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Southampton's first workhouse came into being as a result of a bequest by one John Major who, in 1629, left £200 to provide "a house of twelve rooms for the habitation of poor people". In the next year, his executor arranged with the Corporation, that on the latter building a poor house, the legacy should be devoted to the purchase of land for the poor to work on. Two years later, premises were provided, but by 1668 the buildings had fallen into a bad state. The poorhouse transferred to what became St John's Hospital (founded in 1671 for the instruction of lads in the woollen trade,) on French Street.
In 1753, under the provisions of Knatchbull's Act of 1723, Southampton parishes set up a common workhouse at Bull, or Bugle Hall, a former residence of the Earl of Southampton, that ran from the lower part of Bugle Street towards Westgate.
In 1771, a proposal was instigated to unite (by special Act of Parliament) Southampton's parishes into a poor law Incorporation. This would allow more local autonomy over the management of poor relief, such as the profitable employment of the poor in textile manufacture. An Act for better regulating the Poor, and repairing the Highways, within the Town and County of the Town of Southampton (13 Geo.3. c.50) was passed the following year. A board of guardians was formed which included the Mayor, recorder, bailiffs, senior aldermen, any other resident justices of the peace, and eighteen parish representatives — four each from All Saints, Holy Rood and St Michael's, and two each from St John's, St Lawrence's and St Mary's.
In 1776, the Incorporation decided to build a new workhouse on the north side of St Mary's churchyard. The old St John's Hospital was sold off for £425, and several years later became the site of a theatre.
Eden, in 1797, reported on poor-relief in Southampton:
SOUTHAMPTON contains about 4 square miles, one-fourth of which is common, and about 9,000 inhabitants. There is a mill for twisting silk which employs a few children, and a carpet manufacture, which works 4 or 5 looms, but no other manufacture. The number of houses paying tax could not be estimated, but it may be estimated at 700. Prices of provisions: Beef, 6d. to 8d. per lb. ; mutton, 6d. to 7d. ; veal, 8d. ; bacon, 10½d. ; butter, 15½d. ; bread, 1s. 11½d. the half-peck loaf; potatoes, 10d. a peck; coals, 37s. a chaldron. Wages of bricklayers, carpenters, etc., 15s. to 16s. a week. Common labourers, 1s. 8d. to 2s. a day. Number of ale-houses, 30. One friendly society. The Poor are partly maintained in a Workhouse, partly at their own homes. At present (October, 1795) 120 persons in the house, of whom 50 are children under 14 ; the rest mostly old people. 18 children and 4 old men work at the silk mill.
Bill of fare in the Workhouse: Breakfast—every day except Monday, onion broth; Monday, bread and butter. Dinner—Sunday, beef and vegetables; Monday, broth and bread; Tuesday, pork and vegetables ; Wednesday, Friday, bread and butter; Thursday, salt beef and vegetables; Saturday, bread and cheese. Supper—every day, bread and cheese or butter. Two cups of beer (near 3 pints) are allowed to grown persons, and a proportionate quantity to children. Rates, for the last 18 years, on an average 4s. in the pound on ¾ of the net rental.
Southampton's Local Act Incorporation status mad it largely exempt from the provisions of the 1834 Poor Law Amendment Act. The Incorporation remained in operation until 1909 when it was replaced by a Poor Law Union.
The 1776 workhouse, with its capacity of only 220 inmates, could accommodate only about a fifth of the city's paupers. By 1843, inmates were four to a bed with no segregation of the sexes, and increasingly insanitary conditions. A proposal in 1845 to build a new workhouse on the Common came to nothing, as did a scheme to buy Archer's Lodge in 1863. In 1865, a visiting Poor Law Inspector roundly condemned the "mixing together of all classes, including old, infirm and idiots, in rooms in which it was almost impossible for human beings to live." It was then decided to build a new workhouse on land adjoining the existing site.
Southampton New Workhouse
In 1865, a competition for suitable plans for a new workhouse was held amongst local architects. Although the winning entry appears to have been by Mr A Bidborough, the design actually used was by Mr TA Skelton. The official laying of the foundation stone was on 7th August 1866. The building contract for £20,552-11s. was placed with Bull and Sons. However, by the time of its completion in November 1868, various extra costs involved in digging foundations, providing a new water supply, walls and railings, internal fittings etc. had pushed up the cost to £31,200.
On Monday and Tuesday, the 8th and 9th of November 1868, the city's rate-payers were invited to inspect the new building. Reactions were not overwhelmingly positive — it was judged too small, it contained bad workmanship and inferior materials, it was too extravagant, it was "an incongruous attempt at architecture". The general site layout can be seen on the 1881 map below:
Southampton workhouse site, 1908.
To the west, at the front of the site, was a block of offices and vagrants' wards — these were demolished in 1948.
Southampton workhouse entrance block, 1870.
Behind this were four parallel ranges, the first two linked by a central spine.
Southampton workhouse, 1870.
The front range contained the master's quarters, and accommodation for able-bodied paupers.
Southampton front block from the west, 2000.
The second range contained accommodation for aged paupers and was linked to the first range by a central spine containing the dining hall, kitchen, and chapel.
Southampton rear of front block and dining hall from north, 2000.
Southampton aged paupers' block (left of picture) from north, 2000.
The third range was the infirmary block, and the fourth infectious wards.
Southampton infectious wards and infirmary from north, 2000.
Opposite the workhouse, boys' and girls' schools were located in two converted houses.
In 1902, following the opening of a new infirmary at Shirley Warren, the workhouse accommodation was rearranged and the infectious wards greatly altered.
From 1904, to protect them from disadvantage in later life, the birth certificates for those born in the workhouse gave its address just as 154 St Mary Street, Southampton.
Between 1908 and 1922, the Incorporation absorbed a number of parishes previously belonging to the adjacent South Stoneham Union. In 1926, Southampton took over the South Stoneham workhouse, relieving the Guardians of the union of the financial burden of its operation.
In 1940, the building was taken over as an emergency food centre and for housing homeless families. In 1948, work began on its conversion into a Technical College, with classes beginning the following year. In 1995, the establishment became Southampton City College.
Union Infirmary, Shirley Warren
In 1899, the Southampton Guardians decided to erect a separate union infirmary at Chilworth Road (now Tremona Road), Shirley Warren, to the north-west of Southampton. The site chosen was deliberately chosen to be large enough to also accommodate a new workhouse should that ever be required. The new infirmary was designed by AF Gutteridge and the building contractor was H Cawte. The foundation stone was on March 6th, 1900, by Charles Thomas, Chairman of the Board of Guardians. The hospital was opened in 1902.
The hospital design was based on the typical pavilion-plan layout of the period with a administration block at the centre, linked by covered ways with two female pavilion wards at the west and three male wards at the east. The hospital's location and layout are shown on the 1930s map below.
Southampton Infirmary site, 1930s.
Southampton Union Infirmary from the south, c.1904.
According to a 1902 report by The Builder:
The administrative department contains accommodation for the committee of management, the medical superintendent, the assistant medical officer, the matron, chaplain, porter, and servants generally, kitchen department, with larders, mess-rooms, &c. Steward's department with large general store (two storey), nurses' dining and cloak rooms, matron's office, store, and sewing room, dispensary, anaesthetic operating-rooms, reception wards, and clothes store for male and female patients, &c.;
The ward blocks were two storeys high and could each accommodate sixty patients. The same report in The Builder described the ward blocks as follows:
Each pavilion is provided with two staircases, and wide balconies for convalescents' use. On the floor of each pavilion offices have been provided as follows :— Ward kitchen, with sink, range, larder, &c., bathroom, lavatory, with station for bath on wheels for patients unable to use bathroom, linen-room, day-room for convalescents, slop-sink room, and the usual sanitary accommodation, the latter being in detached sanitary towers, approached from the main buildings by means of enclosed bridges. The pavilion halls are provided with lifts for food, coal, &c., and they and the corridors are heated by hot-water coils, and the wards with central down draught hot-air ventilating stoves with large open fires. The walls throughout are finished with adamant plastering painted. All corners of rooms, ceilings, and floors are rounded
Southampton Union Infirmary from the north-east, c.1905.
An additional ward block was erected to each side in 1911, again designed by AF Gutteridge. A three-storey nurses' home at the east of the site was erected in several phases beginning in 1899 and extended in 1911, 1934, and the 1980s. The Matron's house was located to its north. A lodge stood at the infirmary entrance at the south-east of the site.
Southampton Union Infirmary entrance lodge and nurses' home from the south, date unknown.
As with the workhouse, the infirmary had a euphemistic address for use on birth certificates of 1a Chilworth Road.
In 1930, following the abolition of the workhouse system, the infirmary was appropriated by the Southampton Borough Council and was renamed the Borough Hospital. It joined the new National Health Service in 1948 and continues to operate as Southampton General Hospital. The old administrative and ward blocks were demolished in the 1980s and a new hospital now stands on the site. All that now remains of the old buildings are the nurses' home and a fragment of the covered walkway that once linked the ward pavilions.
Southampton Union Infirmary nurses' home from the west, 2005.
Southampton Union Infirmary nurses' home detail, 2005.
Southampton Union Infirmary covered walkway, 2005.
Hollybrook Children's Homes, Shirley Warren
In 1910, the Guardians paid £3,200 for house and 13 acres of grounds about half a mile to the north east of the Shirley Warren infirmary. The house, which dated from 1835 and known as Hollybrook House, was to be used as a home for 150 boys and opened in August 1911.
Hollybrook House from the south, date unknown.
An additional 13 acres of land was subsequently purchased on which a group of cottage cottage homes was opened in March 1923. The old house was then just used to house seventy boys aged 12 to 16. Children aged eight to twelve were accommodated in cottages, with either ten boys or ten girls under the care of a foster mother and a "house girl" — one of the older girls who did some of the domestic chores.
The location and layout of the homes is shown on the 1930s map below:
Southampton, Hollybrook site, 1930s.
Southampton, Hollybrook House from the south, date unknown.
Southampton, Hollybrook boys' dining room, date unknown.
Two pairs of houses were located on Seymour Road.
Hollybrook — Seymour Road, 2005.
Hollybrook — Seymour Road rear view, 2005.
A house on the corner of Winchester Road and Seagarth Lane was used as a receiving home.
Hollybrook receiving home, 2005.
Two further blocks of houses were located to the north of Seagarth Lane.
Hollybrook childrens's houses, 2005.
The four-to-eight-year-olds housed were housed separately in a home called Oatlands located at the junction of Winchester Road and St. James' Road.
Hollybrook Oatlands site, 1930s.
In the 1930s, a farmhouse on the estate was converted to provide staff accommodation, and a house was built at the Seagarth Lane entrance for accommodating the assistant superintendents.
Hollybrook superintendent's house, 2005.
In October 1936, a two-storey nursery was opened at the east of the site which could hold 27 babies from birth to two years, and 36 children from two to five years of age. There was also accommodation for up to ten residential staff.
Hollybrook 1936 nursery block, 2005.
A further children's house was also erected.
Like many other workhouse children's homes, Hollybrook ran a boys' band.
Hollybrook boys' band, pre-1920.
The site layout at the end of the 1930s is shown on the 1946 map below.
Southampton Hollybrook site, 1946.
Jim Woollison has kindly contributed a picture of one of the Seymour Road cottages taken on George VI's coronation day in 1937. His mother-in-law Rosina is second from the right on the front row.
Hollybrook Seymour Road, 12th May 1937.
Following a decline in demand for places, the nursery closed in the mid 1955. Hollybrook House was demolished in the 1950s and housing estate was built amongst the cottages.
In 1958, the original nursery was converted to become the home of Hollybrook Infants and Junior school which opened on 8th September of that year.
Southampton Hollybrooks entrance gates, 2005
The surviving Hollybrook houses are now used for residential purposes. The former nursery has now become a school.
From 1925 to 1936, William Golding was a resident of the Hollybrook homes. His recollections of life at the homes can be heard in a series of recorded audio clips.
Southampton Archives Service, Civic Centre, Southampton SO14 7LP. has a rich assortment of records going back as far as 1711. Holdings include: Guardians' minute books (1818-1930); visitors' books (1779-1833); settlement examination registers (1711-1901, with gaps); settlement of bastards registers (1811-39); examination and removal orders (1817-42); papers for emigration of children to Canada (1902-07) etc.
A History of the Southampton Technical College Buildings, formerly the Workhouse, with Special Attention To... Architecture by CM Anstey, 1978
A Picture of Southampton by Phillip Brannon, 1850.
The Builder 26th April, 1902.
The Incorporation's early correspondence with the central poor-law authorities is online on the TNA website .


