I don't know about you, but I like my show ip cef and show adjacency detail commands. They give you a bit more information and are rather useful when show ip route and show ip arp are not really enough. But what happens when you jump on a device that doesn't have CEF?
It just happens that I work quite a bit with Nexus 7000s, so I decided I wanted to know how to get the same information from NX-OS (which might have a very similar command syntax, but is a different beast inside).
The easy answer is that they show you the contents of the FIB. The slightly awkward answer is that they show you what the router thinks went into the FIB. Most of the time (99.9%, a number which I totally haven't invented on the spot) it knows what's in there, but very nasty bugs can make it so that the RP thinks it programmed one thing and the (distributed) hardware has another thing.
To give you a snapshot of what the high-level interactions between the RIB, FIB and protocol data is, I took a diagram from the article linked below and decided to include it as a piece of ASCII art. Why, you might ask? Because I keep my notes (mostly) in text format and in this case there's no loss of information.
Here you go. I'll start with the equivalent of show ip cef - the output is quite similar.
I'll throw in two more commands, because I'm feeling generous. show routing memory gives you a boatload of statistics and the one below shows you information for load-balanced prefixes - to which next-hop will a packet with source 1.1.1.1 and destination 2.2.2.2 be sent.
Of course, your best friend is the question mark. Explore show routing and show forwarding and you'll get more than you probably need for usual troubleshooting purposes. But then again, if you need to look at the FIB you don't have an easy one, do you? Good luck.


text:
from: 'Hadi Partovi (Code.org) <hadi_partovi@code.org>'
subject: "Your Amazon.com gift"
---
Thank you for participating in the Hour of Code. Here is your $10.00 Amazon.com gift code.

`<%= prize_code_s %>`

**To redeem:**

1.  Log into your [Amazon account](http://amazon.com).
2.  Click 'Apply a Gift Card to Your Account'.
3.  Enter your claim code and click 'Apply to Your Account'.
3.  Start shopping at Amazon.com

**Note:** You can also enter your claim code during checkout. You won't be able to redeem your gift card using the Amazon.com 1-Click service unless you redeem the gift card funds to Your Account first.

**For help** [go here](https://www.amazon.com/gp/help/customer/display.html).

**Note:** This is a promotional gift card. [See card terms and agreement](https://www.amazon.com/gp/help/customer/display.html?nodeId=3122091&app-nav-type=modal/ref=gcui_d_e_tc_o_d_b_x).

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redpajama_set_name: RedPajamaGithub


BORIYA BASTA is not all that shines and blings. Destination for bohemian and ethno fashion and accessories. Offering a unique mix of tradition, craftsmanship, design and the witty concepts that we see around us in our everyday lives.
© 2019 Boriya Basta. All Rights Reserved.


When and if your child does come clean, make sure you place the emphasis for the deed on the honesty, not the behavior. Of course it makes you mad when your child doesn't empty the dishwasher, but encouraging honesty means taking one for the team and focusing on the positive instead.... There can be nothing as bad as a child lying to a parent. For sure, as parents, we will feel betrayed and hurt by this. We will also feel mad and enraged, as the thought of our child lying to us is just hurtful.
Now, this doesn't mean you should let your children break the rules willy-nilly, says study author Victoria Talwar, Ph.D. (Make sure you're not falling for one of these 6 Easy Ways to Screw Up... There can be nothing as bad as a child lying to a parent. For sure, as parents, we will feel betrayed and hurt by this. We will also feel mad and enraged, as the thought of our child lying to us is just hurtful.
Spread the loveAs a parent, we literally bringing another living, breathing, human being into the world- for the next 18 years or so. it is our responsibility to teach our child to navigate the ups and downs of life, and the society he or she is being brought into by having a strong foundation.


It's pretty cold here in London and everyone's health is at risk, it seems like everyone in the city is ill. It's nothing serious of course, just a cold or the flu but it's everywhere. Just this morning on the tube somebody sneezed on the back of my neck…this was not pleasant. But it got me thinking about the vocabulary a student might need to use in order to describe their cold or flu. Check out the words, idioms and phrases below for help describing your ailment. Pay careful attention to how they are used and if there's anything you don't understand, just click on it for a definition.
"I'm a bit under the weather." = I'm a little bit ill.
"I'm on the mend." = I'm getting better.
So if you're feeling a little under the weather, get out there and use these handy words / phrases to let people know. But please try not to sneeze on anybody's neck on the tube…that's just disgusting.
If you'd like to learn more vocabulary, check out the quick lessons on our website.
If you're thinking about booking some English lessons for adults in the UK, check out our website for more details.


text:
Home LA Showbiz Have Brad Pitt and Jennifer Aniston rekindled their romance following Golden Globes...
Have Brad Pitt and Jennifer Aniston rekindled their romance following Golden Globes reunion?
The pair first divorced in 2005
Clodagh Meaney
Brad Pitt and Jennifer Aniston both attended the Golden Globes in L.A this weekend.
The pair, who divorced in 2005 after five years of marriage were on track for an emotional reunion after Jennifer was spotted gazing adorably at Brad as he gave an acceptance speech at the award ceremony.
It was also reported that pair had attended the same after party, leaving fans hopeful they could rekindle their romance.
Find someone who looks at you the way Jennifer still looks at Brad #GoldenGlobes #BradPitt #JenniferAniston pic.twitter.com/WgNQgBTbQn
— StephaniePC (@mrspanetta) January 6, 2020
However, a source close to the pair has said that the ex-couple are "just friends" following their brief reunion.
The source told TMZ that although they did say hi to each other, there was nothing further to it.
"They did exchange greetings and pleasantries at the after-party they both attended, but that's about it."
The publication was told the couple do not hang out outside of these "rare settings" and there's "simply nothing romantic going on between them."
Jennifer looked on adoringly as Brad accepted a Golden Globe for Best Actor.
During his speech he joked about not bringing a date, wanting to bring his mother along but joked that he feared the press would label them as "dating."
Brad married Angelina Jolie in 2014 after almost 10 years of dating before divorcing in 2016.
Jennifer also moved on and married Justin Theroux in 2015 before splitting two-and-a-half years later in 2017.
Previous articleMarian Finucane's husband John Clarke gives emotional speech at her funeral
Next articleRosie Connolly reveals she suffered from post-natal depression after having daughter Remi
Kim Kardashian calls Kylie Jenner's makeup 'cheap sh*t' in new video
Taylor Swift reveals mother Andrea has been diagnosed with a brain tumour
Ashley Graham announces the arrival of her first child

redpajama_set_name: RedPajamaCommonCrawl


Page 98 - Fayeflam | Wonderful Home Design Ideas Crawl Space Access Door. Reclaimed Wood Wall Decor. Tile For Shower Floor.
best home design ideas, like Crawl Space Access Door. Reclaimed Wood Wall Decor. Tile For Shower Floor. Long Media Console. 42 White Bathroom Vanity. Front Door Awning Ideas.


text:
Home page › The winner of the Grand prize of the deposit campaign is happy with the unexpected holiday present
The winner of the Grand prize of the deposit campaign is happy with the unexpected holiday present
For 50 years residing in Seredžius village in Jurbarkas region, which is well-known not only for its natural beauty but also for its rich historical legend-braided past, Vytautas Pečiukėnas has not even suspected that this Christmas will be so generous to him - the senior has won the Grand prize of the time deposit campaign launched by Šiaulių bankas.
"This is such a big surprise to me as I have never thought of winning anything. Neither I believe in lotteries nor have I won any prizes. Though, long ago, in soviet times I had to acquire some government bonds and, thus, participate in the lottery, so once I was lucky to win 500 roubles - V. Pečiukėnas, 83, the winner of the campaign shared his memories. - News about the prize is a marvellous Christmas present to my children and grandchildren who, as a tradition, visit me on Christmas Eve and at Christmas".
While meeting the representative of Šiaulių bankas during the prize granting ceremony V. Pečiukėnas did not hide his joy for the received prize and, though, he stated that he had no idea how he was going to use this gaining yet, but he was sure that he would not let it go to waste.
The winner was sincerely sharing the story of his life. Born and raised in Panevėžys region he graduated from Panevėžys Boy Gymnasium and was intending to study in Kaunas, however, the political situation directed otherwise at that time. He was not able to seek a high education as he parents were farmers and his father was taking a position of the head of the district for some time while the father's brother and his wife were exiled to Yakutia.
According to V.Pečiukėnas then he started his working career. In 1950 he began working in various reclamation companies in Kaunas, Jurbarkas, and Seredžius where he had worked until the very retirement. In his spare time the man loves working his small kitchen-garden, where he carefully cultivates vegetables and fruits.
Vytautas's family raised a son and two daughters and today he enjoys attention of his five grandchildren. The client of Šiaulių bankas is especially pleased that his grandchildren are seeking a high education and doing well. "Everyone is learning - the oldest granddaughter has just graduated, defended her thesis, while the youngest one is the first-year student at the University of Vilnius." The winner of the Grand prize of the deposit campaign is also happy about the fact that his family has not been touched by emigration - neither his children nor grandchildren have scattered over the wild world, they live, work and study here, in Lithuania.
Šiaulių bankas sincerely congratulates Vytautas Pečiukėnas on winning the Grand prize!
Congratulation to 200 winners of the additional prizes of the deposit campaign "Win the Chest of Golden Coins" as well.
V. Pečiukėnas, the winner of the Grand prize of the deposit campaign "Win the Chest of Golden Coins" congratulated by Giedrė Stulginskienė, Director of Tauragė CSC and Dainius Rekevičius, Project Manager of Jurbarkas CSB.


Welcome to the Psychic Eye's Astrology Services Center.
Click on the price of the desired chart or report for details, samples & purchase information.
All of the astrology charts & reports we offer may be printed and bound and can be picked up at your nearest (or preferred) Psychic Eye location, shipped to you via UPS, or e-mailed to you in PDF format. For one-sheet charts, we offer fax service.
The Psychic Eye also features professional astrologers on-site at each of our store locations who are available for chart interpretation, astrological readings or one-on-one instruction. These sessions can be booked the same way you'd schedule a regular reading. Visit the Psychic Readings section of this site for more information.
Charts and reports purchased from the Psychic Eye are computed using the Tropical Zodiac and the Koch House System.
If you have a preference for a different Zodiac (Heliocentric or Sidereal) or House System (Placidus, Equal, Campanus, Meridian, Porphyry, etc., etc.) for your charts and reports, please let us know in the Special Instructions section of our secure shopping cart, and we will be happy to oblige.


Can I request my textbooks through ILL?
How will I know it is in? Where do I pick it up?
Can my research assistant check out ILL materials on my behalf?
Can you mail books to my home address?
How long it is the borrowing period? Can I renew an ILL book?
Can I request the same book more than once?
How do I clone a previous request?
I am studying at the Rome Center. May I use ILL?
Interlibrary Loan (ILL) obtains research materials such as books, articles, and microfilm that are not available from the Loyola libraries.
Document Delivery is a service that allows you to request scanned copies of articles and book chapters from Loyola's print collections. If the material is not otherwise available online, the library staff will send you pdfs of articles and book chapters within the limitations of copyright law.
Current Loyola undergraduate and graduate students, faculty, and staff may use this service. Patrons who are affiliated with the Law School or the Medical School should direct their requests to the Law Library or Health Sciences Library.
Alumni are not eligible for interlibrary loan services but may request photocopies or delivery of items from Loyola's own collections to the Loyola library of their choice.
Books, theses, periodical articles and some materials reproduced in microform, such as newspapers, may be borrowed in accordance with the US copyright law. Books owned by Loyola that are currently checked out or are designated missing or lost in the library catalog may also be requested. If you are unable to find an item that our catalog indicates is available, please request the item in library catalog. We will conduct a search and notify you of the results. If the item is confirmed missing, you may request a copy through ILL.
All media items (CDs, DVDs, and VHS) requested through interlibrary loan must be for academic or job-related purposes. Undergraduate students may be asked to provide proof of the academic nature of their request via a syllabus, course assignment, or note from their professor.
The following types of materials may not be borrowed through interlibrary loan: books available at a Loyola library including non-circulating materials in the reference collection and on reserve; entire issues or volumes of periodicals; materials for reserve, group, or extended use; and archival materials.
While the ILL department will attempt to fill your request, we recommend against borrowing textbooks through interlibrary loan. Why?
Textbooks ordered through interlibrary loan must be used in the library. You will not be able to check them out for use at home.
There is a good chance that we will not be able to borrow the book you need. Few libraries purchase textbooks and even fewer have the most recent editions. When libraries do purchase textbooks, they usually reserve them for campus-use only and do not lend them to other libraries. Moreover, the high demand for textbooks means that the few library copies that can be borrowed are likely to be checked out.
If we are able to borrow a copy, it can take several weeks to arrive. By then you will already be behind in your reading.
You will not be able to keep the book for the whole semester. Due dates are set by the lending library, not by Loyola. Loan periods are usually between three and six weeks.
ILL books are subject to immediate recall, so you could have to give the book back at any time.
We recommend buying or renting your textbooks when possible. You can also ask your instructor to place a copy in course reserves.
All interlibrary loan and document delivery requests are managed through ILLiad.The first time you log in to ILLiad, you will need to provide your contact information. To make your request, select the appropriate item type from the Request Forms menu. You can also place requests from WorldCat Local and many of our databases by clicking the Find It @ LOYOLA button and then the "Request through Interlibrary Loan" link.
The Loyola University Libraries pay all costs associated with interlibrary loan and document delivery. There is no charge to you.
Factors that affect delivery time include the number of libraries that own the item, the demand for the material, and the distance of the supplying library from Loyola. Articles arrive in an average of 3 calendar days and books arrive in an average of 8 calendar days, but your request may take more or less time. We cannot guarantee the arrival date of any item and suggest that you place your requests well ahead of any deadlines.
Log into ILLiad and look under Pending Requests. If the status is Request Sent, we have not yet received the item. It may be on its way, or we may still be searching for a lender. You are welcome to contact the interlibrary loan office for more information.
See our definitions for more information about ILLiad request statuses.
We will send you an email when your item arrives. Articles and book chapters are usually delivered as pdfs. Books should be picked up promptly at the Cudahy or Lewis circulation desk in order to make use of the entire loan period. Nursing students and faculty who prefer to collect materials at the Maywood campus should place requests through the Health Sciences Library.
In the Online Documents section of your ILLiad account click on the Transaction Number, then on View PDF.
Faculty may request an ILLiad proxy account so that research assistants can request and check out interlibrary loan materials on their behalf. Faculty maintain responsibility for these materials.
Interlibrary loan materials from other institutions must be picked up at Cudahy or Lewis Library. Qualifying Distance Education students may request delivery of books that are available within the Loyola University Libraries using ILLiad.
The loan period is determined by the lending library and is usually between three and six weeks. One renewal for two additional weeks may be requested before the due date subject to the lending library's approval. To request a renewal, log in to your ILLiad account and click on the transaction number of the item you wish to renew. At the top of the page, click "Renew Request." You will be notified and asked to return the item if the lender denies your request.
You may request a single title through ILL twice per semester.
To clone a request, click on the request you would like to clone. Then click "clone request" located at the top left hand corner. Enter or edit any information on the request form. Click "submit request."
Late fines for interlibrary loan are $1.00 a day per item.
Electronic document delivery services are available within copyright limits. Select "Cudahy (LSC)" as your pickup location, even though you will not physically be receiving items.
Your request has copyright restrictions and is waiting for approval.
Your request has been received by the ILL staff.
Your request requires special attention. We are still trying to locate an available copy.
We have sent your request to libraries that own the item. They are either checking to see if they can lend the item to you or they have already shipped the item to Loyola. You are welcome to contact the ILL office for more information.
We have the book chapter or article that you need here at Loyola and will send our staff to retrieve it.
We are in the process of retrieving the book chapter or article you need from one of Loyola's libraries.
Your loan has arrived and is available at the circulation desk.
The item is in your possession.
A pdf of the item you requested is available in your ILLiad account.
None of the lenders that we chose for your request is able to supply the item. We are still looking for other potential lenders.
We were unable to fill your request. You should have received an email with the reason why. A copy of this email is available in the Notifications section of your ILLiad account.


