{l}$, $l=1,\dots,m$, or $\beta=\mu_{k}$, $k=1,2$ we get that

$\displaystyle|\partial_{\beta}[J(W+\phi)-J(W)]|=O(\|\partial_{\beta}R\|_{*}\|\phi\|_{\infty}+\|R\|_{*}\|\partial_{\beta}\phi\|_{\infty}+\|\phi\|_{\infty}^{2}\|\partial_{\beta}\phi\|_{\infty}+\|\phi\|_{\infty}^{3}\|\partial_{\beta}W\|_{\infty})$

by using (8.1), so that,

$\displaystyle|\partial_{\xi_{l}}[J(W+\phi)-J(W)]|=O\left(\big{[}\delta^{2}|\log\delta|\,|\nabla\varphi_{m}^{*}(\xi)|^{2}+\delta^{3-\sigma}|\log\delta|^{2}\big{]}{|\log\delta|\over\delta}\right)$

and
$|\partial_{\mu_{k}}[J(W+\phi)-J(W)]|=O\left(\big{[}\delta^{2}|\log\delta|\,|\nabla\varphi_{m}^{*}(\xi)|^{2}+\delta^{3-\sigma}|\log\delta|^{2}\big{]}|\log\delta|\right)$
in view of (4.6)-(4.7), $\|\partial_{\xi_{l}}W\|_{\infty}=O(\frac{1}{\delta})$
and $\|\partial_{\mu_{k}}W\|_{\infty}=O(1)$. Arguing similarly for the second
derivative in $\mu$, we get that
$\left|\partial_{\mu_{i}\mu_{k}}[J(W+\phi)-J(W)]\right|=O\left(\big{[}\delta^{2}|\log\delta|\,|\nabla\varphi_{m}^{*}(\xi)|^{2}+\delta^{3-\sigma}|\log\delta|^{2}\big{]}|\log\delta|^{2}\right)$.
Combining the previous estimates on the difference $J(W+\phi)-J(W)$ with the
expansion of $J(W)=J_{\lambda_{1},\lambda_{2}}(W)$ contained in Theorem 5.1,
we deduce the validity of the expansion (4.1) with an error term which can be
estimated (in $C^{2}(\mathbb{R}^{2})$ and $C^{1}(\Xi)$) like
$o(\delta^{2})+r_{\lambda_{1},\lambda_{2}}(\mu,\xi)$ as $\delta\to 0$, where
$r_{\lambda_{1},\lambda_{2}}(\mu,\xi)$ does satisfy (4.9). ∎

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