For pennies a day, my husband and I sponsored a child through Compassion International, a Christian organization with charitable projects in third-world countries. This young boy wrote me letters, and I wrote back (inconsistently, I'm afraid). Sometimes, I sent him a few extra dollars for his birthday, and he sent me a Crayon-decorated note to tell me what he bought: a live chicken. For food.
It would help me, in my most shallow moments, to remember this.
But I forget that I already have enough when I'm browsing Pinterest pictures of remodeled houses painted all the right colors. Photos of kitchens with custom-made cabinets and retro ranges that come in a rainbow of pastels. Not that there's anything wrong with having a nice kitchen. But my kitchen is nice enough.
I forget it when my friend shows me the new furniture in her sunroom makeover. When she fills a Lenox teacup from the built-in single-handle instant hot water dispenser, and I think, I need one of those. Need, my foot.
Recently, I read Rick Bragg's All Over but the Shoutin', a beautifully-written, heart-breaking memoir and tribute to his mother, who "went eighteen years without a new dress so her sons could have school clothes." And I recall my over-filled closet where the dresses hang so close, there's no light between them.
Not that I disdain my possessions. I'm thankful for my clothes and furniture and food. I'm supremely thankful for a hard-working, generous husband, because this English Major isn't exactly a money-making machine. I'm thankful for family members who have given us so much, relatives who grew up with Not Enough.
But for me, the scales between Too Much and Not Enough tip heavily to one side. And holding on to five black skirts, because, well, you never know when you might need five black skirts, is just a tad irrational.
Hobbes has a clutter problem.
I'm no different from my little dog who isn't hungry but stands sentry beside his food, growling, for HOURS, to keep the other dogs from eating it. While Calvin and Ollie are chasing toads and digging holes in the yard, Hobbes is shackled to his metal bowl.
I, too, am a captive of my clutter if I allow myself to be. As they say, everything you own, owns you. If, like Hobbes, I spend my limited time, energy and attention on superfluous possessions, I have less of those resources to spend on more important things, such as spirituality, people, or creative pursuits.
If I accumulate stuff motivated by (let's call it what it is) envy, or a false sense of scarcity (rather than true scarcity such as that experienced in the third-world), clutter will fill all the corners of my home and my mind.
Hobbes, you and I need a mindset makeover.
A few months ago I challenged myself to write a story under 300 words. Here's the result at 293.
The day after the tornado fell from a starless sky, and like a mosquito's proboscis, sucked away her house and life as she knew it, Clare sifted through the debris. She searched for irreplaceable things – photos, keepsakes, heirlooms. Two-by-fours and bricks mingled with soggy books and sneakers in a bitter stew.
With a churning stomach, Clare started in what had been the den. She found the cover to The Three Little Pigs, a carboard page from Good Night Moon, Sophie's doll minus the limbs. A key to a house which no longer existed.
"Find anything?" yelled Luke. Which meant, did you find my Legos?
"Find anything?" Sophie echoed, playfully.
Clare hid the mangled doll behind her back and shook her head.
In the roofless garage, under crisscrossed wooden beams, lay a purple box – wet, warped, but uncrushed. Inside was Grandma Bett's Waterford crystal bowl.
The cut angles caught the sun and flashed triangles of color, just the way it had on Grandma Bett's dining room table when Clare, like Sophie, was five. She wasn't allowed to touch it, so she'd walk around the table to examine the sparkles from different perspectives. She'd count all the trapped stars.
"Look!" she called to the children as she walked to the patio. "Not a scratch on it!" Ha! What were the chances?
Clare placed the bowl in her small hands, and Sophie held it up to the sun like an offering.
Then, with the sound of a myriad celestas, the bowl crashed onto the concrete, scattering the light, freeing the stars.
Do You Want to do Great Things?
Encouraging words and a laudable creed for writers. Thank you, M.C. Tuggle.


Where My Will Meets His | Redeemed!
How often must we manage our own competing desires? How often are we required to choose, between multiple desires, which one is a high enough priority to take immediate action to pursue? Daily? Hourly? More frequently than that?
As I sit here typing this post, I am also longing to go back to bed. I did not get my usual sleep time, last night, plus I have a headache, and I would very much like to crawl back under the covers. Yet, instead, I'm up writing a blog post.
Why? Because, as much as I would like to go back to bed, I want even more, to finish this post. I know I will probably have other opportunities for sleep during the week, but this is likely my only opportunity for writing this post. So, for me, right now, writing this post is a higher priority, or a deeper desire, than another hour of sleep.
Yet, even as I type, I also pause to sip my coffee. Granted, the sip of coffee gives me time to think about my choice of words, as well as to combat fatigue. However, I am making a deliberate choice to stop typing in order to sip coffee. I am choosing, momentarily, to make a sip of coffee a higher priority than the typing, although I have already chosen to make typing a higher priority than sleep.
Multiple competing desires! Choosing to forego fulfillment of one desire in order to act on another. These are the multiple decisions we make during the course of our normal daily activities.
So, what are the deepest desires of our hearts? What fundamental desires or priorities motivate us to make the daily choices we act on?
The desire to care and provide for our family is a high priority for many of us. The desire to be a blessing to others may also rate very high. But what deeper motivation caused us to even choose to have a family or to become involved in a ministry? The desire for significance and purpose? The desire to leave a legacy when we pass from this life? The desire to love and be loved? The desire for intimacy?
These are all deep motivators. Yet we're still talking about multiple desires that potentially compete for priority, and sometimes require a choice of one above another.
So, what is the deepest desire of my heart? Is there one over-riding core desire from which all the other desires are born? If so, what is that desire?
Could any desire be more fundamental than this? To be who I was created to be? To fulfill the destiny which I was created to fulfill?
For me, as for other true believers in Christ, the innermost deepest desire of my heart is to become all that God has predestined me to be…to become a true child of God, fully conformed to His image…reflecting His glory in my own unique way, as He created and predestined me to do.
This is the deepest desire of my heart, from which all other desires are born…even those desires that find their expression in distorted perceptions and sinful actions proving counterproductive to the realization of my deepest heart's desire.
Too often, Christianity is viewed as restrictive disciplines and rules imposed on us by God. Too often, we view God's direction as boundaries keeping us from doing the things we really want to do…as being required to miss out on all the fun pleasures that others get to enjoy, but which our God prohibits us from enjoying.
God is not keeping us from doing anything!
Rather, He is holding us accountable to our deepest heart's desire, while providing direction leading to the fulfillment of our destiny, which He alone knows.
Christianity is not about the denial of pleasure and desires. Rather, it is about the relentless pursuit of our deepest heart's desire. It is about trusting God to help us wisely manage our competing desires, to work toward the fulfillment of our heart's deepest desire.
God remains faithful to the vision of my deepest heart's desire, even when I lose focus and stray from the pursuit of the vision.
And it is here, at the very core of my heart, in my deepest desire, that my will meets His!
What happens when your will meets God's will? When the deepest desire of your heart matches the desire of God's heart?
This entry was posted in free will, God's Position toward Us and tagged covenant, desires, desires of my heart, destiny, free will, god, God's will, living, perspective, will by joe. Bookmark the permalink.


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The Residences at Sun Peaks Grand welcomes four-legged guests at their pet friendly accommodation. The pet fee is CAD 35 per accommodation, per week.


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Yeah Yeah Yeahs drummer releases solo album
NME Oct 2, 2007 3:51 pm BST
Brian Chase goes jazz for new album
Yeah Yeah Yeahs drummer Brian Chase has released a solo album.
Chase, with musician Seth Misterka, has named the album 'Duo'.
On Yeah Yeah Yeahs' MySpace the sound of the album is described as: "Free jazz for all of you pop music fans (most of the tracks aren't much longer than 3 mins – just like your favourite songs)! Art Stars? Yes, indeed!"
The album can be purchased via the MySpace.com/heathenskulls.

redpajama_set_name: RedPajamaCommonCrawl


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Long John Silver è il settimo album pubblicato dai Jefferson Airplane, pubblicato il 20 luglio 1972.

L'album

Tracce
Lato A
 Long John Silver (testo: Grace Slick; musiche: Jack Casady) — 4:22
 Aerie (Gang of Eagles) (testo e musiche: Slick) — 3:53
 Twilight Double Leader (testo e musiche: Paul Kantner) — 4:42
 Milk Train (testo: Slick; musiche: Papa John Creach, Roger Spotts) — 3:18
 The Son of Jesus (testo e musiche: Kantner) — 5:27

Lato B
 Easter? (testo e musiche: Slick) — 4:00
 Trial by Fire (testo e musiche: Jorma Kaukonen) — 4:31
 Alexander the Medium (testo e musiche: Kantner) — 6:38
 Eat Starch Mom (testo: Slick; musiche: Kaukonen) — 4:34

Formazione

Jefferson Airplane
 Grace Slick — voce, pianoforte
 Paul Kantner — voce, chitarra ritmica
 Jack Casady — basso
 Jorma Kaukonen — chitarra solista, voce
 Papa John Creach — violino elettrico
 John (Goatee) Barbata – batteria, tamburino, "against the grain stubble scraping"
 Joey Covington — batteria in Twilight Double Leader e The Son of Jesus

Musicisti addizionali
 Sammy Piazza — batteria in Trial by Fire

Crediti
 Jefferson Airplane — produttore, arrangiamenti
 Maurice — coordinatore della produzione
 Don Gooch — ingegnere del suono
 Steve Barncard — special thanks
 Pacific Eye & Ear — album concept, album design
 Bob Tanenbaum, Propella Rotini — illustrazioni
 Bruce Kinch — fotografia
 Borris — weed. AKA Mike Trudnich
 Registrato al Wally Heider Studios, San Francisco