This elegant sculpted horse head resin award is painted in a bright Silver finish and It comes on a gloss black base with plenty of room for personalization. This award is a great way to recognize any champion.
This elegant sculpted horse head resin award is painted in an antique gold finish and It comes on a gloss black base with plenty of room for personalization. This award is a great way to recognize any champion.


Huawei is currently banned from rolling out 5G in some markets which include the United States, Australia and New Zealand due to alleged ties with the Chinese government. The Chinese company has repeatedly denied such claims and they had challenged those that made the allegation to show proof of security threats.
When Malaysian Prime Minister, Tun Dr. Mahathir was asked about the concerns, he mentioned that Huawei has not been found to have done any wrongdoing in the country. Malaysia will not be following other countries without any basis.
During a dialogue session at Oxford Union in the UK, Tun Mahathir was reported saying "We have not caught them doing anything wrong in Malaysia. What Germany wants to do is for Germany to decide, what Canada wants to do also is for them to decide." He also added that Malaysia has to study and find out whether there is a basis for the action they took against this company.
The US government had concerns that Huawei equipment can be utilised as a backdoor for the Chinese government, however, such allegations were not proven. Huawei's rotating Chairman had emphasised that in past 30 years Huawei has never faced a serious cyber security issue and there's no evidence proving that their equipment was a threat.
The Chinese telecommunications giant had called for open lines of communications with these governments in order to take action on any security concerns. In the UK, Huawei has agreed to security demands to address risks found in its equipment and software. Huawei currently has secured more than 25 commercial contracts and they have shipped more than 10,000 5G base stations to date.


text:
Is Ontario becoming the new Quebec?
Konrad Yakabuski
Published June 9, 2014 Updated May 12, 2018
It was le monde à l'envers – the world upside down – last week as Quebec's Finance Minister broke with a long tradition of tabling a budget and immediately accusing Ottawa of shortchanging the province. This stunning about-face came during an Ontario election campaign in which the Liberal Premier has been running almost as much against Ottawa's equalization cuts as against her Tory rival.
As a weaker Ontario sheds its nation-builder role for have-not status, it risks falling into the trap that Quebec fell into five decades ago. It's developing the knee-jerk reaction of looking to Ottawa to mitigate its fiscal woes, fostering a culture of dependence that will be difficult to break.
The new Quebec government of Liberal Premier Philippe Couillard has signalled a desire to try anyway.
"The federal government ultimately decides how it transfers its revenue to the provinces," Finance Minister Carlos Leitao, a former bank economist, said after tabling his budget. "How the federal government will choose to use its surpluses is its choice."
Perhaps he recognizes that the province's fiscal persecution complex is a form of denial that distracts it from tackling real economic challenges and undermines the collective self-confidence that will be critical to its success.
Indeed, we may be witnessing a fiscal spring – a printemps fiscal – in Quebec. Parties of the centre-right captured almost two-thirds of the popular vote in April's provincial election. Mr. Leitao's tough budget slashed business subsidies and imposed a hiring freeze in the public sector.
It's now even possible to question the "Quebec model" of development without being laughed out of the province. Federal Small Business Minister Maxime Bernier, whose political brand is slowly being rehabilitated, received a virtual hero's welcome in Quebec for a gutsy speech he delivered last month rejecting the province's old "never enough" attitude toward Ottawa.
"It's not the rest of Canada's fault that we are a poor province," he said. "It's because of bad economic policies that make Quebec's economy less productive."
Contrast that with the crusade Ontario Premier Kathleen Wynne has been on. She accuses Prime Minister Stephen Harper's government of "willful indifference" toward Ontario as it reels from the erosion of its manufacturing base, a decline she compares (wait for it) to the early 1990s collapse of the Newfoundland cod fishery.
It's a sad day when the Premier of the country's once unstoppable economic engine puts her economically diverse province of 13.5 million in the same boat as a rural outpost of 500,000. Ontario does not have oil, but it has non-natural resources that even newly-rich Newfoundland can only dream of.
Yet Ms. Wynne government's pre-electoral everything-but-kitchen-sink budget accused the Harper government of taking "more than 110 unilateral actions that have hurt" Ontario. It singled out the elimination of a program guaranteeing that no province would see its overall federal cash transfers fall from one year to the next, just at the moment when Ontario qualified for extra cash.
"On transfers, I will stand up for Ontario," Ms. Wynne repeats in one of her campaign stump speeches. "I will stand up to Stephen Harper for the good of our province."
Mr. Harper deserves some of the blame for this turn of events. He first adopted an equalization formula that favoured Quebec in the (unrealized) hope of winning votes there. He then capped the rate of growth in the equalization kitty, creating a zero-sum game that means Quebec's gains become Ontario's losses.
Still, Ontario's new obsession with maximizing federal transfers is the surest sign that a have-not mentality is taking hold.
A low-dollar policy in the 1990s, aimed at offsetting federal budget cuts, made Ontario's manufacturers complacent. The province's share of capital investment, which long matched or exceeded its share of the national economy, began falling. An undervalued dollar boosted exports but made imported machinery and computers costlier. This failure to modernize and transition toward higher value-added production left carnage in its wake. Ontario now accounts for just 31 per cent of overall national capital spending – six points below its share of GDP.
As the Fraser Institute recently noted: "This decline in productive investment suggests that in the future there will be growing economic productivity problems."
It's no coincidence that the least productive provinces depend more on federal transfers. Will that make Ontario the new Quebec?
Hillary writes the book on self-promotion
The self-performed lobotomy of the left
Can Mulroney save Quebecor from Péladeau's heavy fist?
Follow Konrad Yakabuski on Twitter @konradyakabuski


The annual Stratford trip allows students to view various productions at the Stratford Festival and explore the city's rich culture. This year's lineup included the plays An Ideal Husband by Oscar Wilde and Shakespeare's Julius Caesar. Grades 9 and 10 enjoyed a play adaptation of To Kill a Mockingbird, while senior grades were immersed in The Rocky Horror Show.
The AP Physics students worked on their first experiment of the year: an investigation of the behaviours of cans of soup with various consistencies, when rolled down ramps of varying inclinations. Students explored lab equipment such as sonic rangers to optimize the collection of data, which they will analyze to study the cans' energy and fluid dynamics.
This week, the grade 10 math class used TI-Nspire calculators to solve linear systems by graphing. They completed a booklet regarding information on the various uses and shortcuts for the calculator. Finally, students compared features of the TI-Nspire to those of a computer, finding similarities such a cursor, a trackpad, and even a physical alphabetical keyboard.
The Reckoner, the school newspaper of Marc Garneau C.I., is an outlet for student expression which showcases students' skills in writing, illustration, photography, graphic design and coding. This year, The Reckoner has received a total of thirteen awards at the Toronto Star's annual High School Journalism Awards. Winners attended an awards ceremony and luncheon at the Toronto Reference Library.


text:
\section{Introduction}
\setcounter{equation}{0}
The understanding of some large energy, negative specific heat states in the Onsager description of 2D turbulence, seems to require
the analysis of a subtle open problem about bubbling solutions of the mean field equation.
Motivated by this application we prove that, under suitable non degeneracy assumptions on the associated $m$-vortex Hamiltonian,
the $m$-point bubbling solutions of the mean field equation are non degenerate as well. Then we deduce that the Onsager mean field
equilibrium entropy is smooth and strictly convex in the high energy regime on domains of second kind.

\subsection{Motivation.}
In a celebrated paper \cite{On} L. Onsager derived a statistical mechanics description of large vortex structures observed in two
dimensional turbulence. He analysed the microcanonical ensemble relative to the Hamiltonian $N-$vortex model and
concluded (among other things) that the maximum entropy (thermodynamic equilibrium) states could be realized by
highly concentrated configurations,
{where} like vortices attract each other. These highly concentrated states are observed in a negative temperature regime,
meaning that, above a certain energy $E_0\in (0,+\ii)$ (see \rife{e0} below), the equilibrium entropy $S(E)$ is decreasing as a function of the energy $E$.
Those physical arguments has been later turned into rigorous proofs \cite{clmp1}, \cite{clmp2}, \cite{K}, \cite{KL}. We refer to \cite{ESp}, \cite{ESr}
and references therein for a complete discussion about the Onsager theory and a more detailed account of the impact of those ideas. In particular,
the rigorous analysis of the mean field canonical and microcanonical models was worked out in \cite{clmp2}, where it was shown that the mean field
thermodynamics in the energy range $E< E_{8\pi}$ is well described in terms of the unique \cite{BLin3},\cite{CCL},\cite{suz}, solutions with $\lm<8\pi$
of the mean field equation,
\begin{equation*}
\begin{cases}
-\Delta\psi=\dfrac{\dsp e^{\lambda\psi}}{\dsp \int_{\Omega}e^{\lambda\psi}}~&\mathrm{in}~\Omega,\\
\psi=0~&\mathrm{on}~\partial\Omega,
\end{cases}
\eqno \prl
\end{equation*}
where $\om$ is any smooth, bounded and simply connected domain and $E_{8\pi}\in (E_0,+\ii]$ is a critical energy whose value depends in a crucial way by $\om$, see Definition \ref{def1.3} below.  Here $\psi$ is the stream function of the flow,
$$\rl(\psi):=\dfrac{\dsp \e{\lm \psi}}{\dsp \ino \e{\lm \psi}},$$
is the vorticity density and $-\lm=\beta=\frac{1}{\kappa T_{\rm stat}}$ is minus the inverse statistical temperature, $\kappa$ being the Boltzmann constant. This result has been more recently generalized to cover the case of any smooth, bounded and connected domain in \cite{BLin3}.
\bdf
\label{def1}
{\it Let $\om\subset\R^2$ be a smooth and bounded domain. We say that $\om$ is of {\bf second kind} if {\rm $\prl$} admits a solution for $\lm=8\pi$. Otherwise $\om$ is said to be of {\bf first kind}.}
\edf

It follows from the results in \cite{CCL} and  \cite{BLin3} that $\om$ is of second kind if and only if $E_{8\pi}<+\ii$, see {Definition \ref{def1.3}} for the definition of $E_{8\pi}.$  As first proved in \cite{clmp2}, for $E\geq E_{8\pi}$ the Onsager intuitions lead to another amazing result, which is the non concavity of the equilibrium entropy. Although the statistical temperature in this model has nothing to do with the physical temperature, this is still a very interesting phenomenon since, if the entropy $S(E)$ is convex in a certain interval, then the system surprisingly "cools down" when the energy increases in that range. In other words the (statistical) specific heat is negative. With the exception of the results in \cite{clmp2} and more recently in \cite{B2}, we do not know of any progress in the rigorous analysis of this problem. Actually, it has been shown in \cite{clmp2} that $S(E)$ is not concave in $(E_{8\pi},+\ii)$ while, under a suitable set of assumptions, it has been shown in \cite{B2} that it is {strictly convex} in $(E_{*},\ii)$ for some $E_*>E_{8\pi}$ on any strictly star-shaped domain of second kind. One of our aims is to remove the assumptions in \cite{B2} and prove that in fact $S(E)$ is \un{convex} on any convex domain of second kind for any $E>E_{8\pi}$ large enough.

\medskip

The definition of domains of first/second kind was first introduced in \cite{clmp1} with an equivalent but different formulation.
We refer to \cite{CCL} and \cite{BLin3} for a complete discussion about the characterization of domains of first/second kind and related examples.

\brm {\it It is well known that any disk,  say $B_R=B_R(0)$, is of first kind and that in this case {\mbox{\rm $\prl$}} admits a solution if and only if $\lm<8\pi$.
Any regular polygon is of first kind \cite{CCL}. However, there exist domains of first kind with non trivial topology where {\mbox{\rm $\prl$}}
admits solutions also for $\lm>8\pi$. For example $\om=B_R\setminus B_r(x_0)$, with $x_0\neq 0$ is of first kind if $r$ is small enough \cite{BLin3}.
In this case it is well known that for any $N\geq 2$ there are solutions concentrating {\rm (}see \eqref{conc}{\rm )} at $N$ distinct points as $\lm\to 8\pi N$ \cite{EGP, KMdP}, as well as other solutions for any $\lm\notin 8\pi\N$ \cite{BdM2}, \cite{CLin2}.
\medskip

\indent It has been proved in \cite{BdM2} that there exists a universal constant $I_c>4\pi$ such that any convex domain whose isoperimetric ratio $I(\om)$ satisfies $I(\om)>I_c$ is of second kind. Also, if $Q_{a,b}$ is a rectangle whose side are $1\leq a\leq b<+\ii$ then there {exists $\eta_c\in(0,1)$} such that $Q_{a,b}$ is of second kind if and only if $\frac{a}{b}< \eta_c$, see \cite{CCL}.}
\erm

\bigskip

Let us define,
$$\mathcal{P}_{\sscp \om}=\left\{\rh\in L^{1}(\om)\,|\,\rh\geq 0\;\mbox{a.e. in}\;\om \right\},$$
and the entropy, energy and total vorticity of the density $\rh\in \mathcal{P}_{\sscp \om}$ as,
$$\mathfrak{S}(\rh)=-\ino \rh\log(\rh),\quad\mathcal{E}(\rh)=\frac12 \ino \rh G[\rh],\quad \mathfrak{M}(\rh)=\ino\rh,$$
respectively, where,
$$G[\rh](x)=\ino G(x,y)\rh(y)\,dy,\;x\in\om,$$
and $G=G_\om$ is the Green's function of $-\Delta$ with Dirichlet boundary conditions.