Note

Collegamenti esterni

redpajama_set_name: RedPajamaWikipedia


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The Mel and Enid Zuckerman College of Public Health has partnered with the College of Medicine, College of Law, the College of Pharmacy, the Eller College of Management, and the College of Social and Behavioral Sciences to administer six dual degree programs. Students have the opportunity to earn two complementary degrees simultaneously. Dual degree applicants are required to meet the application deadlines, testing requirements, and submit program fees to each program independently. Academically, students are expected to fulfill the basic requirements of each program with the inclusion of course work that satisfies the requirements of both programs.
The DNP/MPH is our latest innovative and collaborative degree program
Doctor of Nursing Practice & Master of Public Health (DNP/MPH)
A dual DNP/MPH addresses the identified need for nurses to have advanced education in public health that prepares them to improve population health services and expand the scope and quality of healthcare for diverse populations of individuals across the lifespan.
The DNP/MPH is offered fully online through the AZ Online campus and combines the DNP with our online MPH.
There are two tracks students may choose from:
MS DNP in Executive Health Systems Leadership (EHSL) specialty with an MPH in Health Services Administration
Download the course of study for the MS DNP in EHSL with an MPH in HSA
DNP in Family Nurse Practitioner specialty with an MPH in Health Behavior Health Promotion
MD/MPH Dual Degrees
There are two MD/MPH dual degree programs with the College of Medicine in Tucson and Phoenix respectively. The programs are designed to prepare future clinicians for community-engaged, evidence-based, and socially-responsible medical practice. These goals are achieved in an interprofessional setting that emphasizes respect for people of all ages, genders, ethnicities, cultures, and national origins. Students' pursuit of knowledge and preparation for leadership will be guided by academic and clinical mentors who foster inquisitiveness, open-mindedness, compassion, and resilience. The dual degree programs provide students with training in core public health competencies. The curricula are intended to enhance the graduate's ability to provide high quality medical care by using public health knowledge and skills to plan, implement, and evaluate policies that affect the health of communities. These skills complement the pathophysiologic, diagnostic, and therapeutic training provided by the College of Medicine that is intended to treat individual patients.
Medicine - Tucson Campus
Dual Degree with Public Health Policy and Management
The MD/MPH in Public Health Policy and Management curriculum comprises 42 units of coursework and experiential learning that leads to a Master's of Public Health (MPH) degree in Public Health Policy and Management (PHPM). All students must complete 16 units of public health courses, 12 units of required PHPM courses, 6 units of applied practice experience, including the internship, and 8 units of elective credit. With approval of the Program Director, students may use up to 8 units of College of Medicine coursework to fulfill the elective requirements. More information about the MD/MPH Dual Degree Program in Tucson is available on the College of Medicine-Tucson website.
Total Minimum Credit Requirement = 42 units to complete the MPH degree in the Public Health Policy and Management concentration (in addition to the medical school curriculum). The MPH requirements are summarized below:
MPH in PHPM - Minimum Credit Requirements (42 units)
Required Core Courses (16 units)
EPID 573A Basic Principles of Epidemiology (3)
PHPM 574 Public Health Policy and Management (3)
EHS 575 Environmental and Occupational Health (3)
BIOS 576A Biostatistics in Public Health (3)
HPS 577 Sociocultural and Behavioral Aspects of Public Health (3)
PHPM 580 Integrated Public Health Learning Experience (1)
Internship Preparation Workshop (required; non-credit)
Interprofessional Education Activities (documentation required; non-credit)
Required PHPM Courses (12 units)
PHPM 510 The US Health Care System (3)
PHPM 567 Public Health Leadership & Management (3)
PHPM 569 Fundamentals of Health Budgeting and Financial Management (3)
PHPM 617 Advanced Public Health Policy Analysis (3)
Required PHPM Experiential Learning Courses (6 units)
PHPM 696P Public Health Policy and Management Field Seminar (2)
PHPM 909 Master's Report "Internship" (3)
Service Learning Course (1)
Electives (minimum 8 units) - Approved by Faculty Advisor
Medicine - Phoenix Campus
Dual Degree with Clinical Leadership
The MD/MPH Clinical Leadership Degree comprises 42 units of coursework and experiential learning that leads to a Master of Public Health with a concentration in Clinical Leadership. More information about the MD/MPH Dual Degree Program in Phoenix is available on the College of Medicine-Phoenix website.
Total Minimum Credit Requirement = 42 units to complete the MPH degree in Clinical Leadership (in addition to the medical school curriculum). The MPH requirements are summarized below:
MDMPH Phoenix Clinical Leadership Concentration Competencies (PDF)
MPH in Clinical Leadership Minimum Credit Requirements (42 units)
PHPM 570 Issues and Trends in Public Health (3)
PHP 696i Clinical Leadership and Administration (4)
PHP 572 Applied Public Health Practice (3)
PHPM 909 Master's Report (internship) (5)
Electives (11) - 2 units MEDP 802 and 9 units chosen in consultation with your faculty advisor
Further information regarding the MD/MPH Dual Degree Program in Phoenix is available on the College of Medicine website.
Offered On Campus in Tucson
The MPH/MA Dual Degree Program allows students to earn a Master of Public Health and a Master of Arts degree in Latin American Studies. This dual degree program offers students the opportunity for interdisciplinary study allowing them to customize their dual degree by selecting any one of eight MPH concentration areas, and by taking electives addressing issues relevant to Latin American populations.
Mexican American Studies
The MPH/MS Dual Degree Program allows students to earn a Master of Public Health and a Master of Science degree in Mexican American Studies. This dual degree program offers students the opportunity for interdisciplinary study allowing them to customize their dual degree by selecting any one of eight MPH concentration areas, and by taking electives addressing issues relevant to Mexican American and Latino populations. Students gain an advanced understanding of the Mexican American and Latino populations in the United States that prepares them to develop culturally-competent public health programs and policies targeting Latinos.
The JD/MPH Dual Degree Program is offered jointly by the University of Arizona College of Law and the Mel and Enid Zuckerman College of Public Health. The dual degree program allows a student to earn a Juris Doctor degree (JD) and a Master of Public Health (MPH). The objective of the program is to support and encourage individuals with an interest in public health studies in the development of interdisciplinary knowledge and skills in preparation for a career in health law, public health policy, or a related field.
This dual degree program offers students the opportunity for interdisciplinary study allowing them to customize their dual degree by selecting any one of eight MPH concentration areas and by taking electives addressing issues relevant to public health law.
The Law and Public Health (JD/MPH) dual degree is designed to train leaders in the field of public health law. Graduates are prepared for positions in a variety of settings, including work in a health law or environmental section of a law firm; positions in local, state, and federal government; research or advocacy jobs in nongovernmental organizations; or posts in academia.
The MPH/MBA Dual Degree Program is offered in partnership by the Mel and Enid Zuckerman College of Public Health and Eller College of Management. The dual degree program allows a student to earn a Master of Public Health (MPH) and a Master of Business Administration (MBA) in three years upon the completion of a minimum of 78-87 units in Public Health and Business Administration. This dual degree program offers students the opportunity for interdisciplinary study by allowing them to customize their dual degree by selecting any one of eight MPH concentration areas.
Offered On Campus in Tucson and Phoenix
The Doctor of Pharmacy/Master of Public Health (PharmD/MPH) is offered jointly by the University of Arizona College of Pharmacy and the Mel and Enid Zuckerman College of Public Health. The dual degree program allows students to simultaneously earn a Doctor of Pharmacy (PharmD) and a Master of Public Health. This PharmD/MPH combines the complementary strengths of the College of Pharmacy and the Zuckerman College of Public Health to provide students with an integrated curriculum focusing on knowledge and ingenuity in the delivery of pharmaceutical care within the broader population based public health environment. View the PharmD/MPH flyer!
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BEIRUT—Turkey warned the Syrian government Monday against entering the Kurdish-controlled enclave in northern Syria where a major Turkish military offensive is underway, saying it would hit back at the troops if their goal is to protect the Kurdish fighters.
The warning sets up a potential clash between Turkish troops and Syrian government forces backed by Russia and Iran, whose deployment would be a first step toward restoring Syrian President Bashar Assad's presence along the border with Turkey.
The warning by the Turkish foreign minister came shortly after Syrian state media said pro-government forces would enter Afrin "within hours" to "bolster" local forces in confronting Turkey's "aggression" after reaching an agreement with the Kurdish militia known as the People's Protection Units, or YPG, which controls Afrin.
Assad's troops have had no presence in Afrin since they pulled out of most of northern Syria in 2012, as nation-wide protests against Assad transformed into a civil war. A return to the area, where a potent mix of regional and international powers have boots on the ground, could further complicate the situation and lead to unwanted confrontations.
But depending on the details of the agreement, it may also serve to defuse the situation in Afrin, where Turkey has been struggling to achieve results in its now month-long offensive to push back YPG fighters from its borders.
Ankara considers the YPG a "terrorist group" linked to the Kurdish insurgency within Turkey's borders. On Jan. 20, it launched a major air and ground offensive, pounding the enclave with airstrikes and artillery on a daily basis.
Turkey's foreign minister, speaking at a news conference in Amman, Jordan, said Turkey would have no problem if Syrian government forces were entering Afrin to clear the area from YPG fighters, but that it would strike back if it turns out the deployment was meant to shore up the Kurds against Turkey.
"If the regime is entering to protect the YPG, then no one can stop us, stop Turkey or the Turkish soldiers," Mevlut Cavusoglu said.
The group has received weapons and training from the U.S. for years and has been Washington's main partner in the war against Daesh, also known as ISIS or ISIL, in Syria.
It was not clear who, exactly, would enter Afrin. Syria's state-run news agency, SANA, said pro-government fighters known as "popular forces" would deploy to the area.
"The popular forces joining the resistance against Turkish occupation in Afrin comes in the framework of supporting residents as well as defending Syria and its sovereignty," SANA said. It added that the deployment aims to "frustrate attempts by Erdogan's regime and its mercenaries of terrorist organizations to occupy the area," referring to Turkish-backed Syrian insurgents.
"The army will deploy in several border areas in co-ordination with the People's Protection Units and the Syrian Democratic Forces," said the official, who spoke on condition of anonymity because he was not authorized to reveal details of the agreement.
"The army will set up military positions in the border area and the agreement is that the Syrian army and the YPG will defend Syria together," the official said.
The Syrian Democratic Forces are a U.S.-allied group led by the YPG that has won a series of major victories against Daesh in Syria.
The Syrian government and Kurdish fighters have clashed on occasion, and Assad technically opposes the Kurds' demands for autonomy. But they have also indirectly worked together in the past, and Turkey represents a common enemy.
Turkish officials said President Recep Tayyip Erdogan held a telephone conversation with his Russian counterpart, Vladimir Putin, during which the two discussed Turkey's military offensive in Afrin. The Turkish officials detailed the call on condition of anonymity in line with government regulations, and did not mention an agreement for government troops to enter Afrin.
Turkish troops and allied Syrian opposition fighters meanwhile continued to pound villages in the enclave with artillery, according to the Britain-based Syrian Observatory for Human Rights and Syria's state media.
The Observatory said Monday's shelling by Turkish troops killed three people, including a child, and that others were wounded.
Turkey's private Haberturk newspaper said the Syrian government forces were expected to deploy to four locations in the next two days and 52 locations within the week. The paper claimed that under the deal, the YPG had agreed to hand over heavy weapons it holds. The paper did not provide a source for the report.
The besieged region known as eastern Ghouta has been subjected to weeks-long bombardment that has killed and wounded hundreds of people. Opposition activists say government forces have brought in reinforcements in preparation for a wide offensive on the last main rebel stronghold near the capital.
The Observatory said Monday's airstrikes and shelling killed 44 people. The Syrian Civil Defence, volunteer first responders known as the White Helmets, said 30 people were killed in eastern Ghouta.


text:
Search Protection
Nigeria: Northeast - Joint Education Needs Assessment (JENA), November 2017
Subject/Objective:
The Joint Education Needs Assessment (JENA) was commissioned on behalf of the Nigerian Federal Ministry of Education, donors, and development and humanitarian partners. It aimed to get an overview of the current situation of education throughout the three states most affected by the insurgency (Adamawa, Borno and Yobe). It also sought to establish which activities could support the resumption of education in a sustained way after years of fighting, displacement, drought and economic crisis.
Methodology:
The focus of the assessment was the education sector in the northeast states of Adamawa, Borno and Yobe. All formal schooling levels (pre-primary, primary, junior secondary and senior secondary) within the government system were included. The assessment consisted of three components: a review of secondary data, key informant interviews at the school level (cross-checked with enumerator observation checklists) and focus group discussions with teachers, parents and youth.
All accessible LGAs within Borno were considered for data collection. All LGAs in Adamawa and Yobe bordering Borno were also considered, as well as affected LGAs in Adamawa bordering Cameroon or hosting significant IDP populations (see Annex F for details of LGAs included in the assessment). Of the LGAs considered, 27 were included in primary data collection (16 in Borno, 8 in Adamawa and 3 in Yobe).
The JENA plan was to visit functional schools. However, several schools that were expected to be open were not holding classes when the assessment teams visited them. Three of the schools were in Konduga LGA, Borno, and all reported the same reason for being closed – 'insecurity in or around the school, real or perceived.' Konduga is a large LGA located to the south and west of Maiduguri/Jere. Most of Konduga is currently inaccessible and the assessment took place mainly in Konduga Town, the main urban centre of the LGA and one of the few accessible areas. Security incidents are a regular occurrence42 in Konduga LGA. The fourth closed school was in Jere where the reason was given that "school was damaged either deliberately or in the fighting."
Sample size:
27092018_nga_jena_report.pdf
05092018_nga_kii_form.pdf
Nigeria: Joint Education Need Assessment (JENA) Validation Workshop Presentation - 12 December 2017
Nigeria: Draft - Joint Education Need Assessment (JENA) Report - 04 December 2017
24 Sep 2017 to 27 Dec 2017
Unit(s) of Measurement:
Baseline data analysis
Displaced population
Returnees
Water Sanitation Hygiene
Assessment Capacities Project
Government of Nigeria - State Universal Basic Education Board
Government of Nigeria - Ministry of Education
Disaster(s)/Emergency:
Nigeria: Complex Emergency - 2014-2020