\bigskip

It has been shown in \cite{CCL} and  \cite{BLin3} that, if for $\lm=8\pi$ a solution of $\prl$ exists, say $\psi_{\sscp 8\pi}$, then it is unique. In particular it has been shown there that $\psi_{\sscp 8\pi}$ exists if and only if it is the uniform limit of the unique solutions of $\prl$ with $\lm<8\pi$. In particular one can see that $\mathcal{E}(\rl(\pl))\to +\ii$ as $\lm\to 8\pi^-$ if and only if $\om$ is of first kind. Consequently the following definition is well posed,

\bdf
\label{def1.3}{\it If $\om$ is of {second kind}, then we define $E_{8\pi}=\mathcal{E}(\mbox{\rm $\rh$}_{8\pi}(\psi_{\sscp 8\pi}))$, while if $\om$ is of {first kind} we set $E_{8\pi}=+\ii$.}
\edf

For any  $E\in(0,+\infty)$, let us consider the Microcanonical Variational Principle,
$$S(E)=\sup \left\{ \mathfrak{S}(\rh),\; \rh \in \mathcal{M}_E \right\},\quad
\mathcal{M}_E=\left\{\rh\in\mathcal{P}_{\sscp \om}\,|\,\mathcal{E}(\rh)=E,\;\;\mathfrak{M}(\rh)=1\right\}.\eqno \mbox{\bf (MVP)}$$

Among many other things which we will not discuss here, the following facts has been proved in \cite{clmp2}:
\medskip

\textbf{(MVP1)}: for each $E> 0$ there exist $\le\in\R$ and a solution $\pl=\ple$ of $\prl$ such that $\rh_{\sscp \le}(\ple)$ solves the ${\bf (MVP)}$, $S(E)=\mathfrak{S}\left(\rh_{\sscp \le}(\ple)\right)$. In particular, if $\rh$ solves the ${\bf (MVP)}$ for a certain energy $E$, then $\psi=G[\rh]$ solves $\prl$ for some $\lm=\le$, where $\pl=\psi$ and $\rh=\rl(\pl)$.
\medskip

\textbf{(MVP2)}: $S(E)$ is continuous and
\beq\label{conc}
\rh_{\sscp \lm(E)}(\ple)\rightharpoonup \dt_{x_0},\quad \mbox{as}\quad E\to +\infty,
\eeq
where $x_0$ is a maximum point of the Robin function $\gamma_{\sscp \om}(x):=R(x,x)$, and $R(x,y)=G(x,y)+\frac{1}{2\pi}\log(|x-y|)$.
\medskip

{For the sake of completeness let us recall that, by the Jensen inequality, we have,  $\mathfrak{S}(\rl(\psi_\lambda))\leq\log(|\om|),$  where the equality holds if and only if  $\rl(\psi_\lambda)(x)\equiv \mbox{\Large \textrho}_{0}(\psi_0)(x)=\frac{1}{|\om|},\,\fo x\in\ov{\om}$.  Therefore $\rh_0$ is the unique solution of the ${\bf (MVP)}$ at energy $E=E_0$, that is,}
\beq
\label{e0}
E_0:=\mathcal{E}\left(\rl(\pl)\right)\left.\right|_{\lm=0}=\mathcal{E}\left(\rh_0\right)=\frac{1}{{2}|\om|^2}\ino \ino G(x,y) dx dy.
\eeq
Moreover, it has been proved in \cite{clmp2} that if $\om$ is simply connected, then for each $E<E_{8\pi}$ there exists $\lm=\lm(E)\in (-\infty,8\pi)$
such that $\lm(E)$ is well defined, continuous and monotonic increasing, $\lm(E)\nearrow 8\pi$ as
$E \nearrow  E_{8\pi}$ and $S(E)=\mathfrak{S}\left(\rh_{\sscp \le}(\ple)\right)$ is smooth, concave and $\frac{dS(E)}{dE}=-\lm(E)$ in $(0,E_{8\pi})$.
This result has been recently generalized to cover the case where $\om$ is any bounded and connected domain in \cite{BLin3}.
In particular, see \cite{clmp2}, $S(E)$ is strictly decreasing for $E>E_0$
and if  $\om$ is a simply connected domain of second kind,
then there exists constants $C_1<C_2$ such that $-8\pi E +C_1\leq S(E)\leq -8\pi E +C_2$, for any $E>E_{8\pi}$ and $S(E)$ \underline{is not concave}
for $E>E_{8\pi}$.

\medskip

Our first main result is the following:
\bte
\label{thm1}
Let $\om$ be a convex domain of second kind.  There exists $E_{*}> E_{8\pi}$ such that $S(E)$ is smooth and strictly convex in $[E_{*},+\ii)$. In particular,
$$ \beta(E):=\frac{dS(E)}{d E}=-\lm(E),\;E\in[E_{*},+\ii), $$
where $\lm(E)$ is smooth, strictly decreasing and $\lm(E)\searrow 8\pi^+$ as $E\to +\ii$.
\ete

\brm{\it Actually the proof shows that a slightly stronger result holds, that is, the claim of Theorem \ref{thm1} is true on any strictly starshaped domain of second kind such that $\gamma_{\sscp \om}$ has a unique and non degenerate maximum point.}
\erm

Even with the aid of the non degeneracy and uniqueness results derived here (see Theorem~\ref{th1.1} below) and in \cite{BJLY}
{(see Theorem 3.B in section \ref{sec5} below),} the proof of Theorem \ref{thm1} is not straightforward, because the monotonicity of
the Lagrange multiplier $\lm(E)$ is not easy to check. We succeed in finding a solution to this problem by showing that
$\lm(E)=\left.\lm_\eps\right|_{\eps(E)}$ where $\lm_\eps$ is a monotone function constructed by using the so called Gelfand equation (see \rife{peps})
and a related non degeneracy property \cite{GlGr}, see {Theorem 3.C} in section \ref{sec5}. The proof of this part also relies on another delicate result about the characterization of domains of second kind \cite{BLin3}. This workaround eventually solves the problem since it turns out that the energy as a function of $\eps$ has the "right" monotonicity property, see Lemma \ref{enrgeps}.

\medskip

\subsection{The mean field equation.}
The proof of Theorem \ref{thm1} is based on various results, see section \ref{sec5}, of independent interest about the mean field equation,
\beq\label{1lm}
\begin{cases}
-\Delta u_{\lambda}=\lambda\dfrac{\dsp he^{u_{\lambda}}}{\dsp \int_{\Omega}he^{u_{\lambda}}}~&\mathrm{in}~\Omega,\\
u_{\lambda}=0~&\mathrm{on}~\partial\Omega,
\end{cases}
\eeq
and about the Gelfand equation \rife{peps}, where $\om$ is a smooth and bounded domain, $h(x)=\hat h(x)e^{-4\pi\sum_{i=1}^\ell \alpha_iG(x,p_i)}\geq0$, the $p_i$'s are distinct points in $\om$, $\alpha_i>-1$ for any $i=1,\cdots,\ell$, $\hat h$ is a positive smooth function in $\ov{\om}$ and $G(x,p)$ is the Green function satisfying,
\begin{equation*}
-\Delta G(x,p)=\delta_p~\mathrm{in}~\Omega,\quad G(x,p)=0~\mathrm{on}~\partial\Omega.
\end{equation*}

Because of the many motivations in pure and applied mathematics \cite{Aub,Ban,bav,beb,clmp1,Gel,On,suzC,T,Troy,yang},
a lot of work has been done to understand these pde's and we just mention few results \cite{BdM2,BdMM,bt2,CCL,CLin5,EGP,Fang,GM1,yy,linwang,Mal2,suz,tar}
and the references quoted therein. Among many other things, the interesting properties of these problems are the lack of compactness \cite{bt,bm,yy},
which in turn causes either non existence or non uniqueness of solutions of \rife{1lm} with $\lm>8\pi$ \cite{CLin2,dem2, Mal1}, see also \cite{B2,clmp2}.
As a consequence, in general, the global bifurcation diagram could be very complicated and in particular solutions may be degenerate.
Here we say that a solution of \rife{1lm} is degenerate if the corresponding linearized problem (see \rife{1.8l} below) admits a non trivial solution.
This is why it is important to understand which are the regions in the bifurcation diagram where one can recover uniqueness and {nondegeneracy.}

\bdf
\label{def1.6}
{\it A sequence of solutions $u_n:=u_{\sscp \lm_n}$ of \eqref{1lm} is said to be an $m$-bubbling sequence if,
$$\lm_n\dfrac{\dsp h\e{u_n}}{ \dsp  \ino h \e{u_n}}\rightharpoonup 8\pi \sum\limits_{j=1}^m \dt_{q_j},\;\mbox{as}\;n\to +\ii,$$
weakly in the sense of measures in $\om$, where $\{q_1,\cdots, q_m\}\subset \om$ are $m$ distinct points satisfying $\{q_1,\cdots, q_m\}\cap \{p_1,\cdots,p_\ell\}=\emptyset$. The points $q_j$'s are said to be the blow up points and $\{q_1,\cdots, q_m\}$ the blow up set.}
\edf

\brm
\label{rem1.7} {\it It is well known \cite{bm,CCL,yy}, that if $u_n$ is an $m$-bubbling sequence, then, as far as the domain is not too irregular, see Remark
\ref{rem1.9}, then
necessarily $\lm_n\to 8\pi m$, as $n\to +\ii$.}
\erm

The uniqueness for $\lm\leq 8\pi$ is a delicate problem which has been discussed in \cite{bl,BLin3, BJLY, CCL, suz}. On the other side, we do not know of any general {nondegeneracy} result for bubbling solutions of \rife{1lm}. Actually
the nondegeneracy of $m$-bubbling solutions of the Gelfand problem (see \rife{peps} below) was first established
in \cite{GlGr} for $m=1$ and then in \cite{GOS} for $m\geq 1$. It seems however that our problem is rather different, since our linearized
equation contains an additional term, see \rife{1.8l}. Therefore, after a suitable translation, we would end up with a solution of an equation
with vanishing weighted mean value and constant and unknown value at the boundary, unlike the Gelfand problem where the solution satisfies the
natural Dirichlet boundary conditions. Let us consider a generic $m$-bubbling sequence which satisfies,
\begin{equation}
\label{1.1l}
\graf{\Delta u_n+\lambda_n\dfrac{\dsp h\e{u_n}}{\dsp \int_{\Omega} h\e{u_n}} =0~&\mathrm{in}~\Omega,\\
u_n=0~&\mathrm{on}~\partial\Omega,}
\end{equation}
for any $n\in\N$, and prove that $u_n$ is non-degenerate, provided that $n$ is sufficiently large, and $h$ and $\om$ satisfy some
other non-degeneracy assumptions. Let
$$R(x,y)=G(x,y)+\frac{1}{2\pi}\log|x-y|,$$
be the regular part of the Green function $G(x,y)$. For $\mathbf{q}=(q_1,\cdots,q_m)\in\om\times\cdots\times\om,$
we set
\begin{equation}
\label{1.2}
G_j^*(x)=8\pi R(x,q_j)+8\pi\sum_{l\neq j}^{1,\dots,m}G(x,q_l),
\end{equation}
\begin{equation}
\label{1.3}
\ell(\mathbf{q})=\sum_{j=1}^m[\Delta \log h(q_j) ]h(q_j)e^{G_j^*(q_j)},
\end{equation}
and
\begin{equation}
\label{1.4}
f_{\mathbf{q},j}(x)=8\pi\Big[R(x,q_j)- R(q_j,q_j)+\sum^{1,\dots,m}_{l\neq j}(G(x,q_l)-G(q_j,q_l))\Big]+\log \frac{h(x)}{ h(q_j)}.
\end{equation}

We will denote by $B_r(q)$ the ball of radius $r$ centred at $q\in\Omega.$ For the case $m\geq2$ we fix a constant $r_0\in(0,\frac12)$ and a family of open sets
$\Omega_j$ satisfying $\Omega_l\cap\Omega_j=\emptyset$ if $l\neq j$, $\bigcup_{j=1}^m\overline \om_j=\overline\om$,
$B_{2r_0}(q_j)\subset\Omega_j,~j=1,\cdots,m$. Then, let us define,
\begin{equation}
\label{1.5}
D(\mathbf{q})=\lim_{r\to0}\sum_{j=1}^m h(q_j)e^{G_j^*(q_j)}\Big(\dsp\int_{\Omega_j\setminus B_{r_j}(q_j)}
e^{\Phi_j(x,\mathbf{q})} \mathrm{d}x-\frac{\pi}{r_j^2}\Big),
\end{equation}
where $\Omega_1=\Omega$ if $m=1$, $r_j=r\sqrt{8h(q_j)e^{G_j^*}(q_j)}$ and
\begin{equation}
\label{1.6}
\Phi_j(x,\mathbf{q})=\sum_{l=1}^m 8\pi  G(x,q_l)-G_j^*(q_j)+\log h(x)-\log h(q_j).
\end{equation}
The quantity $D(\mathbf{q})$ was introduced first in \cite{CCL, clw}. For $(x_1,\cdots,x_m)\in\om\times\cdots\times\om$ we also define,
\begin{equation}
\label{1.7}
f_m(x_1,x_2,\cdots,x_m)=\sum_{j=1}^{m}\Big[\log(h(x_j))+4\pi R(x_j,x_j)\Big]+4\pi\sum_{l\neq j}^{1,\cdots,m}G(x_l,x_j),
\end{equation}
and let $D_\Omega^2f_m$ be its Hessian matrix on $\Omega$. The function $f_m$ is also known in literature as the $m$-vortex Hamiltonian \cite{New}. We consider the linearized problem relative to \eqref{1.1l},
\begin{equation}
\label{1.8l}
\graf{\Delta\phi+\lambda_n\dfrac{\dsp he^{u_n}}{\dsp \int_{\Omega}he^{u_n}\d y}\left(\phi-\dfrac{\dsp\int_{\Omega}he^{u_n}\phi\,\d y}{\dsp\int_{\Omega}he^{u_n}\d y}\right)=0~&\mathrm{in}~\Omega,\\
\phi=0~&\mathrm{on}~\partial\Omega.}
\end{equation}

Our second main result is the following,

\begin{theorem}
\label{th1.1}
Let $u_n$ be an $m$-bubbling sequence of \eqref{1.1l}
with blow up set $\{q_1,\cdots,q_m\}\cap\{p_1,\cdots,p_\ell\}=\emptyset$, where
$\mathbf{q}=(q_1,\cdots,q_m)$ is a critical point of $f_m$ and let $det(D_{\Omega}^2f_m(\mathbf{q}))\neq0$. Assume that either,
\begin{enumerate}
  \item [(1)] $\ell(\mathbf{q})\neq0$, or,
  \item [(2)] $\ell(\mathbf{q})=0$ and $D(\mathbf{q})\neq0$.
\end{enumerate}
Then there exists $n_0\geq1$ such that, for any $n\geq n_0$, \eqref{1.8l} admits only the trivial solution $\phi\equiv 0$.
\end{theorem}

\brm\label{rem1.9} {\it Actually, Theorem \ref{th1.1} holds for a wider class of domains $\om$. More precisely, as in \cite{CCL} we say that $\om$ is regular if its boundary $\partial \om$ is of class $C^2$ but
for a finite number of points $\{Q_1 , . . . , Q_{N_0}\}\subset \partial \om$ such that the following conditions holds at each $Q_j$.
\begin{enumerate}
  \item [(i)] The inner angle $\theta_j$ of $\partial \om$ at $Q_j$ satisfies $0 < {\theta_j \neq \pi} < 2\pi$;
  \item [(ii)] At each $Q_j$ there is an univalent conformal map from $B_\delta (Q_j) \cap \overline{\om}$ to
the complex plane $\mathbb{C}$ such that $\partial \om \cap B_\delta (Q_j)$ is mapped to a $C^2$ curve.
\end{enumerate}
It can be shown by a moving plane argument (see \cite{CCL}) that solutions of \eqref{1lm} are uniformly bounded in a fixed neighborhood of $\partial\om$. Since the argument of Theorem \ref{th1.1} relies on the local estimates for blow up solutions of \eqref{1lm} it can be carried out for the latter class of domains as well. Hence, Theorem \ref{th1.1} holds for any regular domain $\om$. {See also Theorem 3.A below.}
\medskip

On the other hand, we may consider a more general problem than \eqref{1lm} in
which the weight $h$ is replaced by $h_n(x)=\hat h_n(x)e^{-4\pi\sum_{i=1}^\ell \alpha_{i,n}G(x,p_{i,n})}$,
where $p_{i,n}\to p_i$, $\alpha_{i,n}\geq-1+\frac 1C$, $\alpha_{i,n}\to\alpha_i$ for any $i=1,\dots,\ell$ and
$\hat h_n$ are smooth functions such that $\hat h_n\geq \frac 1C$ in $\overline \om$, $\hat h_n\to\hat h_n$ in $C^{2,\sg}(\overline\om)$,
for some fixed constant $C>0$ and $\sg>0$. It is straightforward to check that the latter generalization does not affect the argument of Theorem \ref{th1.1} and thus Theorem \ref{th1.1} holds for this class of problems as well.}
\erm

To prove Theorem \ref{th1.1} we will analyze the asymptotic behavior of the auxiliary sequence,
$$\zeta_n=\dfrac{\phi_n-\frac{\int_{\Omega}he^{u_n}\phi_n\d y}{\int_{\Omega}he^{u_n}\d y}}{\left\|\phi_n-\frac{\int_{\Omega}he^{u_n}\phi_n\d y}{\int_{\Omega}he^{u_n}\d y}\right\|_{L^{\infty}(\Omega)}},$$
where $\phi_n$ is a solution of \eqref{1.8l}. Near each blow up point $q_j$ and after a suitable scaling, $\zeta_n$ converges to an
{element in the kernel space of the linearized operator $L$, where}
\begin{equation}
\label{1.11}
L\phi:=\Delta \phi+\frac{8}{(1+|z|^2)^2}\phi\quad \mathrm{in}\quad \mathbb{R}^2.
\end{equation}
It is well known (\cite[Proposition 1]{bp}) that the kernel of $L$ has {real dimension $3$.} A crucial point in the proof of Theorem \ref{th1.1} is to show that,
after suitable scaling and for large $n$, $\zeta_n$ is orthogonal to {the kernel space}.
However this is not enough, since we also need to figure out the global behaviour of $\zeta_n$ in the zone {far away} from the blow up point.
These problems are rather subtle and require the well known pointwise estimates \cite{CLin1}, together with some refinements recently derived in \cite{BJLY}.
Then we will work out a delicate adaptation of some arguments in \cite{BJLY,ly2} based on the analysis of various suitably defined Pohozaev-type identities.
\medskip

This paper is organized as follows. In section \ref{sec2}, we prove Theorem \ref{th1.1}. In section \ref{sec5}, we review the known results for domains of second kind, the local uniqueness for bubbling solutions of mean field equation and the non degeneracy result for the Gelfand problem. In section \ref{sec6}, we prove Theorem \ref{thm1}. In the Appendix we provide the technical estimation for the proof of Theorem \ref{th1.1}.

\section{{Proof of Theorem \ref{th1.1}}}\label{sec2}
\subsection{{Preliminaries}}
{Suppose} that $u_n$ is a sequence of blow-up solutions of \eqref{1.1l} which blows up at $q_j\notin\{p_1,\cdots,p_\ell\},~j=1,\cdots,m.$ Let
\begin{equation*}
\tilde u_n(x)=u_n(x)-\log\left(\int_{\Omega}he^{u_n}\d y\right),
\end{equation*}
then it is easy to see that,
\begin{equation}
\label{2.1}
\Delta \tilde u_n+\lambda_nh(x)e^{\tilde u_n(x)}=0~\mathrm{in}~\Omega\quad\mathrm{and}\quad \int_{\Omega}he^{\tilde{u}_n}\d y=1.
\end{equation}
We denote by,
\begin{equation}
\label{2.2}
\mu_n=\max_\Omega \tilde{u}_n,\quad \mu_{n,j}=\max_{B_{r_0}(q_j)}\tilde u_n=\tilde u_n(x_{n,j})\quad \mathrm{for}\quad j=1,\cdots,m.
\end{equation}
Let us define,
\begin{equation}
\label{2.3}
U_{n,j}(x)=\log\frac{e^{\mu_{n,j}}}{(1+\frac{\lambda_nh({x}_{n,j})}{8}e^{\mu_{n,j}}|{x}-{x}_{n,j,*}|^2)^2},\quad  {x}\in \mathbb{R}^2,
\end{equation}
where the point $x_{n,j,*}$ is chosen to satisfy,
$$\nabla U_{n,j}({x}_{n,j})=\nabla(\log h({x}_{n,j})).$$
It is not difficult to check that,
\begin{align}
\label{2.4}
|x_{n,j}-x_{n,j,*}|=O(e^{-\mu_{n,j}}).
\end{align}
{Near the blow up point $q_j$, we let $\eta_{n,j}$ denote the "error term",}
\begin{equation}
\label{2.5}
\eta_{n,j}(x)=\tilde{u}_n(x)-U_{n,j}(x)-(G_{j}^{*}(x)-G_{j}^{*}(x_{n,j})), \quad {x}\in B_{r_0}(x_{n,j}).
\end{equation}
{For $x\in \omb\setminus \cup_{j=1}^m B_{\tau}(q_j))$,  $\tilde{u}_n$ should be well approximated by a sum of Green's function, which we denote by,}
\begin{equation}
\label{2.15}
w_n(x)=\tilde{u}_n(x)-\sum_{j=1}^m\rho_{n,j} G(x,x_{n,j})-\tilde{u}_{n,0},
\end{equation}{where   here and in the rest of this work, we set,}
\begin{equation}\label{boundary_u}
{\tilde{u}_{n,0}=\tilde u_n|_{\partial\Omega}.}
\end{equation}
{At each blow up point $q_j,~1\leq j\leq m$, we also define the local mass by}
\begin{equation}
\label{2.12}
\rho_{n,j}=\lambda_n \int_{B_{r_0}(q_j)}he^{\tilde{u}_n}\d y.
\end{equation}

The following sharp estimates for blow up solutions of \eqref{1.1l} have been obtained in \cite{CLin1}. The higher order term in (vi) has been obtained more
recently in \cite{BJLY}.
\medskip

{\bf { Theorem 2.A. (\cite{CLin1})}} \textit{{ Suppose  that $u_n$ is a sequence of blow-up solutions of \eqref{1.1l} which blows up at
$q_j\notin\{p_1,\cdots,p_\ell\},~j=1,\cdots,m.$ Then the following facts hold: }}

 { (i) $\eta_{n,j}=O(\mu_{n,j}^2 e^{-\mu_{n,j}})$ \textit{on} $B_{r_0}(x_{n,j})$;}

 { (ii) $w_n=o(e^{-\frac{\mu_{n,j}}{2}})$ \textit{on} $C^1(\omb\setminus \cup_{j=1}^m B_{\tau}(q_j))$; }

 { (ii) $e^{\mu_{n,j}}h^2(x_{n,j})e^{G_j^*(x_{n,j})}=e^{\mu_{n,1}}h^2(x_{n,1})e^{G_1^*(x_{n,1})}(1+O(e^{-\frac{\mu_{n,1}}{2}}))$; }

 { (iii) $\mu_{n,j}+\tilde{u}_{n,0}+2\log\left(\frac{\lambda_n h(x_{n,j})}{8}\right)+G_j^*(x_{n,j})
=O(\mu_{n,j}^2 e^{-\mu_{n,j}})$;}

 { (iv) $\nabla[\log h(x)+G_j^*(x)]\Big|_{x=x_{n,j}}=O(\mu_{n,j} e^{-\mu_{n,j}})$ \textit{and} $|{x}_{n,j}-{{q}}_j|=O(\mu_{n,j} e^{-\mu_{n,j}})$;}

 { (v) $\rho_{n,j}-8\pi= O(\mu_{n,j}e^{-\mu_{n,j}})$; }

 { (vi) \textit{for a fixed small constant} $\tau>0$,}
\begin{equation*}
\begin{aligned}
\label{2.14}
\lambda_n-8\pi m=~&\frac{2\ell(\mathbf{q})e^{-\mu_{n,1}}}{m h^2(q_1)e^{G_1^*(q_1)}}\Big(\mu_{n,1}+\log\left(\lambda_n h^2(q_1)e^{G_1^*(q_1)}\tau^2\right)
-2\Big)\\
&+\frac{8e^{-\mu_{n,1}}}{h^2(q_1)e^{G_1^*(q_1)}\pi m}\Big(D(\mathbf{q})
+O(\tau^\sigma)\Big)+O(\mu_{n,1}^2e^{-\frac{3}{2}\mu_{n,1}})+O(e^{-(1+\frac{\sigma}{2})\mu_{n,1}}),
\end{aligned}
\end{equation*}
\textit{where} $\sigma>0$ \textit{is a positive number that satisfies} $\hat h\in C^{2,\sigma}(\ov{\om})$.

\subsection{{The proof of Theorem \ref{th1.1}}} We shall prove Theorem \ref{th1.1} by contradiction. Suppose that \eqref{1.8l} admits a nontrivial solution $\phi_n,$ where $u_n$ is an $m$-bubbling sequence of solutions of \eqref{1.1l} with blow up points $q_j\notin\{p_1,\cdots,p_\ell\}$, $j=1,\cdots,m$, such that $\mathbf{q}=(q_1,\cdots,q_m)$ is a critical point of $f_m$ and $\textrm{det}(D_{\Omega}^2f_m(\mathbf{q}))\neq 0$. Next, let us define,
\begin{equation}
\label{3.1}
\zeta_n(x)=\frac{\phi_n(x)-\frac{\int_{\Omega}he^{u_n}\phi_n\mathrm{d}y}{\int_{\Omega}h e^{u_n}\mathrm{d}y}}
{\left\|\phi_n-\frac{\int_{\Omega}he^{u_n}\phi_n\mathrm{d}y}{\int_{\Omega}he^{u_n}\mathrm{d}y}\right\|_{L^\infty(\Omega)}}
\quad\mathrm{and}\quad
f_n^*(x)=\lambda_nh(x)e^{\tilde{u}_n(x)}\zeta_n(x).
\end{equation}
It is easy to see that,
\begin{equation}
\label{3.2}
\int_{\Omega}f_n^*dx=0,
\end{equation}
and $\zeta_n$ satisfies,
\begin{equation}
\label{3.3}
\begin{aligned}
\Delta \zeta_n+f_n^*(x)=\Delta \zeta_n+\lambda_nh(x)c_n(x)\zeta_n(x)=0,
\end{aligned}
\end{equation}
where\begin{equation}
\label{cn}
\begin{aligned}{c_n(x)=e^{\tilde{u}_n}(x).}\end{aligned}
\end{equation}
{In order to prove Theorem \ref{th1.1}, we shall follow the argument in \cite{ly2, BJLY}. In \cite{ly2,BJLY},
the authors used various Pohozaev identities to show the local uniqueness of bubbling solutions.
In our case, $\zeta_n$ plays the role of the difference of two solutions in \cite{ly2,BJLY} (see Step 3 below).   }

\medskip
\noindent
{Step 1. The asymptotic behavior of $\zeta_n$ near blow up points.} \\
{In this step we shall prove that there exist constants $b_{j,0}$, $b_{j,1}$, and $b_{j,2}$ such that,}
\begin{equation}\label{nearblow}
{ \zeta_n\left(e^{-\frac{\mu_{n,j}}{2}}z+{x}_{n,j}\right)\to  b_{j,0}\psi_{j,0}(z)+b_{j,1}\psi_{j,1}(z)+b_{j,2}\psi_{j,2}(z)
~\mbox{in}~C^0_{\mathrm{loc}}(\mathbb{R}^2),}
\end{equation}
{where,}
\begin{equation*}
{\psi_{j,0}(z)=\frac{1-\pi m h(q_j)|z|^2}{1+\pi m h(q_j)|z|^2},~\psi_{j,1}(z)=\frac{\sqrt{\pi mh(q_j)}z_1}{1+\pi m h(q_j)|z|^2},~
\psi_{j,2}(z)=\frac{\sqrt{\pi mh(q_j)}z_2}{1+\pi m h(q_j)|z|^2}.}
\end{equation*}
{Indeed,  it is well that, after a suitable scaling, $\zeta_{n}$ converges to a function
$\zeta_j(z)$ in $C^0_{\mathrm{loc}}(\mathbb{R}^2)$,
where $\zeta_j(z)$ satisfies,}
\begin{equation*}{
\Delta \zeta_j+\frac{8\pi m h({q}_j)}{(1+\pi m h({q}_j)|z|^2)^2}\zeta_j(z)=0~\mathrm{in}~\mathbb{R}^2,\quad |\zeta_j|\leq 1.}
\end{equation*}
Then  \eqref{nearblow} follows by \cite[Proposition 1]{bp}.
\medskip

\noindent
{Step 2. The global behavior of $\zeta_n$ far away from the blow up points.}\\
{We claim that there exists  a constant $b_0$ such that,}
\begin{equation}\label{farblow}
{\zeta_n\to-b_0\ \ \textrm{in}\ \ C^0_{\textrm{loc}}( \ov{\om}\setminus\bigcup_{j=1}^m \{q_j\}), \ \ \textrm{and} \  b_0=b_{j,0} \ \ \textrm{for}\  j=1,\cdots, m.}
\end{equation}
{By Theorem 2.A and  $\|\zeta_n\|_{L^\infty(\Omega)}\leq 1$,
we see that $\zeta_n\to\zeta_0$ in $C^0_{\mathrm{loc}}(\omb\setminus\{q_1,\cdots,q_m\})$, where $\zeta_0$ satisfies,}
\begin{equation}
\label{3.14}
\Delta\zeta_0=0~\mathrm{in}~\Omega\setminus\{q_1\cdots q_m\}.
\end{equation}
{By $\|\zeta_n\|_{L^\infty(\Omega)}\leq 1$,  we can extend \eqref{3.14} to $\Omega$.
Then we conclude that,}
\begin{equation}
\label{limit}
\zeta_0\equiv-b_0 \ \textrm{in}\ \  \Omega, \ \textrm{ where}\  b_0\ \textrm{ is a constant}.
\end{equation}
{In order to prove $b_{j,0}=b_0$ for $j=1,\cdots, m$, let us set,}  \[\psi_{n,j}(x)=\frac{1-\frac{\lambda_n}{8}h({x}_{n,j})|x-x_{n,j}|^2e^{\mu_{n,j}}}
{1+\frac{\lambda_n}{8}h(x_{n,j})|x-x_{n,j}|^2e^{\mu_{n,j}}}\ \ \textrm{and}\ \ \zeta_{n,j}^*(r)=\int_0^{2\pi}\zeta_n(r,\theta)\mathrm{d}\theta, \ \textrm{ where}\  r=|x-x_{n,j}|.\]
{Fix large $R>0$. By \eqref{3.3} and Theorem 2.A, we can derive  the following estimate  (see \cite{ly2} and \cite[Lemma  3.4]{BJLY} for details) }
\begin{equation}\label{int_est}
\begin{aligned}&{\frac{1}{r}\int_{\partial B_d(x_{n,j})}\left(\psi_{n,j}\frac{\partial\zeta_n}{\partial\nu}
-\zeta_n\frac{\partial\psi_{n,j}}{\partial \nu}\right)\d\sigma
=(\zeta_{n,j}^*)'(r)\psi_{n,j}(r)-\zeta_{n,j}^*(r)\psi_{n,j}'(r)=\frac{o\left(\frac{1}{\mu_{n,j}}\right)}{r},}\,\end{aligned}\end{equation} for $r\in (Re^{-\mu_{n,j}/2},r_0).$
By using \eqref{int_est}, we conclude that,
\begin{equation}
\label{3.19}
\zeta_{n,j}^*(r)=-2\pi b_{j,0}+o_R(1)+o_n(1)(1+O(R))~\mbox{for all}~r\in (Re^{-\mu_{n,j}/2},r_0),
\end{equation}where $\lim_{n\to+\infty}o_n(1)=0$ and  $\lim_{R\to+\infty}o_R(1)=0$.
This fact together with \eqref{limit}, shoes that $b_{j,0}=b_0$ for $j=1,\cdots, m$ and thus the claim \eqref{farblow} follows.
\medskip