Rep. David Taylor completed a rare trifecta in the legislative arena by having three bills receive public hearings today in the House Environment Committee. Taylor, R-Moxee, says his legislation would reduce the regulatory burden on the public, increase job creation, reduce the costs of transportation construction projects and save taxpayer dollars.
Taylor said his bill would help relieve the staff and financial pressure many jurisdictions face with duplicative and overlapping reviews, saving time and money in the process.
Taylor's other bills, House Bill 2095 and House Bill 2269, are designed to create a 'mitigation bank' from lands previously purchased by state agencies to mitigate transportation and other infrastructure construction projects.
All three bills remain in the House Environment Committee where they await potential executive action.
The 60-day 2014 regular legislative session is scheduled to end on March 13.


text:
Virginia Mayo/AP
NATO Secretary General Jens Stoltenberg speaks during a media conference at NATO headquarters in Brussels on Wednesday.
NATO hopes to impress Trump with new headquarters and defense policies
On the campaign trail, Trump called the alliance 'obsolete.'
By Lorne Cook Associated Press
May 24, 2017 | Brussels
NATO is not only rolling out the red carpet for President in Brussels Thursday, the military alliance – which Mr. Trump once declared obsolete – has been busy repackaging its image and is ready to unveil a new headquarters worth more than 1 billion euros.
In recent months, member nations have strained to show they are ramping up defense spending as Trump has demanded, even though they have been doing so for a few years in response to an aggressive Russia. And while they agree with the chief of the alliance's most powerful member that NATO can do more to fight terrorism, they say it can be achieved with more of the same; training and mentoring troops in Afghanistan, and equipping local forces in Iraq so they can better fight the Islamic State group themselves.
"They'll only talk about what he cares about, so really he should come out of this meeting feeling as though NATO responds to him," said Kristine Berzina, NATO analyst at the German Marshall Fund think tank. "At least that's what they hope here."
Indeed, as part of the repackaging to be announced during Trump's 24-hour visit to the city he branded a "hellhole," NATO is likely to agree to join the 68-nation international coalition fighting IS. The move is symbolically important, especially since the group claimed responsibility Tuesday for a deadly explosion at an Ariana Grande concert in Manchester, England.
An anti-terror coordinator may also be named, but most changes will be cosmetic, as NATO allies have no intention of going to war against IS.
Reading, writing, and paychecks. What's a teacher worth?
"It's totally out of the question for NATO to engage in any combat operations," NATO Secretary-General Jens Stoltenberg said Wednesday, on the eve of the meeting.
The 28 member nations, plus soon-to-join Montenegro, will renew an old vow to move toward spending 2 percent of their gross domestic product on defense by 2024. Still, many are skeptical about this arbitrary bottom line that takes no account of effective military spending where it's needed most. Germany would have to virtually double its military budget and spend more than Russia.
Putting some meat on the pledge, the leaders will agree to prepare action plans by the end of the year, plotting how to reach 2 percent over the next seven years, and show how they will use the money and contribute troops to NATO operations.
Only five members currently meet the target: Britain, Estonia, debt-laden Greece, Poland, and the United States, which spends more on defense than all the other allies combined.
"It's not fair that we're paying close to 4 percent and other countries that are more directly affected are paying 1 percent when they're supposed to be paying 2 percent," Trump told the Associated Press in an interview last month.
Tomas Valasek from the Carnegie Europe think tank says the president's demands on overdue debts have shaken up the other allies.
"Trump has challenged the idea that active engagement in Europe is a core US interest," Mr. Valasek said. "He appears to regard all foreign relations as zero-sum transactions, in which each contribution to someone else's security represents a net loss to the United States."
The Europeans, Valasek said, should respond in two ways: "In the short term, focus on preventing the president from abandoning the alliance and, in the long term, prepare to assume a bigger role in defending the European continent."
The short working-dinner meeting will be high on symbolism. At the entry to the new premises – a village-sized complex that should be in full use early next year – Trump and Mr. Stoltenberg will unveil a piece of the World Trade Center.
After the September 11, 2001, attacks in the United States, NATO activated its collective defense clause for the first and only time, with member nations pledging to help their beleaguered ally.
Stoltenberg and German Chancellor Angela Merkel will also unveil a part of the Berlin Wall that once divided East and West Germany.
But the ceremonies and symbolism will do little to hide the divisions running through NATO. Trump wants more from the alliance, while countries such as Poland, Lithuania, Latvia and Estonia want iron-clad assurances that they won't be left alone should Russia cross their borders.
Turkey's President Recep Tayyip Erdogan, meanwhile, has purged around 11,000 military personnel from its armed forces since last July's thwarted coup. Hundreds of Western-educated senior officers were removed from posts at NATO, severely weakening the army.
Yet it's a subject that is almost taboo at NATO headquarters; a national affair to be dealt with internally.
Tensions have also mounted between Erdogan and Merkel since Germany offered asylum to some of the officers. Belgium has publicly warned against any pro-Erdogan rallies during his visit.
Outside the heavily guarded security perimeter near the city's airport and in downtown Brussels, peace groups have planned rallies of their own.
But, as the Manchester bombing remains fresh in mind, Belgium will remain on security Level 3 – meaning that the threat of an extremist attack "is possible and likely" – as it has since the suicide-bomb attacks on the Brussels airport and subway killed 32 people last year.
Test your knowledge How much do you know about NATO? Take our quiz
Trump's first trip: Can he offer leadership on both security and values?
Trump call on Afghan surge: Another step away from 'America First'?
https://www.csmonitor.com/World/2017/0524/NATO-hopes-to-impress-Trump-with-new-headquarters-and-defense-policies