\noindent
{Step 3. The Pohozaev type identities.}\\
{The following Pohozaev type identities play a crucial role in the proof of Theorem \ref{th1.1}. {For any} fixed $r\in(0,r_0)$, the followings hold:}
\begin{equation}
\begin{aligned}
\label{4.3} &\int_{\partial B_r(x_{n,j})}r\langle Dv_{n,j},D\zeta_n\rangle\mathrm{d}\sigma
-2\int_{\partial B_r({x}_{n,j})}r\langle\nu,D v_{n,j}\rangle\langle\nu,D\zeta_n\rangle\mathrm{d}\sigma\\
&=\int_{\partial B_r({x}_{n,j})}r\lambda_nhe^{\tilde{u}_n}\zeta_n\mathrm{d}\sigma
-\int_{B_r(x_{n,j})}\lambda_nh({x})e^{\tilde{u}_n}\zeta_n(2+\langle D(\log h+\phi_{n,j}),x-x_{n,j}\rangle)\mathrm{d}x,
\end{aligned}
\end{equation}and for $i=1,2$, \begin{equation}
\begin{aligned}
\label{4.47} &\int_{\partial B_r(x_{n,j})}\langle\nu, D\zeta_n\rangle D_iv_{n,j}+\langle\nu, Dv_{n,j}\rangle D_i\zeta_n\mathrm{d}\sigma
-\int_{\partial B_r(x_{n,j})}\langle D {v_{n,j}} ,D\zeta_n\rangle\frac{(x-x_{n,j})_i}{|x-x_{n,j}|}\mathrm{d}\sigma\\
&=-\int_{\partial B_r(x_{n,j})}\lambda_nh(x) e^{\tilde{u}_n}\zeta_n\frac{(x-x_{n,j})_i}{|x-x_{n,j}|}\mathrm{d}\sigma
+\int_{B_r(x_{n,j})}\lambda_nh(x)e^{\tilde{u}_n}\zeta_nD_i(\phi_{n,j}+\log h)\mathrm{d}{x}.
\end{aligned}
\end{equation}
where\begin{equation}
\label{4.2}
v_{n,j}(y)=\tilde{u}_n(y)-\phi_{n,j}(y)\ j=1,\cdots,m, \ \textrm{and}
\end{equation}
\begin{equation}
\label{4.1}
\phi_{n,j}(y)=\frac{\lambda_n}{m}\Big[(R(y,x_{n,j})-R(x_{n,j},x_{n,j}))+ \sum_{l\neq j}(G(y,x_{n,l})-G(x_{n,j},x_{n,l}))\Big].
\end{equation}
The proof of \eqref{4.3} is obtained by using,
\begin{equation*}
\Delta\zeta_n+\lambda_nh(x)e^{\tilde{u}_n}\zeta_n=0,\quad
\Delta v_{n,j}+\lambda_nhe^{\tilde{u}_n}=0,
\end{equation*}
together with,
\begin{equation}
\label{4.4}
\begin{aligned}
&\mathrm{div}\left(\nabla\zeta_n\left(\nabla v_{n,j}\cdot(x-x_{n,j})\right)+\nabla v_{n,j}\left(\nabla\zeta_n\cdot(x-x_{n,j})\right)-  \nabla\zeta_n\cdot\nabla v_{n,j}(x-x_{n,j})\right)\\
&=\Delta \zeta_n\left(\nabla v_{n,j} \cdot({x}-{x}_{n,j})\right)+\Delta v_{n,j}\left(\nabla\zeta_n\cdot({x}-{x}_{n,j})\right)\\
&=-\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h}\zeta_n\left(\nabla v_{n,j}\cdot(x-x_{n,j})\right)
-\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h}\left(\nabla\zeta_n\cdot(x-x_{n,j})\right)\\
&=-\mathrm{div}\left(\lambda_n e^{v_{n,j}+\phi_{n,j}+\log h}\zeta_n(x-x_{n,j})\right)+2\lambda_n e^{v_{n,j}+\phi_{n,j}+\log h}\zeta_n\\
&~\quad+\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h}\zeta_n\left(\nabla(\phi_{n,j}+\log h)\cdot(x-x_{n,j})\right).
\end{aligned}
\end{equation}
While the proof of \eqref{4.47} follows by using the equation $\Delta v_{n,j}+\lambda_nh e^{\tilde{u}_n}=0$ and
\begin{equation*}
\begin{aligned}
&\mathrm{div}\left(\nabla\zeta_nD_lv_{n,j}+\nabla v_{n,j}D_l\zeta_n-\nabla\zeta_n\cdot\nabla v_{n,j}e_l\right)\\
&=\Delta\zeta_nD_lv_{n,j}+\Delta v_{n,j} D_l\zeta_n=-\lambda_nh(x) e^{\tilde{u}_n} \zeta_nD_lv_{n,j}-\lambda_nh(x)e^{\tilde{u}_n}D_l\zeta_n\\
&=-\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h}\zeta_nD_l(v_{n,j}+\phi_{n,j}+\log h)-\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h} D_l\zeta_n\\
&\quad+\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h}\zeta_nD_l(\phi_{n,j}+\log h)\\
&=-\mathrm{div}(\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h}\zeta_ne_l)
+\lambda_ne^{v_{n,j}+\phi_{n,j}+\log h}\zeta_nD_l(\phi_{n,j}+\log h),
\end{aligned}
\end{equation*}
where $e_l=\frac{(x-x_{n,j})_l}{|x-x_{n,j}|}$, $l=1,2$.

\medskip
\noindent
{Step 4. Based on the identities established in the last step, we can prove that
\begin{equation}
\label{bequal0}
b_0=b_{j,0}=b_{j,1}=b_{j,2}=0,~j=1,\cdots,m.
\end{equation}
The proof of the above fact is long and technical and so we postpone it to the last section \ref{appendix}.}
Once we get that \eqref{bequal0} holds, then the proof of Theorem \ref{th1.1} is almost done. Indeed, if $x_n^*$ is a maximum point of $\zeta_n$, then we have,
\begin{equation}
\label{4.56}
|\zeta_n(x_n^*)|=1.
\end{equation}
In view of Step 1-2 and \eqref{bequal0}, we find that $\lim_{n\to+\infty}x_n^*=q_j$ for some $j$ and
\begin{equation}
\label{4.57}
\lim_{n\to+\infty}e^{\frac{\mu_{n,j}}{2}}s_n=+\infty,~\mathrm{where}~s_n=|x_n^*-x_{n,j}|.
\end{equation}
Setting $\tilde{\zeta}_n(x)=\zeta_n(s_n {x}+x_{n,j})$, then from \eqref{3.3} and Theorem 2.A we see that $\tilde{\zeta}_n$ satisfies,
\begin{equation*}
\begin{aligned}
\Delta\tilde{\zeta}_n+\frac{\lambda_nh(x_{n,j})s_n^2e^{\mu_{n,j}}(1+ O(s_n|x|)+o(1))\tilde{\zeta}_n}
{(1+\frac{\lambda_nh(x_{n,j})}{8}e^{\mu_{n,j}}|s_nx+x_{n,j}-x_{n,j,*}|^2)^2}=0.
\end{aligned}
\end{equation*}
On the other hand, by \eqref{4.56} we also have,
\begin{equation}
\label{4.58}
\Big|\tilde{\zeta}_n\Big(\frac{x_n^*-x_{n,j}}{s_n}\Big)\Big|=|\zeta_n(x_n^*)|=1.
\end{equation}
In view of \eqref{4.57} and $|\tilde{\zeta}_n|\leq 1$ we see that $\tilde{\zeta}_n\to\tilde{\zeta}_0$
on any compact subset of $\mathbb{R}^2\setminus\{0\}$, where $\tilde{\zeta}_0$ satisfies $\Delta \tilde{\zeta}_0=0$
in $\mathbb{R}^2\setminus\{0\}$. Since $|\tilde{\zeta}_0|\leq 1$, we have $\Delta \tilde{\zeta}_0=0$ in $\mathbb{R}^2$, whence
$\tilde{\zeta}_0$ is a constant. At this point, since $ \frac{|x_n^*-x_{n,j}|}{s_n}=1$ and in view of \eqref{4.58}, we find that,
$\tilde{\zeta}_0\equiv1$ or $\tilde{\zeta}_0\equiv-1$. As a consequence we conclude that,
$$|\zeta_n(x)|\ge \frac{1}{2}~\mathrm{if}~s_n\leq |{x}-{x}_{n,j}|\leq 2s_n,$$
which contradicts \eqref{3.19},  since $e^{-\frac{\mu_{n,j}}{2}}<< s_n$, $\lim_{n\to+\infty}s_n=0$, and $b_0=b_{j,0}=0$.
This fact concludes the proof of Theorem \ref{th1.1}.\hfill$\Box$

\medskip
\section{The proof of Theorem \ref{thm1}: preliminary results}\label{sec5}
First of all, we will need the following characterization of domains of second kind recently derived in \cite{BLin3}.
Let us recall that here $\gamma_{\sscp \om}(x)=R(x,x)$.

\medskip

{\bf Theorem 3.A. (\cite{BLin3})} {\it A domain $\om$ is of second kind if and only if $\gamma_{\sscp \om}$ admits at least one
maximum point $q$ such that $D_{\sscp \om}(q)>0$, where,
$$D_{\sscp \om}(q)=\lim_{\eps \to 0^+} \int_{\om\setminus B_{\eps}(q)} {\dsp \frac{\e{8\pi(R(x,q)-\gamma_{\sscp \om}(q)) }-1}{|x-q|^4}}-\int_{\R^2\setminus \om}\frac{1}{|x-q|^4}\,.$$}

\medskip

\brm{\it It is easy to check that the above defined quantity $D_{\sscp \om}(q)$ is proportional to the $D(q)$ as introduced in \eqref{1.5},
i.e. $D_{\sscp \om}(q)=c D(q)$ for some constant $c>0$ {provided that $h$ is a constant function and $m=1$. }}
\erm

Next we will need the following particular case of a uniqueness result recently derived in \cite{BJLY}.

\medskip

{\bf Theorem 3.B. (\cite{BJLY})} {\it Let $\om$ be any smooth and bounded domain,  $h$ a constant function and let  $u_{n,i}$, $i=1,2$ be two sequences of solutions of \eqref{1lm} which satisfy, $$\lm_{n,1}=\lm_n=\lm_{n,2},\;n\in \N,$$ and,}
\beq
\label{blowup}
\lm_n\dfrac{\dsp\e{u_n}}{\dsp\ino \e{u_n}}\rightharpoonup 8\pi \dt_{x=q},\mbox{ as }n\to +\ii,
\eeq
{\it weakly in the sense of measures in $\om$, where ${q}$ is a critical point of $\gamma_{\sscp \om}$ such that $\textrm{det}(D^2\gamma_{\sscp \om}({q}))\neq 0$. If $D_{\sscp \om}({q})\neq 0$, then there exists $n_0\in\N$ such that $u_{n,1}=u_{n,2}$ for all $n\ge n_0$.}

\medskip

\brm\label{seckind} {\it A crucial estimate needed to make our argument work has been first obtained in \cite{CCL} for $1$-point blow up solutions
and more recently generalized to the general $m$-point blow up case in \cite{BJLY}. In particular, if $u_n$ satisfies \eqref{blowup}, then it holds,
$$\lm_n-8\pi={c_0}(D_{\sscp \om}(q)+o(1))\left(\max\limits_{x\in \ov{\om}} \dfrac{\dsp \e{u_n(x)}}{\dsp \ino \e{u_n}}\right)^{-1}, \;\ainf,$$
where $c_0>0$ is a constant, see \cite{CCL,BJLY} and Theorem 2.A\mbox{\rm (vi)}.}
\erm

Finally we will need a non degeneracy result about solutions of the Gelfand problem derived in \cite{GlGr}.

\medskip
{\bf {Theorem 3.C. } (\cite{GlGr})} {\it Let $\ue$ be a family of solutions of,}
\beq\label{peps}
\begin{cases}
-\Delta u_{\varepsilon}=\varepsilon^2e^{u_{\varepsilon}}~&{\it in}~\Omega,\\
u_{\varepsilon}=0~&{\it on}~\partial\Omega,
\end{cases}
\eeq {\it which satisfy,}
\beq\label{peps1}
\eps^2\e{\ue}{\rightharpoonup}  8\pi \dt_{q},\,\eps \to 0^+,
\; \mbox{\it weakly in the sense of measures in }\om,
\eeq
{\it where $q$ is  is a critical point of $\gamma_{\sscp \om}$ such that $\textrm{det}(D^2\gamma_{\sscp \om}({q}))\neq 0$. Then there exists $\eps_0>0$ such that the linearized problem relative to \eqref{peps} admits only the trivial solution  for any $\eps\in (0,\eps_0)$.}
\medskip

\section{The proof of Theorem \ref{thm1}}\label{sec6}
In this section we prove Theorem \ref{thm1}.

\medskip

{\it The Proof of Theorem \ref{thm1}.} Let $\pl$ denote any solution of $\prl$. {For a fixed $E_{\sscp \om}>E_{8\pi}$
and a fixed function $\lm_{\sscp\ii}:(E_{\sscp \om},+\ii)\to (0,+\ii)$, let $\rh_{\sscp \ii, E}:=\rh_{\sscp \lm_{\sscp \ii}(E)}(\psi_{\sscp \lm_{\sscp \ii}(E)})$
be the corresponding density for $E\in(E_{\sscp \om},+\ii)$, where $\psi_{\sscp \lm_{\sscp \ii}(E)}$ is a solution of $\prl$ with
$\lm=\lm_{\sscp \ii}(E)$.} If, for a fixed $E\in(E_{\sscp \om},+\ii)$, we assume that $\pl$ and $\lm_{\sscp\ii}$ are differentiable at the points $\lm=\lm_{\sscp\ii}(E)$ and $E$ respectively, then, by defining,
\beq\label{1.1en}
S_{\ii}(E)=\mathfrak{S}\left(\rh_{\sscp \ii, E}\right),
\eeq
it can be shown that,
\beq\label{diff}
\frac{dS_{\ii}(E)}{d E}=-\lm_{\sscp\ii}(E),
\eeq
see (5.1) in \cite{B2}. Theorem \ref{thm1} follows immediately from the next result which has its own interest, as it describes some properties of the
entropy maximizers in the high energy regime.

\bte\label{thm2} $\mathbf{(a)}$ Under the assumptions of Theorem \ref{thm1}, there exist $E_*> E_{8\pi}$ and a smooth, strictly decreasing function $\lm_{\sscp\ii}(E)$, $E\in[E_*,+\ii)$,
satisfying $\lm_{\sscp\ii}(E)\searrow 8\pi^+$ as $E\to +\ii$,
and there exists a smooth map $(8\pi, \lm_{\sscp \ii}(E_*))\ni \lm \mapsto \psi^{(\ii)}_{\lm}$,
such that $S(E)=S_{\ii}(E)$ takes the form \eqref{1.1en} with
$\psi_{\sscp \lm_{\sscp \ii}(E)}:=\psi^{(\ii)}_{\sscp \lm_{\sscp \ii}(E)}$, $E\in [E_*,+\ii)$.\\
$\mathbf{(b)}$ $S(E)$ is smooth and strictly convex,
and $\frac{dS(E)}{d E}=-\lm_{\sscp\ii}(E)$ for any $E\in [E_*,+\ii)$.
\ete

\brm\label{secep0}
{\it Since $\om$ is convex, then $\gamma_{\sscp \om}$ admits a unique critical point which coincides with its unique
maximum point, say $x_{\sscp \om}\in \om$, which is also non degenerate, that is, $\textrm{det}(D^2\gamma_{\sscp \om}({x_{\sscp \om}}))\neq 0$, see \cite{Gu}.
We will often use this fact throughout the proof.}
\erm

\medskip

{\it The proof of Theorem \ref{thm2}.} {Step 1:} Let $E_n\to +\ii$ and let $\lm_n$ and $\psi_n:=\psi_{\sscp \lm_n}$ be any sequence of entropy maximizers with energy $E_n$. Then, as a consequence of \textbf{(MVP1)}-\textbf{(MVP2)}, $\{\psi_n\}$ solves $\prl$ with
$\lm=\lm_n$ and satisfies \rife{conc}. By the Pohozaev identity jointly with the convexity of $\om$, see for example \cite{clmp1}, we have $\lm_{n}\leq \ov{\lm}<+\ii$ for any $n\in \N$ {and some $\ov{\lm}\in\mathbb{R}$}. Since $\om$ is convex, then a well known moving plane argument shows that $\psi_{n}$ is uniformly bounded from above near $\pa\om$. Therefore, in view of \rife{conc} and the results in \cite{bm} and \cite{yy}, it is not difficult to see that,
\beq\label{lmn8}
\lm_n\to 8\pi \mbox{ and }u_n=\lm_n\psi_n \mbox{ satisfies }\rife{1lm},\rife{blowup}\;\ainf,
\eeq
with blow up point $q=x_{\sscp \om}$ as defined in Remark \ref{secep0}. We point out that here the convexity is used with the Pohozaev identity to provide
a bound from above for the Lagrange multipliers relative to the high energy regime.
\medskip

{Step 2:} By Remark \ref{secep0} we can apply a result in \cite{suzD}, and conclude that there exists $\eps_*>0$ such that there exists a continuous map from $(0,\eps_*)$ to $C^{2}_0(\ov{\om})$: $\eps\mapsto u_{\eps}$ such that $\ue$ solves \rife{peps} and satisfies \rife{peps1} with $q=x_{\sscp \om}$. Obviously we have,
\beq\label{inj1}
\eps_1\neq \eps_2\quad \Rightarrow \quad u_{\scp \eps_1}\neq u_{\scp \eps_2}.
\eeq

By {Theorem 3.C}, and taking a smaller $\eps_*$ if necessary, then $\ue$ is non degenerate for any $\eps\in (0,\eps_*)$, that is, the linearized problem
relative to \rife{peps} has only the trivial solution. Therefore the implicit function theorem implies that the map $\eps\mapsto \ue$ is smooth in the
$C^{2}_0(\ov{\om})$ topology. Let,
$$
\lep=\eps^2 \ino \e{\ue},\;\eps\in (0,\eps_*),
$$
which therefore is smooth in $(0,\eps_*)$ as well.