text:
\section{Introduction}
	In this paper, we discuss the following fractional elliptic problem with a singularity, a critical exponent and a Radon measure.
	\begin{equation}\label{4p1}\tag{$P_\lambda$}
			\begin{split}
			(-\Delta)^su&= \frac{1}{u^\gamma}+\lambda u^{2_s^*-1}+\mu ~\text{in}~\Omega,\\
			u&>0~\text{in}~\Omega,\\
			u&= 0~\text{in}~\mathbb{R}^N\setminus\Omega,
			\end{split}
			\end{equation}
			where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary, $s\in(0,1)$, $N>2s$, $0<\gamma<1$, $\lambda>0$, $\mu$ is a nonnegative bounded Radon measure and $(-\Delta)^s$ is the fractional Laplacian defined by
			$$(-\Delta)^su=\text{P. V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}dy.$$
Problems involving nonlocal operators have theoretical applications as well as real life applications in various fields of science. The applications of fractional order Laplacian can be found in L\'{e}vy stable diffusion process, chemical reactions in liquids, geophysical fluid dynamics, electromagnetism etc (refer \cite{Applebaum} for further details). Nonlocal problems containing singular or irregular data are used in dislocation problems \cite{Cacase}, quasi-geostrophic dynamics \cite{Caffarelli}, image reconstruction problems \cite{Gilbao} etc. The problem of denoising a image is to find a clear image $u$ from a noisy $f$. In the deblurring problem, a given image $f$ is considered as a blurry version of an unknown exact image $u$, which is to be determined. For further details refer Kinermann et al. \cite{kinder}. Readers may refer to the work in \cite{toma}, \cite{yaro1}, \cite{yaro2}.\\
In general, the presence of a measure data in the problem weakens the class of solution space, i.e. we lose some degrees of differentiability or/and integrability of the solution space. Solutions to problems involving measure data or $L^1$ data are obtained by approximations and usually by working in Marcinkiewicz spaces. Readers may refer \cite{Benilian}, \cite{Boccardo 1},\cite{Boccardo 2},\cite{Kuusi} and the references therein for further readings on these types of problems. Boccardo et al. (\cite{Boccardo 1},\cite{Boccardo 2}) proved that the solution to a nonlinear elliptic equation involving $p$-Laplacian and a Radon measure lies in $W^{1,q}_0(\Omega)$ for every $q<\frac{N(p-1)}{N-1}$, where $1<p<N$. Recently, in 2015, Kuusi et al. \cite{Kuusi} considered the problem, as in \cite{Boccardo,Boccardo 1}, in a fractional set up with fractional $p$-Laplacian and established the existence of a solution in $W^{s_1,q}(\Omega)$ for every $s_1<s<1$, $q<\min\{\frac{N(p-1)}{N-s},p\}$. Purely singular problems both in the local and nonlocal cases are studied in \cite{Boccardo},\cite{Canino},\cite{Lazer}, etc. and the references therein. In all these articles the choice of a solution space depends on the power $\gamma$ of the singular term (whether $\gamma\leq1$ or $\gamma>1$). Further, we refer \cite{Bisci 1},\cite{Bisci 2},\cite{Servadei}, etc. to survey Brezis$-$Nirenberg type critical exponent problems (without the singular term and measure data).\\
The problem $(P_\lambda)$ for $\lambda=0$ and the limiting case of $s=1$ has been analyzed by Panda et al. in \cite{Panda}. The authors have guaranteed the existence of a weak solution in $W_0^{1,q}(\Omega)$ if $0<\gamma\leq1$  and in $W_{loc}^{1,q}(\Omega)$ if $\gamma>1$ for every $q<\frac{N}{N-1}$. Ghosh et al. in \cite{Ghosh} extended this result and studied the problem $(P_\lambda)$ with $s\in(0,1)$ and $\lambda=0$. \\
We use the relation among the fractional Sobolev space, Bessel potential space, Marcinkiewicz space to find a solution in a function space weaker than $H_0^s(\Omega)$. Such solutions are called as SOLA (see Definition $\ref{SOLA}$). Due to the presence of nonlinearities with a critical exponent, singularity and a measure data, difficulties arise in the study of $(P_\lambda)$. Thus, it is not easy to directly approach the problem with any commonly used tools like variational method, Nehari manifold method, etc. We study our main problem via a sequence of approximating problems. It is very challenging to prove the existence of a solution to the approximating problems and simultaneously showing the boundedness of the sequence of solution to these approximating problems in $L^{2_s^*}(\Omega)$. To overcome these difficulties we take the help of two auxiliary problems. For that, we apply the concentration compactness principle as in \cite{Mosconi} and Ekeland's variational principle as in \cite{Ekeland}. Precisely, we guarantee that the approximating problem admits at least one solution in a complete Hilbert manifold $H=\{u\in H_0^s(\Omega):\|u\|_{L^{2_s^*}(\Omega)}=1 \}$. We follow some of the arguments of \cite{Panda} to prove our main result stated in the following theorem.
	\begin{theorem}\label{main theorem}
	There exists $0<\Lambda<\infty$ such that for $\lambda\in(0,\Lambda)$ the problem $(P_\lambda)$ admits a positive SOLA $u\in W_{0}^{s_1,q}(\Omega)$ for every $s_1<s$ and $q<\frac{N}{N-s}$ in the sense of  Definition $\ref{SOLA}$.
\end{theorem}
\noindent Before ending this section we describe the {organization} of the paper. In Section $\ref{2}$, we introduce suitable function spaces to deal with our problem and also provide some auxiliary results which will play important roles throughout the article. In Section $\ref{sec 2}$, we prove the existence of a weak solution to the approximating problem for a certain range of $\lambda$. Section $\ref{sec 3}$ is devoted to the proof of Theorem $\ref{main theorem}$. Further, in the Appendix, we show the multiplicity of solutions {using the Nehari manifold.}
\section{Functional settings and auxiliary results}\label{2}
Let $\Omega$ be a bounded domain in $\mathbb{R}^N$, $1\leq p<\infty$ and $s\in (0,1)$. The fractional order Sobolev space (refer \cite{Valdinocci}) is defined as
$$W^{s,p}(\mathbb{R}^N)=\left\{u\in L^p(\mathbb{R}^N):\int_{\mathbb{R}^{2N}}\frac{{|u(x)-u(y)|^p}}{|x-y|^{N+sp}}dxdy<\infty\right\}$$ and $W_0^{s,p}(\Omega)$ is a subspace of $W^{s,p}(\mathbb{R}^N)$ given by
$$W_0^{s,p}(\Omega)=\{u\in W^{s,p}(\mathbb{R}^N): \int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dydx<\infty,~u=0 \text{ in }\mathbb{R}^N\setminus\Omega\}$$ equipped with the norm $$\|u\|_{W_0^{s,p}(\Omega)}^p=\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dydx.$$
\noindent Further, the space $(W_0^{s,p}(\Omega),\|\cdot\|_{W_0^{s,p}(\Omega)})$ with $p>1$ is a reflexive separable Banach space. The following classical theorem will be used frequently in this article.
\begin{theorem}[Theorem 6.5, \cite{Valdinocci}]\label{constant}
	Let $0<s<1$ and $p\in[1,\infty)$ with $sp<N$. Then there exists constant $C=C(N,s,p)>0$ such that for any $u\in W_0^{s,p}(\Omega)$,
	$$\|u\|_{L^r(\Omega)}\leq C \|u\|_{W_0^{s,p}(\Omega)} $$
	for any $r\in[p,p_s^*]$, where $p_s^*=\frac{Np}{N-sp}$. Moreover, the space $W^{s,p}_0(\Omega)$ is continuously embedded in $L^r(\Omega)$ for every $r\in[1,p_s^*]$ and compactly embedded in $L^r(\Omega)$ for every $r\in[1,p_s^*)$.
\end{theorem}
\noindent Denote
\begin{equation}\label{best}
\mathbb{S}_{s,p}=\underset{u\in W_0^{s,p}(\Omega)\setminus\{0\}}{\inf}\frac{\|u\|_{W_0^{s,p}(\Omega)}^p}{\|u\|_{L^{p_s^*}(\Omega)}^p}
\end{equation}
which is the best Sobolev constant in the Sobolev embedding (Theorem $\ref{constant}$). We now define some function spaces which will be further used in this article.
\begin{remark}
For $p=2$, we denote the Sobolev space $H^s(\mathbb{R}^N)=W^{s,2}(\mathbb{R}^N)$. These spaces are Hilbert spaces. Proposition 3.6 of \cite{Valdinocci} provides the relationship between the fractional Sobolev space $H^s(\mathbb{R}^N)$ and the fractional Laplacian $(-\Delta)^s$. It states that the norms $\|\cdot\|_{H^s(\mathbb{R}^N)}$ and $\|(-\Delta)^{s/2}\cdot\|_{L^2(\mathbb{R}^N)}$ are two equivalent norms.
\end{remark}
\begin{definition}[\cite{Shieh}]\label{Bessel}
For $s\in (0,1)$ and $p\in [1,\infty)$, the Bessel potential space $L^{s,p}(\mathbb{R}^N)$ is defined as
$$L^{s,p}(\mathbb{R}^N)=\{u\in L^{p}(\mathbb{R}^N):|\nabla^s u|\in L^{p}(\mathbb{R}^N)\}$$
where $\nabla^s u=\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+s}}\frac{x-y}{|x-y|}dy$ is the fractional gradient of order $s$.
\end{definition}
\noindent We refer Theorem 2.2 of \cite{Shieh} to see the relation between the fractional Sobolev spaces and the Bessel potential spaces.
\begin{theorem}\label{properties}
\begin{enumerate}
	\item For $s\in(0,1)$ and $p=2$, $L^{s,2}(\mathbb{R}^N)=W^{s,2}(\mathbb{R}^N)$.
	\item For $s\in (0,1)$, $1<p<\infty$ and $0<\epsilon<s$, the following continuous embedding holds
	$$L^{s+\epsilon,p}(\mathbb{R}^N)\subset W^{s,p}(\mathbb{R}^N)\subset L^{s-\epsilon,p}(\mathbb{R}^N).$$
\end{enumerate}
\end{theorem}
\begin{definition}
A measurable function $u:\Omega\rightarrow\mathbb{R}$ is said to be in the Marcinkiewicz space $M^q(\Omega)$ $(0<q<\infty)$ if $$m(\{x\in\Omega:|u(x)|>t\})\leq \frac{C}{t^q},~~\text{for }t>0 \text{ and }0<C<\infty.$$
\end{definition}
\begin{remark}
For $\Omega$ bounded,
\begin{enumerate}
	\item $M^{q_1}(\Omega)\subset M^{q_2}(\Omega)$ for every $q_1\geq q_2>0$.
	\item For $1\leq q<\infty$ and $0<\epsilon<q-1$, the following continuous embedding holds
	\begin{equation}\label{continuous}
	L^q(\Omega)\subset 	M^q(\Omega)\subset 	L^{q-\epsilon}(\Omega).
	\end{equation}
\end{enumerate}
\end{remark}
\noindent For a fixed $k>0$, we denote the truncation functions $T_k:\mathbb{R}\rightarrow \mathbb{R}$ by
\begin{eqnarray}
\begin{split}
T_k(s)=&\begin{cases}
s&\text{if}~ |s|\leq k\\
k~sign{s} &\text{if}~ |s|>k.\nonumber
\end{cases}
\end{split}
\end{eqnarray}
\noindent Since our problem, defined in $(P_\lambda)$, involves a measure data as a nonhomogeneous term in the right hand side, we need to introduce the notion of convergence in measure.
\begin{definition}\label{measure}
	Let $\mathcal{M}(\Omega)$ be the set of all finite Radon measures on $\Omega$ and $(\mu_n)$ be a sequence of measurable functions in $\mathcal{M}(\Omega)$. Then we say $(\mu_n)$ converges to $\mu\in \mathcal{M}(\Omega)$ in the sense of measure if $$\int_{\Omega}\mu_n \phi\rightarrow\int_\Omega \phi d\mu,~~\forall \phi \in C_0(\bar{\Omega}).$$
\end{definition}
\noindent In the following theorem we state a commonly used variational principle, introduced by Ekeland in \cite{Ekeland}. Ekeland variational principle is also used to show multiplicity.
	\begin{theorem}\label{eke}
	(Ekeland Variational Principle $\cite{Ekeland}$) Let $V$ be a Banach space and $\Psi:V\rightarrow\mathbb{R}\cup\{+\infty\}$ is a lower semicontinuous,  G$\hat{a}$teaux-differentiable and bounded from below function. Then for every $\epsilon>0$, every $u\in V$ satisfying $ \Psi(u)\leq\inf\Psi+\epsilon$, every $\delta>0$, there exists $v\in V$ such that $\Psi(v)\leq\Psi(u)$, $\|u-v\|\leq\delta$ and $\|\Psi^\prime(v)\|^*\leq\frac{\epsilon}{\delta}$
	where $\|.\|$ and $\|.\|^*$ are the norm of $V$ and the dual norm of $V$, respectively.
\end{theorem}
\noindent We now introduce a suitable notion of solution to $(P_\lambda)$ that in general do not lie in the natural energy space corresponding to the operator $(-\Delta)^s$, i.e. $H_0^s(\Omega)$, but has a lower degree of differentiability and integrability. They are called SOLA (Solutions Obtained as Limits of Approximations) and the procedure of construction of SOLA is through a sequence of approximating problems.
\begin{definition}[SOLA for ($P_\lambda$)]\label{SOLA}
	Let $\mu\in \mathcal{M}(\Omega)$ and $0<\gamma<1$. Then we say that a function $u\in W_0^{s_1,q}(\Omega)$ for $s_1<s$, $q<\frac{N}{N-s}$, is a SOLA to $(P_\lambda)$ if
	 \begin{equation}\label{SOLA weak}
	 \int_{\mathbb{R}^{N}}(-\Delta)^{s/2}u\cdot(-\Delta)^{s/2}\phi=\int_{\Omega}\frac{1}{u^\gamma}\phi+\int_{\Omega}\lambda u^{2_s^*-1}\phi+\int_{\Omega} \phi d\mu,~~\forall \phi\in C_c^\infty(\Omega)
	 \end{equation}
	 and for every $\omega\subset \subset\Omega$, there exists a $C_\omega$ such that
	 \begin{equation}\label{comapct}
	 u\geq C_\omega>0.
	 \end{equation}
Furthermore,  there exists a sequence of weak solutions $(u_n)\subset H_0^s(\Omega)$ to the approximate Dirichlet problems
\begin{equation}\label{4p2}\tag{$P_{\lambda,n}$}
\begin{split}
(-\Delta)^su_n&= \frac{1}{(u_n+\frac{1}{n})^\gamma}+\lambda u_n^{2_s^*-1}+\mu_n ~\text{in}~\Omega,\\
u_n&>0~\text{in}~\Omega,\\
	u_n&= 0~\text{in}~\mathbb{R}^N\setminus\Omega,
		\end{split}
		\end{equation}
	in the sense of Definition $\ref{weak form of apprpx}$, such that $u_n$ converges to $u$ a.e. in $\mathbb{R}^N$ and locally in $L^q(\mathbb{R}^N)$. Here $(\mu_n)\subset L^\infty(\Omega)$ is a positive $L^1$ bounded sequence which converges to $\mu$ in the sense of measure as defined in Definition $\ref{measure}$.
	\end{definition}
		\begin{definition}\label{weak form of apprpx}
			A function $u_n\in H_0^s(\Omega)$ is said to be a positive weak solution of ($P_{\lambda,n}$) if for every $\phi\in C_c^\infty(\Omega)$
			 \begin{equation}\label{approximating weak}
		\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}u_n\cdot(-\Delta)^{s/2}\phi=\int_{\Omega}\frac{1}{(u_n+\frac{1}{n})^\gamma}\phi+\int_{\Omega}\lambda u_n^{2_s^*-1}\phi+\int_{\Omega}\mu_n \phi
			\end{equation}
			and for every $\omega\subset \subset\Omega$, there exists a $C_\omega$ such that $u_n\geq C_\omega>0$.
		\end{definition}
\noindent We begin with the following sequence of problems.
	\begin{equation}\label{4p3}\tag{$P^1_{\lambda,n}$}
	\begin{split}
	(-\Delta)^sw_n&= \frac{1}{(w_n+\frac{1}{n})^\gamma}+\mu_n ~\text{in}~\Omega,\\
	w_n&>0~\text{in}~\Omega,\\
	w_n&= 0~\text{in}~\mathbb{R}^N\setminus\Omega.
	\end{split}
	\end{equation}
Let us define the spaces $H$ and $\bar{H}$ as follows.
\begin{equation}\label{H}
H=\{u\in H_0^s(\Omega):\|u\|_{L^{2_s^*}(\Omega)}=1\}~\text{and}~\bar{H}=\{u\in H_0^s(\Omega):\|u\|_{L^{2_s^*}(\Omega)}<1\}.
\end{equation}
 We now look for a weak solution to $(P_{\lambda,n}^1)$ in $\bar{H}$. Following the proof of {Lemmata 2.3 and 2.4} of Ghosh et al. \cite{Ghosh}, the problem $(P^1_{\lambda,n})$ admits a positive weak solution $w_n$ in $\bar{H}\cap L^\infty(\Omega)$. Furthermore, for every $n\in\mathbb{N}$ and for every relatively compact set $\omega\subset \Omega$, there exists a constant $C_\omega$ independent of $n$ such that $w_n\geq C_\omega>0$.
\begin{remark}
	The solution to the problem $(P^1_{\lambda,n})$ is unique. To prove this, assume that the problem has two different solutions $w_n$ and $\bar{w}_n$. Let us consider $(w_n-\bar{w}_n)^+$ as a test function in the weak formulation of $(P^1_{\lambda,n})$. Then by \cite{Leonori}, we have
	\begin{align}
	0&\leq\int_{\mathbb{R}^N}|(-\Delta)^{s/2}(w_n-\bar{w}_n)^+|^2\nonumber\\&\leq\int_{\mathbb{R}^N}(-\Delta)^{s/2}(w_n-\bar{w}_n)\cdot(-\Delta)^{s/2}(w_n-\bar{w}_n)^+\nonumber\\&=\int_{\Omega}\Big(\frac{1}{(w_n+\frac{1}{n})^\gamma}-\frac{1}{(\bar{w}_n+\frac{1}{n})^\gamma}\Big)(w_n-\bar{w}_n)^+\nonumber\\&\leq 0.
	\end{align}
	This implies $(w_n-\bar{w}_n)^+=0$ a.e in $\Omega$ and $w_n\leq \bar{w}_n$ a.e in $\Omega$. In a similar manner, taking $(\bar{w}_n-w_n)^+$ as a test function we can show that $w_n\geq \bar{w}_n$ a.e in $\Omega$. This proves the claim.
\end{remark}
\noindent We observe that $w_n+v_n$ is a solution to $(P_{\lambda,n})$ if and only if $w_n$ is a weak solution to $(P^1_{\lambda,n})$ and $v_n$ is a weak solution to the following problem.
	\begin{equation}\label{4p4}\tag{$P^2_{\lambda,n}$}
	\begin{split}
	(-\Delta)^sv_n+\frac{1}{(w_n+\frac{1}{n})^\gamma}-\frac{1}{(v_n+w_n+\frac{1}{n})^\gamma}&=\lambda(w_n+v_n)^{2_s^*-1}  ~\text{in}~\Omega,\\
	v_n&>0~\text{in}~\Omega,\\
	v_n&= 0~\text{in}~\mathbb{R}^N\setminus\Omega.
	\end{split}
	\end{equation}
The following theorem guarantees the existence of a weak solution to $(P^2_{\lambda,n})$ in the set $H_n$ where $H_n$ is defined by $$H_n=\{u\in H_0^s(\Omega):\|u+w_n\|_{L^{2_s^*}(\Omega)}=1\}.$$
Clearly, $H_n\subset \bar{H}$, where $\bar{H}$ is given in $\eqref{H}$.
\begin{theorem}\label{second solution}
There exists $\Lambda>0$ such that for $\lambda\in(0,\Lambda)$, the problem $(P^2_{\lambda,n})$ has a positive weak solution $v_n$ in $H_n$.
\end{theorem}
\noindent We will prove this theorem in Section $\ref{sec 2}$. Denote $u_n=w_n+v_n$, where $w_n\in \bar{H}$ and $v_n\in H_n$ are the positive weak solutions to $(P^1_{\lambda,n})$ and $(P^2_{\lambda,n})$, respectively. Thus, $u_n\in H$ is a weak solution to $(P_{\lambda,n})$ where $H$ is defined in $\eqref{H}$. We are now in a position to state the following existence theorem.
\begin{theorem}\label{approx solution}
	There exists $0<\Lambda<\infty$ such that for $\lambda\in (0,\Lambda)$  the problem $(P_{\lambda,n})$ admits a positive weak solution $u_n\in H$, in the sense of Definition $\ref{weak form of apprpx}$.
\end{theorem}
\section{Existence of positive weak solution to $(P_{\lambda,n}^2)$ }\label{sec 2}
Define a function $g_n:\Omega\times\mathbb{R}\rightarrow \mathbb{R}\cup\{-\infty\}$ by
\begin{eqnarray}\label{gn}
\begin{split}
g_n(x,t)=&\begin{cases}
\frac{1}{(w_n(x)+\frac{1}{n})^\gamma}-\frac{1}{(t+w_n(x)+\frac{1}{n})^\gamma}&\text{if}~ t+w_n(x)+\frac{1}{n}>0\\
-\infty&\text{otherwise}.
\end{cases}
\end{split}
\end{eqnarray}
Clearly, $g_n$ is nonnegative and non decreasing in $t$. The required measurability of $g_n(\cdot,t)$ follows
from Lemmata 1 and 2 of \cite{Hirano 2}.\\
 Denote $G_n(x,t)=\int_{0}^{t}g_n(x,\tau)d\tau$ for $(x,t)\in \Omega\times\mathbb{R}$. We define the energy functional $I_{\lambda,n}:H_0^s(\Omega)\rightarrow (-\infty,\infty]$, corresponding to $(P^2_{\lambda,n})$, by
\begin{eqnarray}
\begin{split}
I_{\lambda,n}(v)=&\begin{cases}
\frac{1}{2}\int_{\mathbb{R}^{2N}}\frac{|v(x)-v(y)|^2}{|x-y|^{N+2s}}dxdy+\int_{\Omega}G_n(x,v)dx-\frac{\lambda}{2_s^*}\int_{\Omega}|v+w_n|^{2_s^*}dx&\text{if}~ G_n(.,v)\in L^1(\Omega)\\
\infty&\text{otherwise}.
\end{cases}
\end{split}
\end{eqnarray}
Further,
$$\langle I^\prime_{\lambda,n}(v),\bar{v}\rangle=\int_{\mathbb{R}^{2N}}\frac{(v(x)-v(y))(\bar{v}(x)-\bar{v}(y)}{|x-y|^{N+2s}}dxdy+\int_{\Omega}g_n(x,v)\bar{v}dx-\lambda\int_{\Omega}|v+w_n|^{2_s^*-1}\bar{v}dx$$