\brm\label{secep}
{\it Since $\om$ is of second kind, then Remark \ref{secep0} and Theorem {3.A} imply that $D_{\sscp \om}(x_{\sscp \om})>0$ and so, as a consequence of  Remark \ref{seckind},
we see in particular that $\lm_{\eps}\to 8\pi^+$ as $\eps\searrow 0^+$. This is a crucial point which will play a major role in the rest of the proof.}
\erm

Next, we have the following,
\ble\label{lem1}
By taking a smaller $\eps_*$ if necessary, then the map $\eps\mapsto \lep$ is  injective.
\ele
\proof If not there exist $\eps_{n,i}\searrow 0^+$, $i=1,2$ such that, $\eps_{n,1}\neq\eps_{n,2}$, $\lm_n:=\lm_{\eps_{n,1}}\equiv \lm_{\eps_{n,2}}\to 8\pi$ and
$u_{n,i}:=u_{\scp \eps_{n,i}}$, $i=1,2$ satisfy,
\begin{equation*}
\begin{cases}
\Delta u_{n,i}+\lambda_n\dfrac{\dsp e^{u_{n,i}}}{\dsp \int_{\Omega}e^{u_{n,i}}}=0~&\mathrm{in}~\Omega,\\
u_{n,i}=0~&\mathrm{on}~\partial\Omega,\\
\lambda_n\dfrac{\dsp e^{u_{n,i}}}{\dsp \int_{\Omega}e^{u_{n,i}}}\rightharpoonup 8\pi\delta_{x_{\Omega}}~&\mathrm{as}~n\to+\infty.
\end{cases}
\end{equation*}
In view of  \rife{inj1} we have $u_{n,1}\neq u_{n,2}$. On the other side, by Remark \ref{secep}, we have that
$D_{\sscp \om}(x_{\sscp \om})>0$ and then {Theorem 3.B} yields the desired contradiction.
\finedim

\bigskip

At this point, since $\lep$ is continuous and injective in $(0,\eps_*)$, then it must be strictly monotone in $(0,\eps_*)$ and then, in view of Remark \ref{secep}, we conclude that $\lep$ is strictly increasing. Therefore, it is well defined $\lm_*=\lim_{\eps\to \eps_*^-} \lep$ which in particular is finite, $\lm_*\leq \ov{\lm}$, see step 1. Thus, it is well defined the inverse map $\lm\mapsto \eps_{\scp \lm}$, which is also continuous, strictly increasing and differentiable almost everywhere in $(8\pi,\lm_*)$. As a consequence, by defining,
\beq\label{branch}
\pl^{(\ii)}:=\frac1\lm u_{\scp \eps_{\sscp \lm}},\;\lm \in (8\pi,\lm_*),
\eeq
then the branch,
$$\mathcal{G}_\ii:=\{(\lm,\pl^{(\ii)}),\lm\in (8\pi,\lm_*)\},$$
is a continuous branch of solutions of $\prl$ which also satisfies,
\beq
\label{conc1}
\dfrac{\dsp \e{\lm \pl}}{\dsp \ino \e{\lm \pl}}\rightharpoonup \dt_{ x_{\sscp \sscp \om}},\;\mbox{ as }\lm \searrow 8\pi.
\eeq
Clearly, since $\ue$ is smooth in $\eps$ with respect to the $C^{2}_0(\om)$ topology, then we conclude that $\pl^{(\ii)}$
is differentiable almost everywhere as a function of $\lm$ in $(8\pi,\lm_*)$, with respect to the $C^{2}_0(\om)$ topology as well.
At this point we can prove that Theorem \ref{th1.1} implies the following fact:
\medskip

{$\mathbf{(i)}$}  By taking a smaller $\lm_*$ if necessary, then $\pl^{(\ii)}$ is smooth as a function of $\lm$ in $(8\pi,\lm_*)$ with respect to the $C^{2}_0(\om)$ topology.

Indeed, clearly if $\pl^{(\ii)}$ is not differentiable at some point $\lm$, then we must have at least one vanishing eigenvalue in the spectrum of \rife{1.8l}.
Therefore, if {$\mathbf{(i)}$} were not true, then we could find a bubbling sequence as $\lm_n\to 8\pi$ such that \rife{1.8l} admits a non trivial solution for any $n$,
which is a contradiction to Theorem \ref{th1.1}. In particular for any $\lm\in (8\pi,\lm_*)$ the linearized problem \rife{1.8l} has only the trivial solution, whence $\lm\mapsto \pl^{(\ii)}$ is smooth as a function of $\lm$ in $(8\pi,\lm_*)$.
\medskip

{Step 3:} We will use  {Theorem 3.B,} \rife{lmn8} in step 1, Remark \ref{secep} and  \rife{branch}, \rife{conc1} to identify
the stream functions corresponding to the entropy maximizing densities at fixed and large enough energy $E$.
More exactly we have,
\ble Let $\ple$ denote the stream function corresponding to any entropy maximizing density at fixed energy $E$, whose Lagrange multiplier is $\lm(E)$.
There exists $E_{*}>E_{8\pi}$ such that, for any
$E\geq E_{*}$,  there exists $\eps(E)\in (0,\eps_*)$ such that,
$$
\lm(E)=\lm_{\sscp \ii}(E):= \left.\lep\right|_{\eps=\eps(E)},\;
\ple=\psi^{(\ii)}_{\sscp \lm_{\sscp\ii}(E)}\equiv \left.\frac1\lep \ue\right|_{\eps=\eps(E)},\mbox{ and in particular } (\lm,\pl^{(\ii)})\in \mathcal{G}_\ii.
$$
\ele
\proof Since $\eps_{\sscp \lm}$, $\lm\in (8\pi,\lm_*)$ is an homeomorphism, then it is enough to check that, by taking a smaller $\lm_*$ if necessary,
there exists $E_*>E_{8\pi}$ such that, for any $E>E_*$, there exists $\lm\in (8\pi,\lm_*)$ such that
the stream function $\ple$ corresponding to any entropy maximizing density at fixed $E$ satisfies,
$$
\ple\equiv \pl^{(\ii)}=\frac1\lm u_{\scp \eps_{\sscp \lm}},\quad(\lm,\pl^{(\ii)})\in \mathcal{G}_\ii.
$$
By ${\bf (MVP1)}$ we know that $(\lm(E), \ple)$ exists for any $E>0$. We argue by contradiction and suppose that there exists $E_n\to +\ii$ and $\lm_n:=\lm(E_n)$ such that $\psi_n:=\psi_{\sscp \lm_n}$, $n\in \N$, are the stream
functions of some entropy maximizing densities at energies $E_n$ but $(\lm_n,\psi_n)\notin \mathcal{G}_\ii$ for any $n$.
In view of \rife{lmn8} in step 1 and Remark \ref{secep} we see that $\lm_n\to 8\pi^+$ and $v_n=\lm_n\psi_n$ satisfies \rife{blowup} with $q=x_{\sscp \om}$.
Therefore, for any $n$ large enough, we also find that $\lm_n\in (8\pi,\lm_*)$ and we denote by $u_n=\lm_n \psi^{(\ii)}_{\sscp \lm_n}$ the sequence of solutions
of \rife{1lm}, $(\lm_n,\psi^{(\ii)}_{\sscp \lm_n})\subset \mathcal{G}_\ii$, as defined in \rife{branch}, which in view of \rife{conc1} also satisfy \rife{blowup}.
Therefore we have found two distinct sequences of solutions of \rife{1lm} sharing
the same $\lm_n$ and satisfying \rife{blowup} which is a contradiction to {Theorem 3.B.}
\finedim

\medskip

Next we claim that:
\medskip

{$\mathbf{(ii)}$} By taking a {larger} $E_*$ if necessary, then $\lm_{\sscp\ii}(E)$ is  smooth, strictly decreasing in $[E_*,+\ii)$, and $\lm_{\sscp\ii}(E)\searrow 8\pi^+$ as $E\to +\ii$.\\

{\it Proof of $\mathbf{(ii)}$.} Since $\lm_\eps$ is smooth and strictly increasing in $(0,\eps_*)$ and $\lep\to 8\pi^+$ as $\eps\to 0^+$, then it is enough to show that the energy $E_{\scp \eps}$ is smooth as a function of $\eps$, with $\frac{d E_\eps}{d\eps}<0$ in $(0,\eps_*)$ and $E_{\scp \eps}\to +\ii$ as $\eps\searrow 0^+$. This is the content of the following,

\ble
\label{enrgeps}
By taking a smaller $\eps_*$ if necessary, then, for $\eps\in (0,\eps_*)$, is well defined the map,
$$E_{\scp \eps}:=\mathcal{E}(\mbox{\rm $\rh_{\sscp \lep}(\psi_{\scp \lep})$}),$$
which is also smooth and strictly decreasing, with $\frac{d E_\eps}{d\eps}\to -\ii$ and $E_{\scp \eps}\to +\ii$ as $\eps\searrow 0^+$.
\ele

\proof Let $\we=\frac{d\ue}{d\eps}$ which is well defined for $\eps\in (0,\eps_*)$ for some $\eps_*$ small enough by { Theorem 3.C.}
It is easy to check that,
\beq\label{ue1}
\begin{cases}
\Delta w_{(\eps)}+2\eps e^{u_{\eps}}+\eps^2e^{u_{\eps}}w_{(\eps)}=0~&\mathrm{in}~\Omega,\\
w_{(\eps)}=0~&\mathrm{on}~\partial\Omega,
\end{cases}
\eeq
and
\beq\label{ue2}
\frac{d\lep}{d\eps}=2\eps \ino \e{\ue}+\eps^2\ino \e{\ue}\we,
\eeq
and in particular that,
$$
E_\eps=\dfrac 12\dfrac{\eps^2\ino \e{\ue}\ue}{\lep^2},
$$
where we used
\begin{equation*}
u_{\eps}=\lambda_{\eps}\psi_{\eps}\quad\mathrm{and}\quad\lambda_{\eps}=\eps^2\int_{\Omega}e^{u_{\eps}}.
\end{equation*}
Multiplying the equation  in \rife{ue1} by $\ue$, integrating by parts and using \rife{peps} we find that,
$$
\eps^2 \ino \e{\ue} \we=2\eps \ino \e{\ue}\ue + \eps^2\ino \e{\ue}\we \ue,
$$
which readily implies that,
\beq\label{ue4}
\frac{d}{d\eps}\left(\eps^2\ino \e{\ue}\ue\right)=2\eps \ino \e{\ue}\ue + \eps^2\ino \e{\ue}\we \ue+\eps^2 \ino \e{\ue} \we=2\eps^2 \ino \e{\ue} \we.
\eeq
In view of \rife{ue2} and \rife{ue4} we find that,
\begin{align*}
\lep^4\frac{d E_\eps}{d\eps}
=~&\frac12 \frac{d}{d\eps}\left(\eps^2\ino \e{\ue}\ue\right)\lep^2-\lep \frac{d\lep}{d\eps}\eps^2\ino \e{\ue}\ue\\
=~&\left(\eps^2 \ino \e{\ue} \we\right)\lep^2-\lep \frac{d\lep}{d\eps}\eps^2\ino \e{\ue}\ue\\
=~&\left(\frac{d\lep}{d\eps}-\frac{2\lep}{\eps}\right)\lep^2-\lep \frac{d\lep}{d\eps}\eps^2\ino \e{\ue}\ue
=\frac{d\lep}{d\eps}\left( \lep^2- \lep \eps^2\ino \e{\ue}\ue\right)-\frac{2\lep^3}{\eps}.
\end{align*}
At this point, by using the fact that $\frac{d\lep}{d\eps}\geq 0$,  $\eps^2\ino \e{\ue}\ue\to +\ii$ and that $\lep\to 8\pi$ as $\eps\to 0^+$,
then we conclude by the last equality that $\frac{d E_\eps}{d\eps}\to -\ii$ as $\eps\to 0^+$.
The fact that $\eps^2\ino \e{\ue}\ue\to +\ii$ is a straightforward consequence of well known blow up arguments, see for example
\cite{yy}, which however could be easily deduced from \rife{2.5} and Theorem 2.A. Clearly {\it $\mathbf{(i)}$} and {\it $\mathbf{(ii)}$} imply that $\lm_{\sscp \ii}(E)$ and $\psi^{(\ii)}_{\sscp \lm_{\sscp\ii}(E)}$ satisfy all the properties needed in $[E_*,+\ii)$ and in $(8\pi,\lm_{\sscp \ii}(E_*))$ respectively, which concludes the proof of $\mathbf{(a)}$.
\medskip

The conclusion $\mathbf{(b)}$ is a straightforward consequence of $\mathbf{(a)}$ and \rife{diff}.
\finedim
\medskip

\section{Appendix: the proof of $b_0=b_{j,i}=0$}\label{appendix}
In this section, we are going to show $b_0=b_{j,i}=0$ by making use of the Pohozaev type identities \eqref{4.3} and \eqref{4.47}.
The argument is an adaptation of those in \cite{ly2} and \cite{BJLY}. Therefore we shall discuss the main steps and refer the readers to \cite{ly2,BJLY}
for further details.
\medskip