for any $\bar{v}\in H_0^s(\Omega)$. We now define the weak solution of $(P^2_{\lambda,n})$ as follows.
\begin{definition}
A function $v_n\in H_n$ is said to be a weak solution of $(P^2_{\lambda,n})$ if $v_n$ is a critical point of the functional $I_{\lambda,n}$.
\end{definition}
\begin{lemma}\label{P-S}
The functional $I_{\lambda,n}$ satisfies the Palais-Smale condition in $H_n$ for energy level
$$c<\frac{s}{N}\frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*}.$$
\begin{proof}
Let $(v_{n,m})\subset H_n$ be a Palais-Smale sequence of $I_{\lambda,n}$, i.e. $I_{\lambda,n}(v_{n,m})\rightarrow c$ and $I^\prime_{\lambda,n}(v_{n,m})\rightarrow 0$. Clearly, the functional $I_{\lambda,n}$ is coercive restricted to $H_n$ and hence the sequence $(v_{n,m})$ is bounded in $H_0^s(\Omega)$. Thus, there exists $v_n\in H_0^s(\Omega)$ and a subsequence of $v_{n,m}$, which is still denoted as $(v_{n,m})$, such that $v_{n,m}\rightarrow v_n$ weakly in $H_0^s(\Omega)$.\\
{\it Claim:} $v_{n,m}\rightarrow v_n$ strongly in $H_0^s(\Omega)$ and $v_n\in H_n$.\\
By the concentration compactness principle [Theorem 2.5, \cite{Mosconi}] for the case $p=2$, there exist two positive Borel regular measures $\nu_1,\nu_2$ such that
\begin{equation}\label{first}
\int_{\mathbb{R}^N}\frac{|v_{n,m}(x)-v_{n,m}(y)|^{2}}{|x-y|^{N+2s}}dy\overset{t}{\rightharpoonup}\nu_1\geq\int_{\mathbb{R}^N}\frac{|v_n(x)-v_n(y)|^{2}}{|x-y|^{N+2s}}dy+\sum_{j\in I}\nu_{1,j}\delta_{x_j},~\nu_{1,j}=\nu_1\{x_j\},
\end{equation}
\begin{equation}\label{second}
|v_{n,m}|^{2_s^*}\overset{t}{\rightharpoonup}\nu_2=|v_n|^{2_s^*}+\sum_{j\in I}\nu_{2,j}\delta_{x_j},~\nu_{2,j}=\nu_2\{x_j\}
\end{equation}	and
\begin{equation}\label{third}
\mathbb{S}_{2,s}\nu_{2,j}^{\frac{2}{2_s^*}}\leq\nu_{1,j}, ~\forall j\in I
\end{equation}
where $\{x_j:j\in I\}$, $I$ countable, is a set of distinct points in $\mathbb{R}^N$,  $\{\nu_{1,j}:j\in I\}\subset(0,\infty),~ \{\nu_{2,j}:j\in I\}\subset (0,\infty)$ and $\mathbb{S}_{2,s}$ is the best Sobolev constant given in  $\eqref{best}$. Here the symbol $\overset{t}{\rightharpoonup}$ denotes the tight convergence. Hence, if $I=\emptyset$ then $v_{n,m}\rightarrow v_n$ strongly in $L^{2_s^*}(\Omega)$ and $v_n\in H_n$.\\
Suppose $I\neq \emptyset$. Then choose $\zeta\in C_c^\infty(\mathbb{R}^N)$ with support in a unit ball of $\mathbb{R}^N$ such that $0\leq\zeta\leq1$ and $\zeta(0)=1$. Let us define for any $\epsilon>0$, the function $\zeta_{\epsilon,j}$ as $\zeta_{\epsilon,j}=\zeta(\frac{x-x_j}{\epsilon})$. Then
\begin{align}\label{1}
\langle I_{\lambda,n}^\prime(v_{n,m}), \zeta_{\epsilon,j}v_{n,m}\rangle&=\int_{\mathbb{R}^{2N}}\frac{(v_{n,m}(x)-v_{n,m}(y))(\zeta_{\epsilon,j}v_{n,m}(x)-\zeta_{\epsilon,j}v_{n,m}(y)}{|x-y|^{N+2s}}dxdy\nonumber\\&~~~+\int_{\Omega}g_n(x,v_{n,m})\zeta_{\epsilon,j}v_{n,m}-\lambda\int_{\Omega}|v_{n,m}+w_n|^{2_s^*-1}\zeta_{\epsilon,j}v_{n,m}\nonumber\\&\geq\int_{\mathbb{R}^{2N}}\frac{(v_{n,m}(x)-v_{n,m}(y))v_{n,m}(y)(\zeta_{\epsilon,j}(x)-\zeta_{\epsilon,j}(y))}{|x-y|^{N+2s}}dxdy\nonumber\\&~~~+\int_{\mathbb{R}^{2N}}\frac{|v_{n,m}(x)-v_{n,m}(y)|^{2}\zeta_{\epsilon,j}(x)}{|x-y|^{N+2s}}dxdy+\int_{\Omega}g_n(x,v_{n,m})\zeta_{\epsilon,j}v_{n,m}\nonumber\\&~~~-\lambda\int_{\Omega}|v_{n,m}+w_n|^{2_s^*}\zeta_{\epsilon,j}dx\nonumber\\&\geq\int_{\mathbb{R}^{2N}}\frac{(v_{n,m}(x)-v_{n,m}(y))v_{n,m}(y)(\zeta_{\epsilon,j}(x)-\zeta_{\epsilon,j}(y))}{|x-y|^{N+2s}}dxdy\nonumber\\&~~~+\int_{\mathbb{R}^{2N}}\frac{|v_{n,m}(x)-v_{n,m}(y)|^{2}\zeta_{\epsilon,j}(x)}{|x-y|^{N+2s}}dxdy+\int_{\Omega}g_n(x,v_{n,m})\zeta_{\epsilon,j}v_{n,m}\nonumber\\&~~~-\frac{\lambda2^*_s}{2}\int_{\Omega}\left(|v_{n,m}+w_n|^{2_s^*-1}w_n+v_{n,m}^{2_s^*-1}w_n\right)\zeta_{\epsilon,j}-\lambda\int_{\Omega}|v_{n,m}|^{2_s^*}\zeta_{\epsilon,j}.
\end{align}
By Mosconi \& Squassina \cite{Mosconi}, we have $\lim\limits_{\epsilon\rightarrow 0}\int_{\mathbb{R}^{2N}}\frac{|v_{n}(y)|^2|\zeta_{\epsilon,j}(x)-\zeta_{\epsilon,j}(y)|^2}{|x-y|^{N+2s}}dxdy=0$. Thus, on using the H\"{o}lder's inequality we have
\begin{align}\label{use1}
&\lim\limits_{\epsilon\rightarrow 0}\lim\limits_{m\rightarrow\infty}\left|\int_{\mathbb{R}^{2N}}\frac{(v_{n,m}(x)-v_{n,m}(y))v_{n,m}(y)(\zeta_{\epsilon,j}(x)-\zeta_{\epsilon,j}(y))}{|x-y|^{N+2s}}dxdy\right|\nonumber\\&\leq \lim\limits_{\epsilon\rightarrow 0}\lim\limits_{m\rightarrow\infty} \|v_{n,m}\|_{H_0^s(\Omega)}\left(\int_{\mathbb{R}^{2N}}\frac{|v_{n,m}(y)|^2|\zeta_{\epsilon,j}(x)-\zeta_{\epsilon,j}(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{1/2}\nonumber\\&\leq \lim\limits_{\epsilon\rightarrow 0}\left(\int_{\mathbb{R}^{2N}}\frac{|v_{n}(y)|^2|\zeta_{\epsilon,j}(x)-\zeta_{\epsilon,j}(y)|^2}{|x-y|^{N+2s}}dxdy\right)^{1/2}\nonumber\\&=0.
\end{align}
Since $\zeta(0)=1$ and for $x\neq x_j$, $\zeta_{\epsilon,j}(x)\rightarrow 0$ as $\epsilon\rightarrow 0$, by using $\eqref{first}$ and $\eqref{second}$ we have
\begin{align}
\lim\limits_{\epsilon\rightarrow 0}\lim\limits_{m\rightarrow\infty}\int_{\Omega}\left(|v_{n,m}+w_n|^{2_s^*-1}w_n+|v_{n,m}|^{2_s^*-1}w_n\right)\zeta_{\epsilon,j}&=\lim\limits_{\epsilon\rightarrow 0}\int_{\Omega}\left(|v_{n}+w_n|^{2_s^*-1}w_n+|v_{n}|^{2_s^*-1}w_n\right)\zeta_{\epsilon,j}=0,\\ \lim\limits_{\epsilon\rightarrow 0}\lim\limits_{m\rightarrow\infty}\int_{\mathbb{R}^{2N}}\frac{|v_{n,m}(x)-v_{n,m}(y)|^{2}\zeta_{\epsilon,j}(x)}{|x-y|^{N+2s}}dxdy&=\lim\limits_{\epsilon\rightarrow 0}\int_{\mathbb{R}^{N}}\zeta_{\epsilon,j}d\mu=\nu_{1,j},\\\lim\limits_{\epsilon\rightarrow 0}\lim\limits_{m\rightarrow\infty}\int_{\Omega}g_n(x,v_{n,m})\zeta_{\epsilon,j}v_{n,m}&=\lim\limits_{\epsilon\rightarrow 0}\int_{\mathbb{R}^{N}}g_n(x,v_n)v_n\zeta_{\epsilon,j}=0,\\\lim\limits_{\epsilon\rightarrow 0}\lim\limits_{m\rightarrow\infty}\int_{\Omega}|v_{n,m}|^{2_s^*}\zeta_{\epsilon,j}&=\lim\limits_{\epsilon\rightarrow 0}\int_{\Omega}\zeta_{\epsilon,j}d\nu=\nu_{2,j}.\label{use2}
\end{align}
Thus, with the consideration of the equations $\eqref{use1}-\eqref{use2}$, from $\eqref{1}$ we have $0\geq \nu_{1,j}-\lambda\nu_{2,j}$. This further implies that $\nu_{1,j}\leq\lambda\nu_{2,j}$. From $\eqref{third}$, we already have $\mathbb{S}_{2,s}\nu_{2,j}^{\frac{2}{2^*_s}}\leq\nu_{1,j}$. Hence, we obtain $\nu_{2,j}\geq \left(\frac{\mathbb{S}_{2,s}}{\lambda}\right)^{\frac{N}{2s}}$.\\
On the other hand,
\begin{align}
c&=\lim\limits_{m\rightarrow\infty}I_{\lambda,n}(v_{n,m})\nonumber\\&= \lim\limits_{m\rightarrow\infty}\left(\frac{1}{2}\int_{\mathbb{R}^{2N}}\frac{|v_{n,m}(x)-v_{n,m}(y)|^2}{|x-y|^{N+2s}}dxdy+\int_{\Omega}G_n(x,v_{n,m})dx\right)-\frac{\lambda}{2_s^*}\nonumber\\&\geq\frac{1}{2}\int_{\mathbb{R}^{N}}\left(\int_{\mathbb{R}^N}\frac{|v_n(x)-v_n(y)|^{2}}{|x-y|^{N+2s}}dy+\sum_{j\in I}\nu_{1,j}\right) dx-\frac{\lambda}{2_s^*}\nonumber\\&\geq \frac{s}{N}\nu_{1,j}-\frac{\lambda}{2_s^*}\nonumber\\&\geq\frac{s}{N}\mathbb{S}_{2,s}\nu_{2,j}^{\frac{2}{2_s^*}}-\frac{\lambda}{2_s^*}\nonumber\\&\geq \frac{s}{N} \frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*},\nonumber
\end{align}
which is a contradiction to our assumption $c<\frac{s}{N} \frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*}$. Hence, the indexing set $I$ is empty and $v_{n,m}\rightarrow v_n$ strongly in $L^{2_s^*}(\Omega)$ and $v_n\in H_n$. \\
It remains to prove that $v_{n,m}\rightarrow v_n$ strongly in $H_0^s(\Omega)$. We use a standard method to prove this claim. Recall
\begin{align}\label{conv}
\langle I^\prime_{\lambda,n}(v_{n,m}),v_{n,m}-v_n\rangle&=\int_{\mathbb{R}^{2N}}\frac{(v_{n,m}(x)-v_{n,m}(y))((v_{n,m}-v_n)(x)-(v_{n,m}-v_n)(y))}{|x-y|^{N+2s}}dxdy\nonumber\\&~~~+\int_{\Omega}g_n(x,v_{n,m})(v_{n,m}-v_n)-\lambda\int_{\Omega}|v_{n,m}+w_n|^{2_s^*-1}(v_{n,m}-v_n).
\end{align}
Since $(v_{n,m})$ is a bounded Palais-Smale sequence, on passing the limit $m\rightarrow\infty$ in $\eqref{conv}$ we have
$$\lim\limits_{m\rightarrow\infty}\int_{\mathbb{R}^{2N}}\frac{(v_{n,m}(x)-v_{n,m}(y))((v_{n,m}-v_n)(x)-(v_{n,m}-v_n)(y))}{|x-y|^{N+2s}}dxdy=0.$$
On using a simple calculation we get
\begin{align}\label{G}
\|v_{n,m}-v_n\|_{H_0^s(\Omega)}^{2}&=\int_{\mathbb{R}^{2N}}\frac{|(v_{n,m}-v_n)(x)-(v_{n,m}-v_n)(y)|^{2}}{|x-y|^{N+2s}}dxdy\nonumber\\&\leq C\int_{\mathbb{R}^{2N}}
\left\{\frac{(v_{n,m}(x)-v_{n,m}(y))((v_{n,m}-v_n)(x)-(v_{n,m}-v_n)(y))}{|x-y|^{N+2s}}\right. \nonumber \\
&~~~~\left.\kern-\nulldelimiterspace -\;\frac{(v_n(x)-v_n(y))((v_{n,m}-v_n)(x)-(v_{n,m}-v_n)(y))}{|x-y|^{N+2s}}\vphantom{}\right\}.\nonumber
\end{align}
Thus, $\lim\limits_{m\rightarrow\infty}\|v_{n,m}-v_n\|_{H_0^s(\Omega)}=0$ and hence $v_{n,m}\rightarrow v_n$ strongly in $H_0^s(\Omega)$. Therefore, $v_n\in H_n$ is a critical point of $I_{\lambda,n}$ and hence a weak solution to $(P^2_{\lambda,n})$.
\end{proof}
\end{lemma}
\noindent Consider the sequence $(V_\epsilon)$ which is given by $$V_\epsilon=\epsilon^{-\frac{N-2s}{2}} v^*\left(\frac{x}{\epsilon}\right), ~~x\in \mathbb{R}^N.$$