To analyse the various terms in \eqref{4.3} and \eqref{4.47} we need a delicate estimate about $\nabla v_{n,j}$ and $\nabla\zeta_n$. Let
\begin{equation}
\label{4.5}
\widetilde{G}(x)=\frac{\lambda_n}{m}\sum_{l=1}^m G(x,x_{n,l}){.}
\end{equation}{
In view of Theorem 2.A, it is not difficult to see that,}
\begin{equation}
\label{4.7}
\begin{aligned}
\nabla v_{n,j}=\nabla(\widetilde{G}-\phi_{n,j})+o(e^{-\frac{\mu_{n,j}}{2}})
{=-4\frac{x-x_{n,j}}{|x-x_{n,j}|^2}+ o(e^{-\frac{\mu_{n,j}}{2}})}~\mathrm{in}~\bigcup_{l=1}^mB_{2r_0}({x}_{n,l})\setminus B_{r/2}({x}_{n,l}).
\end{aligned}
\end{equation}
{For the estimate of $\nabla\zeta_n$ we will apply the Green representation formula and a suitable scaling.
It is convenient to introduce the following notations,}
\begin{equation}
\label{3.6}
\Lambda_{n,j,r}^{-}=re^{-\mu_{n,j}/2},\quad \Lambda_{n,j,r}^{+}=re^{\mu_{n,j}/2},
\ \ \textrm{for any}\ \  r>0, \end{equation}
and
\begin{equation}
\label{sc}\overline{f}(z)=f(e^{-\frac{\mu_{n,j}}{2}}z+x_{n,j}),~|z|<r_0 e^{\frac{\mu_{n,j}}{2}}
~\mbox{for any function}~f:B_{r_0}({x}_{n,j})\to\mathbb{R}.
\end{equation}
\medskip

\begin{lemma}
\label{le3.2}
\begin{equation}
\label{3.7}
\begin{aligned}
\zeta_n(x)-\zeta_{n,0}=&\sum_{j=1}^mA_{n,j}G(x_{n,j},x)+\sum_{j=1}^m\sum_{h=1}^2B_{n,j,h}\partial_{y_h}G(y,x)|_{y=x_{n,j}}
+o(e^{-\frac{1}{2}\mu_{n,1}}),
\end{aligned}
\end{equation}
holds in $C^1(\ov{\om}\setminus\cup_{j=1}^m B_{\delta}(x_{n,j}))$,
for a suitably defined positive constant $\delta>0$, where $\zeta_{n,0}=\zeta_n|_{\partial\Omega}$,
$\partial_{y_h} G({y},{x})=\frac{\partial G({y},{x})}{\partial {y}_h}$, $y=(y_1,y_2)$,
\begin{equation*}
A_{n,j}=\int_{\Omega_j}f_n^*(y)\mathrm{d}y,
\quad\mbox{and}\quad
B_{n,j,h}=e^{-\frac{1}{2}\mu_{n,j}}\frac{b_{j,h}4\sqrt{8}}{\sqrt{\lambda_n h(q_j)}}
\int_{\mathbb{R}^2}\frac{|z|^2}{(1+|z|^2)^3}\mathrm{d}z.
\end{equation*}
Moreover, there exists a constant $C>0$ such that,
\begin{equation}
\label{3.8}
\begin{aligned}
\left|\zeta_n(x)-\zeta_{n,0}-\sum_{j=1}^m A_{n,j}G(x_{n,j},x)\right| \leq {C}\sum_{j=1}^m e^{-\frac{\mu_{n,j}}{2}}\frac{1}{|x-x_{n,j}|},
\end{aligned}
\end{equation}
for $x\in \Omega\setminus\bigcup_{j=1}^m B_{\Lambda_{n,j,R}^{-}}(x_{n,j})$, for any $R>0$ (where $C$ do not depend by $R$).
\end{lemma}

\begin{proof}{The proof of Lemma \ref{le3.2} is based on the Green representation formula}
\begin{equation}
\label{3.9}
\begin{aligned}
\zeta_n(x)-\zeta_{n,0}&=\int_{\Omega}G(y,x)f_n^*(y)\mathrm{d}y\\
&=\sum_{j=1}^m A_{n,j}G(x_{n,j},x)+\sum_{j=1}^m\int_{\Omega_j}(G(y,x)-G(x_{n,j},x))f_n^*(y)\mathrm{d}y{,}
\end{aligned}
\end{equation}
and the estimates of $\tilde{u}_n$ in Theorem 2.A. We refer to \cite[Lemma 2.3]{ly2} and  \cite[Lemma 3.3]{BJLY} for further details. \end{proof}

\begin{lemma}
\label{le4.2}
\begin{equation*}
\begin{aligned}
\mathrm{L.H.S.~of}~\eqref{4.3}
=& -4A_{n,j}-\frac{256b_0e^{-\mu_{n,1}}h(q_j)e^{G_j^*(q_j)}}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}
\int_{\Omega_j\setminus B_{r}(q_j)}e^{\Phi_j(x,\mathbf{q})}\mathrm{d}x\\ & +o(e^{-\frac{\mu_{n,j}}{2}}\sum_{l=1}^m|A_{n,l}|)+o(e^{-\mu_{n,j}}),
\end{aligned}
\end{equation*}
for fixed $r\in (0,r_0)$, with $r_0$ as defined right below \eqref{1.4}.
\end{lemma}
\begin{proof} We shall first derive a refined estimate about  $\nabla\zeta_n$. By the Green representation formula, we see that,
\begin{equation}
\label{4.10}
\begin{aligned}
\zeta_n(x)-\zeta_{n,0}
=\int_{\Omega}G(y,x)f_n^*(y)\mathrm{d}y
&=\sum_{l=1}^mA_{n,l}G(x_{n,l},x)+\sum_{l=1}^m\sum_{h=1}^2B_{n,l,h}\left(\partial_{y_h}G(y,x)|_{y=x_{n,l}}\right)\\
&\quad+\frac{1}{2}\sum_{l=1}^m\sum_{h,k=1}^2C_{n,l,h,k}\left(\partial^2_{y_hy_k}G(y,x)|_{y=x_{n,l}}\right)\\
&\quad+\sum_{l=1}^m\int_{\Omega_l}\Psi_{n,l}(y,x)f_n^*(y)\mathrm{d}y,
\end{aligned}
\end{equation}
where
\begin{align*}
A_{n,l}=&\int_{\Omega_{l}}f_n^*(y)\mathrm{d}y,~B_{n,l,h}=\int_{B_{r_0}(x_{n,l})}(y-x_{n,l})_h f_n^*(y)\mathrm{d}{y},
\\\quad &{C_{n,l,h,k}}=\int_{B_{r_0}(x_{n,l})}(y-x_{n,l})_h(y-x_{n,l})_{k}f_n^*({y})\mathrm{d}{y},
\end{align*}
and
\begin{align*}
\Psi_{n,l}(y,x)=~&G(y,x)-G(x_{n,l},x)-\langle\partial_{y}G(y,x)|_{y=x_{n,l}},y-x_{n,l}\rangle\\
&-\frac{1}{2}\langle\partial^2_yG({y},{x})|_{y=x_{n,l}}(y-x_{n,l}),y-x_{n,l}\rangle.
\end{align*}

Fix $\bar r\in(0,\frac{r}{2})$. By using \eqref{farblow} and Theorem 2.A, we see that,
\begin{equation}
\begin{aligned}
\label{4.13}
f_n^*{(y)}=\frac{64 e^{-\mu_{n,j}}}{\lambda_nh(x_{n,j})}{e^{\Phi_j(y,\mathbf{x}_n)}}(-b_0+o(1))~\mathrm{for}~
y\in\ov{\om_j}\setminus B_{\bar r}(x_{n,j}),
\end{aligned}
\end{equation}
where $\mathbf{x}_n=(x_{n,1},\cdots,x_{n,m})$ and
$$\Phi_j(y,\mathbf{x}_n)=\sum_{l=1}^m 8\pi G(y,x_{n,l})-G_j^*(x_{n,j})+\log h(y)-\log h(x_{n,j}).$$

Let us define,
\begin{equation}
\label{4.16}
\begin{aligned}
\overline{G}_n(x)=~&\zeta_{n,0}+\sum_{l=1}^mA_{n,l}G(x_{n,l},x)
+\sum_{l=1}^m\sum_{h=1}^2B_{n,l,h}\partial_{y_h}G(y,x)|_{y=x_{n,l}}\\
&+\frac{1}{2}\sum_{l=1}^m\sum_{h,k=1}^2C_{n,l,h,k}\partial^2_{y_hy_k}G(y,x)|_{y=x_{n,l}},
\end{aligned}
\end{equation}
and
\begin{equation}
\label{4.18}
\zeta_n^*(x)=-b_0\sum_{l=1}^m\int_{\Omega_l\setminus B_{\bar r}(x_{n,l})}
\frac{64e^{-\mu_{n,l}}\Psi_{n,l}(y,x)}{\lambda_nh(x_{n,l})}e^{\Phi_l({y},\mathbf{x}_n)}\mathrm{d}y.
\end{equation}
By  \eqref{4.10} and \eqref{4.13}, we conclude that for $x\in\partial B_r(x_{n,j})$, it holds,
\begin{equation}
\label{4.17}
\begin{aligned}
 \zeta_n(x)-\overline{G}_n(x)
 &{
=} \zeta_n^*(x)
+O\left(\frac{\bar re^{-\mu_{n,j}}}{|x-x_{n,j}|^3}\right)  +o(e^{-\mu_{n,j}})~\mathrm{in}~C^1(\partial B_r(x_{n,j})).
\end{aligned}\end{equation}
Substituting \eqref{4.7} and \eqref{4.17} into \eqref{4.3}, we see that,
\begin{equation}
\label{4.19}
\begin{aligned}
\mathrm{L.H.S.~of}~\eqref{4.3}=~&\int_{\partial B_r(x_{n,j})}4\langle\nu,D(\overline{G}_n+\zeta_n^*)(x)\rangle\mathrm{d}\sigma(x)
+o(e^{-\frac{\mu_{n,j}}{2}}\sum_{l=1}^m|A_{n,l}|)\\
&+O\left(\frac{{\bar{r}}e^{-\mu_{n,j}}}{r^3}\right)+o(e^{-\mu_{n,j}}),
\end{aligned}
\end{equation}
for any $\bar r\in(0,\frac{r}{2}).$ The estimate about the right hand side of \eqref{4.19} depends on the following identity
\begin{equation}
\label{4.20}
\begin{aligned}
&\Delta u\left(\nabla v\cdot(x-x_{n,j})\right)+\Delta v\left(\nabla u\cdot(x-x_{n,j})\right)\\
&=\mathrm{div}\left(\nabla u(\nabla v\cdot(x-x_{n,j}))
+\nabla v(\nabla u\cdot(x-x_{n,j}))-\nabla u\cdot\nabla v(x-x_{n,j})\right).
\end{aligned}
\end{equation}
Letting $u=\overline{G}_n$ and $v=\widetilde{G}-\phi_{n,j}$ (resp. $u=\Psi_{n,j}(y,\cdot)$ and $v=\widetilde{G}-\phi_{n,j}$),
then we are able to compute the right hand side of \eqref{4.19}
$\int_{\partial B_r(x_{n,j})} \langle\nu,D \overline{G}_n \rangle\mathrm{d}\sigma $
(resp. $\int_{\partial B_r(x_{n,j})} \langle\nu,D \zeta_n^* \rangle\mathrm{d}\sigma$). See \cite{ly2} and \cite[Lemma 4.2]{BJLY} for further details.
\end{proof}

The following result is needed to estimate the right hand side of \eqref{4.3}.

\begin{lemma}
\label{le4.3}
\begin{equation*}
\begin{aligned}
(i)~\int_{\partial B_r(x_{n,j})}rf_n^*\mathrm{d}\sigma
=&-\frac{32\pi e^{-\mu_{n,1}}b_0h(q_j)e^{G_j^*(q_j)}(\Delta\log h(q_j))}{\lambda_n(h({q}_1))^2e^{G_1^*(q_1)}}\\
&-\frac{128e^{-\mu_{n,1}}b_0h(q_j)e^{G_j^*(q_j)}}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}\frac{\pi}{r^2}
+O(re^{-\mu_{n,1}})+\frac{o(e^{-\mu_{n,1}})}{r^2}.
\end{aligned}
\end{equation*}
\begin{equation*}
(ii)~\sum_{j=1}^m\int_{B_r(x_{n,j})}f_n^*\mathrm{d}x
=\frac{64b_0e^{-\mu_{n,1}}}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}\sum_{j=1}^m
\int_{\Omega_j\setminus B_r(q_j)}h(q_j)e^{G_j^*(q_j)}e^{\Phi_j(x,\mathbf{q})}\mathrm{d}x+o(e^{-\mu_{n,1}}).
\end{equation*}
\begin{equation*}
\begin{aligned}
(iii)~&\int_{B_r(x_{n,j})}f_n^*\langle D(\log h+\phi_{n,j}),x-x_{n,j}\rangle\mathrm{d}{x}\\
&=\Bigg[\frac{32\pi\zeta_{n,0}(\Delta\log h(q_j))}{\lambda_n(h(q_1))^2e^{G_{1}^*(q_1)}}h(q_j)e^{G_j^*(q_j)}e^{-\mu_{n,1}}\Bigg(\mu_{n,1}
+\log\Big(\frac{\lambda_n(h({q}_1))^2e^{G_1^*(q_1)}}{8h(q_j)e^{G_j^*(q_j)}}r^2\Big)-2\Bigg)\Bigg]\\
&\quad+O(1)(\frac{|\log r|e^{-\mu_{n,j}}}{(\mu_{n,j})^2})+O(re^{-\mu_{n,j}})+o(e^{-\mu_{n,j}})|\log R|+O\left(\frac{e^{-2\mu_{n,j}}}{r^2}\right)+O\left(\frac{e^{-\mu_{n,j}}}{R^2}\right)\\
&\quad+O(1)\Bigg(\sum_{l=1}^m(|A_{n,l}|+e^{-\frac{\mu_{n,j}}{2}})\Big(\frac{e^{-\frac{\mu_{n,j}}{2}}}{R}+e^{-\mu_{n,j}}
(\mu_{n,j}+|\log r|)\Big)\Bigg)~\mbox{for any}~R>1,
\end{aligned}
\end{equation*}
where $O(1)$  denotes any quantity uniformly bounded with respect to $r, R$ and $n$.
\end{lemma}
\begin{proof} The proof of Lemma \ref{le4.3} is based on Theorem 2.A  and  $\int_{\Omega}f^*_ndy=0$ as in \cite{ly2,BJLY}.
We sketch the proof of $(iii)$ for reader's convenience. By Theorem 2.A, the expansion of $D(\log h+\phi_{n,j})$,
and  the scaling $x=e^{-\frac{\mu_{n,j}}{2}}z+x_{n,j}$,   we see that,
\begin{equation}
\label{4.39}
\begin{aligned}
&\int_{B_r(x_{n,j})}f_n^*\langle D(\log h+\phi_{n,j}),x-x_{n,j}\rangle\d x\\
&=\int_{B_{\Lambda_{n,j,r}^{+}}(0)}\frac{\lambda_n h(x_{n,j})\overline{\zeta_n}}{(1+\frac{\lambda_nh(x_{n,j})}{8}|z|^2)^2}
\langle D^2(\log h+\phi_{n,j})(x_{n,j})z,z\rangle e^{-\mu_{n,j}}\d z\\
&\quad+O(1)\left(\frac{|\log r|e^{-\mu_{n,j}}}{(\mu_{n,j})^2}\right)+O(re^{-\mu_{n,j}})+o(e^{-\mu_{n,j}}).
\end{aligned}\end{equation}
At this point, by using the following identity $\int\frac{(1-r^2)r^3}{(1+r^2)^3}\mathrm{d}r=\frac{1}{2}\left(\frac{-3r^2-2}{(1+r^2)^2}-\log(1+r^2)\right)+C,$
together with \eqref{nearblow} and \eqref{farblow}, then for any fixed and large $R>0$ we see that,
\begin{align*}
&\int_{B_R(0)}\frac{\lambda_nh(x_{n,j})\overline{\zeta_n}}{(1+\frac{\lambda_nh(x_{n,j})}{8}|z|^2)^2}
\langle D^2(\log h+\phi_{n,j})(x_{n,j})z,z\rangle e^{-\mu_{n,j}}\d z\\
&=\frac{32\pi b_0\Delta(\log h+\phi_{n,j})(x_{n,j})}{\lambda_n h(x_{n,j})}e^{-\mu_{n,j}}
\Big(\frac{\frac{\lambda_n h(x_{n,j})}{8}R^2(1+\frac{\lambda_n h(x_{n,j})}{4}R^2)}{(1+\frac{\lambda_n h(x_{n,j})}{8}R^2)^2}
-\log(1+\frac{\lambda_nh(x_{n,j})}{8}R^2)\Big)\\
&\quad+o(e^{-\mu_{n,j}})|\log R|.
\end{align*}
Next, let us observe that,
\begin{equation}
\label{4.40}
\int\frac{r^3}{(1+r^2)^2}\d r=\frac{1}{2}\Big(\frac{1}{1+r^2}+\log(1+r^2)\Big)+C.
\end{equation}
In view of \eqref{3.8}, we also see that if $|z|\ge R$ then it holds,
\begin{equation}
\label{4.41}
\overline{\zeta_n}(z)=\zeta_{n,0}+O(1)\Big(\sum_{l=1}^m(|A_{n,l}|+e^{-\frac{\mu_{n,j}}{2}})(\frac{e^{\frac{\mu_{n,j}}{2}}}{|z|}+1)\Big).
\end{equation}
Since $\zeta_{n,0}$ is constant, we also conclude from \eqref{4.40} and \eqref{4.41} that,
{\allowdisplaybreaks
\begin{align*}
&\int_{B_{\Lambda_{n,j,r}^{+}}(0)\setminus B_R(0)}\frac{\lambda_nh(x_{n,j})\overline{\zeta_n}}{(1+\frac{\lambda_nh(x_{n,j})}{8}|z|^2)^2}
\langle D^2(\log h+\phi_{n,j})(x_{n,j})z,z\rangle e^{-\mu_{n,j}}\d z
\\
=&-\left[\frac{1}{1+\frac{\lambda_nh(x_{n,j})}{8}R^2}+\log(1+\frac{\lambda_nh(x_{n,j})}{8}R^2)
-\mu_{n,j}-\log(\frac{\lambda_nh(x_{n,j})}{8}r^2)\right]\\
&\times\frac{32\pi\zeta_{n,0}\Delta(\log h+\phi_{n,j})(x_{n,j})}{\lambda_nh(x_{n,j})}e^{-\mu_{n,j}}+O(\frac{e^{-2\mu_{n,j}}}{r^2})\\&+O(1)\sum_{l=1}^m(|A_{n,l}|+e^{-\frac{\mu_{n,j}}{2}})\Big(\frac{e^{-\frac{\mu_{n,j}}{2}}}{R}+e^{-\mu_{n,j}}(\mu_{n,j}+|\log r|)\Big).
\end{align*}}
By  \eqref{farblow}, it is easy to check that,
\begin{equation}
\label{4.43}
\zeta_{n,0}=-b_0+o(1).
\end{equation}
The estimates \eqref{4.39}-\eqref{4.43} used together with Theorem 2.A conclude the proof of Lemma~\ref{le4.3}-(iii).
\end{proof}

Now we are able to prove that $b_{j,0}=0$ for all $j=1,\cdots,m$.

\begin{lemma}
\label{le4.4}$ $\\
$(i)$ $A_{n,j}=\int_{\Omega_j}f_n^*(y)\d y= o(e^{-\frac{\mu_{n,j}}{2}})$.\\
$(ii)$ $b_0=0$ and in particular $b_{j,0}=0$, $j=1,\cdots,m$.
\end{lemma}
\begin{proof}
$(i)$ This is an immediate consequence of Theorem 2.A and Lemmas \ref{le4.2}-\ref{le4.3}.

\noindent $(ii)$  For any $r>0$, let us set,
\begin{equation}
\label{4.45}
r_j=r\sqrt{8h(q_j)G_j(q_j)}\quad \mathrm{for}\quad j=1,\cdots,m.
\end{equation}
Note that $\sum_{j=1}^m A_{n,j}=0$. This fact, together with the Pohozaev type indentity \eqref{4.3}, Lemmas \ref{le4.2}-\ref{le4.3}, and $(i)$, implies that,
\begin{equation}
\label{4.46}
\begin{aligned}
&-\frac{256b_0e^{-\mu_{n,1}}}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}\sum_{j=1}^mh(q_j)e^{G_j^*(q_j)}
\int_{\Omega_j\setminus B_{r_j}(q_j)}e^{\Phi_j(y,\mathbf{q})}\d y\\
&=-\frac{128e^{-\mu_{n,1}}b_0}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}\sum_{j=1}^mh(q_j)e^{G_j^*(q_j)}\frac{\pi}{r_j^2}
-\frac{32\pi e^{-\mu_{n,1}}b_0\ell(\mathbf{q})}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}\\
&\quad-\frac{128e^{-\mu_{n,1}}b_0}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}\sum_{j=1}^mh(q_j)e^{G_j^*(q_j)}
\int_{\Omega_j\setminus B_{r_j}(q_j)}e^{\Phi_j(y,\mathbf{q})}\d y\\
&\quad-\frac{32\pi\zeta_{n,0}\ell(\mathbf{q})e^{-\mu_{n,1}}}{\lambda_n(h(q_1))^2e^{G_1^*(q_1)}}
\Big(\mu_{n,1}+\log\Big(\lambda_n(h(q_1))^2e^{{G_1^*}(q_1)}r^2\Big)-2\Big)\\
&\quad+O(e^{-\mu_{n,1}})(r+R^{-1})+o(e^{-\mu_{n,1}})(\frac{1}{r^2}+\log R).
\end{aligned}
\end{equation}
If  either $\ell(\mathbf{q})\neq0$ or $D(\mathbf{q})\neq0$, then \eqref{4.46} implies  $b_0=0$.
In view of \eqref{nearblow},  we also obtain $b_{j,0}=b_0=0$, $j=1,\cdots, m$. This fact concludes the proof of $(ii)$.
\end{proof}

Next, by using \eqref{4.47}, we shall  prove that $b_{j,1}=b_{j,2}=0$.
\begin{lemma}
\label{le4.6}
\begin{equation*}
\mathrm{R.H.S.~of}~\eqref{4.47}=\tilde{B}_j\Big(\sum_{h=1}^2D^2_{hi}(\phi_{n,j}+\log h)(x_{n,j})e^{-\frac{\mu_{n,j}}{2}}b_{j,h}\Big)
+o(e^{-\frac{\mu_{n,j}}{2}}),
\end{equation*}
where $$\tilde{B}_j=4\sqrt{\frac{8}{\lambda_nh(x_{n,j})}}\int_{\mathbb{R}^2}\frac{|z|^2}{(1+|z|^2)^3}\mathrm{d} z.$$
\end{lemma}
\begin{proof} Since $\mathbf{q}$ is a critical point of $f_m$, then by using Theorem 2.A,  we find that,
\begin{equation}
\label{4.50}
\begin{aligned}
D_i(\phi_{n,j}+\log h)(x_{n,j})=D_i(G_j^*+\log h)(x_{n,j})+O(\mu_{n,j}e^{-\mu_{n,j}})
=O(\mu_{n,j}e^{-\mu_{n,j}}).
\end{aligned}
\end{equation}
As a consequence, in view of the blow up profile of $\tilde{u}_n$ in Theorem 2.A and \eqref{nearblow}, we conclude that,
\begin{equation}
\label{4.48}
\begin{aligned}
&\mathrm{R.H.S.~of}~\eqref{4.47}\\&=-\int_{\partial B_r(x_{n,j})}\lambda_nh(x)e^{\tilde{u}_n}\zeta_n\frac{(x-x_{n,j})_i}{|x-x_{n,j}|}\mathrm{d}\sigma+\int_{B_r(x_{n,j})}\lambda_nh(x)e^{\tilde{u}_n}\zeta_nD_i(\phi_{n,j}+\log h)\d x
\\
&=4\sqrt{\frac{8}{\lambda_nh(x_{n,j})}}\int_{\mathbb{R}^2}\frac{|z|^2}{(1+|z|^2)^3}\d z\Big( \sum_{h=1}^2D^2_{hi}(\phi_{n,j}+\log h)(x_{n,j})e^{-\frac{\mu_{n,j}}{2}}b_{j,h}\Big)+o(e^{-\frac{\mu_{n,j}}{2}}),
\end{aligned}
\end{equation}which implies that the conclusion of Lemma \ref{le4.6} holds.
\end{proof}

\begin{lemma}
\label{le4.7}
\begin{equation*}
\begin{aligned}
\mathrm{L.H.S.~of}~\eqref{4.47}
=&-8\pi\Bigg[\sum_{l\neq j}e^{-\frac{\mu_{n,l}}{2}}D_iG_{n,l}^*(x_{n,j})+ e^{-\frac{\mu_{n,j}}{2}}D_i\sum_{h=1}^2\partial_{y_h}R(y,x)|_{x=y=x_{n,j}}b_{j,h}\tilde B_j\Bigg]\\
&+o(e^{-\frac{\mu_{n,j}}{2}}),
\end{aligned}
\end{equation*}
where $G_{n,l}^*(x)=\sum_{h=1}^2{\partial_{y_h}}G(y,x)|_{y=x_{n,l}}b_{l,h}\tilde{B}_l.$
\end{lemma}
\begin{proof}
By the definition of $G_{n,i}^*$, we have for any $\theta\in(0,r)$,
$$\Delta G_{n,l}^*=0~\mathrm{in}~B_r(x_{n,j})\setminus B_{\theta}(x_{n,j}).$$
Then for $x\in B_r(x_{n,j})\setminus B_{\theta}(x_{n,j})$, and setting $e_i=\frac{x_i}{|x|}$, $i=1,2$, we have,
\begin{equation*}
\begin{aligned}
0=~&\Delta G_{n,l}^*D_i\log\frac{1}{|x-x_{n,j}|}+\Delta\log\frac{1}{|x-x_{n,j}|}D_iG_{n,l}^*\\
=~&\mathrm{div}\Bigg(\nabla G_{n,l}^*D_i\log\frac{1}{|x-x_{n,j}|}+\nabla\log\frac{1}{|x-x_{n,j}|}D_iG_{n,l}^*
-\nabla G_{n,l}^*\cdot\nabla\log\frac{1}{|x-x_{n,j}|}e_i\Bigg),
\end{aligned}
\end{equation*}
which readily implies that,
\begin{equation}
\label{4.52}
\begin{aligned}
\int_{\partial B_r(x_{n,j})}\frac{\nabla_iG_{n,l}^*}{|x-x_{n,j}|}\d\sigma
&=\int_{\partial B_\theta(x_{n,j})}\frac{\nabla_iG_{n,l}^*}{|x-x_{n,j}|}\mathrm{d}\sigma.
\end{aligned}
\end{equation}
In view of  Lemma \ref{le3.2} and Lemma \ref{le4.4}, we also have,
\begin{equation}
\label{4.53}
\zeta_n(x)-\zeta_{n,0}=\sum_{l=1}^me^{-\frac{\mu_{n,l}}{2}}G_{n,l}^*(x) +o(e^{-\frac{\mu_{n,j}}{2}})\quad\mathrm{in}\quad C^1(B_r(x_{n,j})\setminus B_{\theta}(x_{n,j})).
\end{equation}
By using $D_iD_h(\log|z|)=\frac{\delta_{ih}|z|^2-2z_iz_h}{|z|^4},$ we see that,
\begin{equation*}
\begin{aligned}
\int_{\partial B_{\theta}(x_{n,j})}\frac{\nabla_iG_{n,j}^*}{|x-x_{n,j}|}\mathrm{d}\sigma
=2\pi D_i\sum_{h=1}^2\partial_{y_h}R(y,x)|_{x=y=x_{n,j}}b_{j,h}\tilde{B}_j+o_\theta(1).
\end{aligned}
\end{equation*}
This fact, together with \eqref{4.52}-\eqref{4.53} and  the estimate about $\nabla v_{n,j}$ in \eqref{4.7},  we see that, for any $\theta\in(0,r)$,
\begin{equation*}
\begin{aligned}
\mathrm{L.H.S.~ of}~\eqref{4.47}=&-4\int_{\partial B_r(x_{n,j})}
\sum_{l=1}^me^{-\frac{\mu_{n,l}}{2}}\frac{\nabla_iG_{n,l}^*}{|x-x_{n,j}|}\d\sigma+o(e^{-\frac{\mu_{n,j}}{2}})\\
=&-8\pi\Bigg[\sum_{l\neq j} e^{-\frac{\mu_{n,l}}{2}}D_iG_{n,l}^*(x_{n,j})
+e^{-\frac{\mu_{n,j}}{2}}D_i\sum_{h=1}^2\partial_{y_h}R({y},{x})|_{x=y=x_{n,j}}b_{j,h}\tilde{B}_j\Bigg]\\
&+o(e^{-\frac{\mu_{n,j}}{2}})+o_\theta(1) e^{-\frac{\mu_{n,j}}{2}},
\end{aligned}
\end{equation*}
which proves the claim of Lemma \ref{le4.7}.
\end{proof}

Finally, we have the following,
\begin{lemma}
\label{le4.8}
$b_{j,1}=b_{j,2}=0$, $j=1,\cdots, m$.
\end{lemma}

\begin{proof}
From the Pohozaev type indentitly \eqref{4.47} and Lemma \ref{le4.6}-Lemma \ref{le4.7}, we have, for $i=1,2$,
\begin{equation}
\label{4.55}
\begin{aligned}
&\tilde{B}_j\sum_{h=1}^2(D_{hi}^2(\phi_{n,j}+\log h)(x_{n,j})b_{j,h})e^{-\frac{\mu_{n,j}}{2}}\\
&=-8\pi\Bigg[\sum_{l\neq j}e^{-\frac{\mu_{n,l}}{2}}D_iG_{n,l}^*(x_{n,j})
+e^{-\frac{\mu_{n,j}}{2}}D_i\sum_{h=1}^2\partial_{y_h}R(y,x)|_{y=y=x_{n,j}}b_{j,h}\tilde{B}_j\Bigg]+o(e^{-\frac{\mu_{n,j}}{2}})\\
&=-8\pi\sum_{l\neq j} e^{-\frac{\mu_{n,l}}{2}}\sum_{h=1}^2D_{x_i}\partial_{y_h}G(y,x)|_{(y,x)=(x_{n,j},x_{n,l})}b_{lh}\tilde{B}_l\\
&\quad-8\pi e^{-\frac{\mu_{n,j}}{2}}\sum_{h=1}^2D_{x_i}\partial_{y_h}R(y,x)|_{x=y=x_{n,j}}b_{j,h}\tilde{B}_j+o(e^{-\frac{\mu_{n,j}}{2}}).
\end{aligned}
\end{equation}
Set $\vec{b}=({\hat{b}}_{1,1}\tilde{B}_1,{\hat{b}}_{1,2}\tilde{B}_1,\cdots,{\hat{b}}_{m,1}\tilde{B}_m,{\hat{b}}_{m,2}\tilde{B}_m)$,
where $\hat{b}_{lh}=\lim_{n\to+\infty}(e^{\frac{\mu_{n,j}-\mu_{n,l}}{2}}b_{lh})$.
Then, by using Theorem 2.A and passing to the limit as $n\to +\infty$, we conclude from \eqref{4.55} that,
$$
D^2f_m({q}_1,{q}_2,\cdots,{q}_m) \cdot \vec{b}= 0.
$$
By the non degeneracy assumption $\mathrm{det}(D_{\Omega}^2f_m(\mathbf{q}))\neq 0$, we  conclude $b_{j,1}=b_{j,2}=0$, $j=1,\cdots, m$.
\end{proof}
\bigskip

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