Here $v^*(x)=\bar{v}\left(\frac{x}{\mathbb{S}_{2,s}^{\frac{1}{2s}}}\right)$, $\bar{v}(x)=\frac{\tilde{v}(x)}{\|\tilde{v}\|_{L^{2_s^*}(\Omega)}}$ and $\tilde{v}(x)=\beta(\alpha^2+|x|^2)^{-\frac{N-2s}{2}}$ with two fixed constants $\beta\in \mathbb{R}^N\setminus\{0\},~\alpha>0$. According to Servadei \& Valdinoci \cite{Sevadei 1}, for each $\epsilon>0$ the corresponding $V_\epsilon$ satisfies the problem
$$(-\Delta)^s v=|v|^{2_s^*-2}v~~\text{ in }\mathbb{R}^N$$ and $$\int_{\mathbb{R}^{2N}}\frac{|V_\epsilon(x)-V_\epsilon(y)|^2}{|x-y|^{N+2s}}dxdy=\int_{\mathbb{R}^N}|V_\epsilon|^{2_s^*}dx=\mathbb{S}_{2,s}^{N/2s}.$$
Without loss of generality we can assume $0\in\Omega$. Consider the function $\zeta\in C_c^{\infty}(\mathbb{R}^N)$ such that $0\leq\zeta\leq 1$ and for a fixed $\delta>0$ with $B_{4\delta}\subset\Omega$, $\zeta\equiv 0$ in $\mathbb{R}^N\setminus B_{2\delta}$, $\zeta\equiv1$ in $B_\delta$. Let us define a function $\Psi_\epsilon(x)=\zeta(x)V_\epsilon(x)$, which is zero in $\mathbb{R}^N\setminus\Omega$. By Giacomoni et al. \cite{Giacomoni}, there exists $a_1,a_2,a_3>0$ such that for $1<q<\min\{2,\frac{N}{N-2s}\}$ we have the following estimates.
\begin{align}
\int_{\mathbb{R}^{2N}}\frac{|\Psi_\epsilon(x)-\Psi_\epsilon(y)|^2}{|x-y|^{N+2s}}dxdy&\leq \mathbb{S}_{2,s}^{N/2s}+a_1\epsilon^{N-2s},\nonumber\\
\int_{\Omega}|\Psi_\epsilon|^{2_s^*}dx&\geq \mathbb{S}_{2,s}^{N/2s}-a_2\epsilon^N,\nonumber\\
\int_{\Omega}|\Psi_\epsilon|^{q}dx&\leq a_3\epsilon^{(N-2s)q/2}.\nonumber
\end{align}
\begin{lemma}\label{upper bound}
There exists $\Lambda>0$ such that for sufficienty small $\epsilon>0$ and for $\lambda\in (0,\Lambda)$,
$$\sup\{I_{\lambda,n}(t\Psi_\epsilon): t\geq 0\}<\frac{s}{N}\frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*}.$$
\begin{proof}
Clearly for $\lambda<\left(\frac{2_s^*s}{N}\right)^{2s/N}\mathbb{S}_{2,s}$, we have $\left(\frac{s}{N}\frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*}\right)>0$. Consider $\epsilon>0$ to be sufficiently small. Then for any $t\geq0$,
\begin{align}
I_{\lambda,n}(t\Psi_\epsilon)&=\frac{t^2}{2}\int_{\mathbb{R}^{2N}}\frac{|\Psi_\epsilon(x)-\Psi_\epsilon(y)|^2}{|x-y|^{N+2s}}dxdy+\int_{\Omega}G_n(x,t\Psi_\epsilon)dx-\frac{\lambda}{2_s^*}\int_{\Omega}|t\Psi_\epsilon+w_n|^{2_s^*}dx\nonumber\\&=\frac{t^2}{2}\int_{\mathbb{R}^{2N}}\frac{|\Psi_\epsilon(x)-\Psi_\epsilon(y)|^2}{|x-y|^{N+2s}}dxdy+\int_{\Omega}\frac{|t\Psi_\epsilon|}{(w_n+1/n)^\gamma}\nonumber\\&~~~-\frac{1}{1-\gamma}\int_{\Omega}(t\Psi_\epsilon+w_n+1/n)^{1-\gamma}-(w_n+1/n)^{1-\gamma}-\frac{\lambda}{2_s^*}\int_{\Omega}|t\Psi_\epsilon+w_n|^{2_s^*}dx\nonumber\\&\leq \frac{t^2}{2}(\mathbb{S}_{2,s}^{N/2s}+a_1\epsilon^{N-2s})+tn^\gamma\int_{\Omega}|\Psi_\epsilon|+\frac{\lambda}{2_s^*}-\frac{\lambda}{2_s^*}\nonumber\\&~~~-\frac{1}{1-\gamma}\int_{\Omega}(t\Psi_\epsilon+w_n+1/n)^{1-\gamma}-(w_n+1/n)^{1-\gamma}-\frac{\lambda t^{2_s^*}}{2_s^*}\int_{\Omega}|\Psi_\epsilon|^{2_s^*}dx\nonumber\\&\leq \frac{t^2}{2}(\mathbb{S}_{2,s}^{N/2s}+a_1\epsilon^{N-2s})+tn^\gamma a_3^{1/q}\epsilon^{(N-2s)/2}+\frac{\lambda}{2_s^*}-\frac{\lambda}{2_s^*}\nonumber\\&~~~-\frac{1}{1-\gamma}\int_{\Omega}(t\Psi_\epsilon+w_n+1/n)^{1-\gamma}-(w_n+1/n)^{1-\gamma}-\frac{\lambda t^{2_s^*}}{2_s^*}(\mathbb{S}_{2,s}^{N/2s}-a_2\epsilon^N).
\end{align}
Assume $\lambda\leq1$ and denote a function $h:\mathbb{R}^+\rightarrow\mathbb{R}$ as follows.
\begin{align}
h(t)&=\frac{\lambda}{2_s^*}-\frac{1}{1-\gamma}\int_{\Omega}(t\Psi_\epsilon+w_n+1/n)^{1-\gamma}-(w_n+1/n)^{1-\gamma}\nonumber\\&\leq \frac{1}{2_s^*}-\frac{1}{1-\gamma}\int_{\Omega}(t\Psi_\epsilon+w_n+1/n)^{1-\gamma}-(w_n+1/n)^{1-\gamma}\nonumber\\&\leq \frac{1}{2_s^*}-\frac{1}{1-\gamma}\int_{\Omega}(t\Psi_\epsilon)^{1-\gamma}+C.
\end{align}
Since $w_n\in \bar{H}$, we can bound $\int_{\Omega} (w_n+1/n)^{1-\gamma}$ uniformly by a constant $C>0$, independent of $n$. Clearly as $t\rightarrow\infty$, $h(t)\rightarrow -\infty$. Hence, there exists $T>0$ such that for every $t\geq T$, $h(t)\leq0$. Thus, for $t\geq T$ we get
\begin{align}
I_{\lambda,n}(t\Psi_\epsilon)&\leq \frac{t^2}{2}(\mathbb{S}_{2,s}^{N/2s}+a_1\epsilon^{N-2s})+tn^\gamma a_3^{1/q}\epsilon^{(N-2s)/2}-\frac{\lambda t^{2_s^*}}{2_s^*}(\mathbb{S}_{2,s}^{N/2s}-a_2\epsilon^N)-\frac{\lambda}{2_s^*}\nonumber\\&= \bar{h}_\epsilon(t).\nonumber
\end{align}
A simple use of basic calculus yields that the maximum value of $\bar{h}_\epsilon$ is attained at $$t_\lambda=\left(\frac{1}{\lambda}\right)^{\frac{N-2s}{4s}}+o(\epsilon^{(N-2s)/2}).$$
Thus, we have
\begin{align}
I_{\lambda,n}(t\Psi_\epsilon)&\leq \frac{s}{N}\frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}+o(\epsilon^{(N-2s)/2})-\frac{\lambda}{2_s^*}\nonumber\\&< \frac{s}{N}\frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*}.
\end{align}
For any $t<T$,
\begin{align}
I_{\lambda,n}(t\Psi_\epsilon)&\leq\frac{t^2}{2}\int_{\mathbb{R}^{2N}}\frac{|\Psi_\epsilon(x)-\Psi_\epsilon(y)|^2}{|x-y|^{N+2s}}dxdy+\int_{\Omega}\frac{|t\Psi_\epsilon|}{(w_n+1/n)^\gamma}\nonumber\\&\leq \frac{t^2}{2}(\mathbb{S}_{2,s}^{N/2s}+a_1\epsilon^{N-2s})+tn^\gamma a_3^{1/q}\epsilon^{(N-2s)/2}\nonumber\\&<\frac{T^2}{2}(\mathbb{S}_{2,s}^{N/2s}+a_1\epsilon^{N-2s})+Tn^\gamma a_3^{1/q}\epsilon^{(N-2s)/2}.\nonumber
\end{align}
Choose $\lambda^*>0$ depending on $T,N,2s,\mathbb{S}_{2,s}$ such that for $\lambda\in(0,\lambda^*)$ we obtain
$$I_{\lambda,n}(t\Psi_\epsilon)<\frac{s}{N}\frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*}.$$
Denote $\Lambda=\min\{1,\left(\frac{s}{N}2_s^*\mathbb{S}_{2,s}\right)^{2s/N},\lambda^*\}$. Then for $0<\lambda<\Lambda$ we have $$\sup\{I_{\lambda,n}(t\Psi_\epsilon): t\geq 0\}<\frac{s}{N}\frac{\mathbb{S}_{2,s}^{\frac{N}{2s}}}{\lambda^{\frac{N-2s}{2s}}}-\frac{\lambda}{2_s^*.}$$ Hence the result.
\end{proof}
\end{lemma}
\begin{proof}[Proof of Theorem $\ref{second solution}$]
We at first need to produce a Palais-Smale sequence named $(v_{n,m})$ of $I_{\lambda,n}$ in $H_n$ using the Ekeland variational principle (see Theorem $\ref{eke}$). Then by  Lemmata $\ref{P-S}$ and $\ref{upper bound}$, there exists $\Lambda>0$ such that for any $\lambda\in (0,\Lambda)$, the functional $I_{\lambda,n}$ satisfies the Palais-Smale compactness condition in $H_n$. This guarantees the existence of a critical point $v_n$ of $I_{\lambda,n}$ in $H_n$ for any $\lambda\in (0,\Lambda)$.\\
Observe that $H_n\subset H_0^s(\Omega)$ is a complete Hilbert manifold. Since the functional $I_{\lambda,n}$ is $C^1$ and bounded from below on $H_n$, we denote $k_n=\underset{v\in H_n}{\inf}I_{\lambda,n}(v)$. Hence, there exists a sequence $(u_{n,m})\subset H_n$ such that $I_{\lambda,n}(u_{n,m})\rightarrow k_n$ as $m\rightarrow\infty$ and for every $\epsilon>0$ there exists $m_0\in \mathbb{N}$ such that $I_{\lambda,n}(u_{n,m})<k_n+\epsilon$ for every $m\geq m_0$. The functional $I_{\lambda,n}$ satisfies the hypotheses of Ekeland variational principle stated in Theorem $\ref{eke}$. By choosing $\delta=\sqrt{\epsilon}$ in Theorem $\ref{eke}$, we guarantee the existence of a sequence $(v_{n,m})\subset H_n$ such that $(I_{\lambda,n}(v_{n,m}))$ is uniformly bounded and $ I_{\lambda,n}^\prime(v_{n,m})\rightarrow 0$. This implies $(v_{n,m})$ is a Palais-Smale sequence and we conclude our proof.
\end{proof}
\section{Existence of SOLA to $(P_\lambda)$}\label{sec 3}
Let us denote $u_n=v_n+w_n$, where $w_n\in \bar{H}$ and $v_n\in H_n$ are the positive weak solutions to $(P^1_{\lambda,n})$ and $(P^2_{\lambda,n})$, respectively. Thus, $u_n\in H$ is a positive weak solution to $(P_{\lambda,n})$ and for every $\omega\subset\subset\Omega$, there exists a constant $C_\omega$ such that $u_n\geq C_\omega>0$. Refer Section $\ref{2}$ for these notations of the function spaces. \\
In this section, we discuss the boundedness of the sequence of solutions $(u_n)$ in a suitable fractional Sobolev space and also prove the existence of SOLA to $(P_\lambda)$.
\begin{lemma}\label{lemma1}
	Let $0<\gamma<1$ and $u_n\in H$ be a weak solution to $(P_{\lambda,n})$ as given in Theorem $\ref{approx solution}$. Then the sequence $(u_n)$ is bounded in $W_0^{s_1,q}(\Omega)$ for every $0<s_1<s$ and $1\leq q<\frac{N}{N-s}$.
	\begin{proof}
We follow Panda et al. in \cite{Panda} to prove this lemma. Let $u_n\in H$ be a weak solution of $(P_{\lambda,n})$. Then for any $k\geq1$, consider $\phi=T_k(u_n)$ as a test function in the weak formulation $\eqref{approximating weak}$ of $(P_{\lambda,n})$ and we get
	\begin{align}
		\int_{\mathbb{R}^N}|(-\Delta)^{s/2}T_k(u_n)|^2&\leq\int_{\mathbb{R}^{N}}(-\Delta)^{s/2}u_n\cdot(-\Delta)^{s/2}T_k(u_n)\nonumber\\&=\int_{\Omega}\frac{1}{(u_n+\frac{1}{n})^\gamma}T_k(u_n)+\int_{\Omega}\lambda u_n^{2_s^*-1}T_k(u_n)+\int_{\Omega}\mu_n T_k(u_n).
\end{align}
Clearly,
 $\frac{T_k(u_n)}{(u_n+\frac{1}{n})^\gamma}\leq \frac{u_n}{(u_n+\frac{1}{n})^\gamma}\leq u_n^{1-\gamma}$. Since  $\|u_n\|_{L^{2_s^*}(\Omega)}=1$ and the sequence $(\mu_n)$ is $L^1$ bounded we have
\begin{align}\label{4p6}
	\int_{\mathbb{R}^N}|(-\Delta)^{s/2}T_k(u_n)|^2&\leq \int_{\Omega} u_n^{1-\gamma}+k\|\mu_n\|_{L^1(\Omega)}+\lambda k\|u_n\|_{L^{2_s^*-1}(\Omega)}^{2_s^*-1}\nonumber\\&\leq C_1\|u_n\|_{L^{2_s^*}(\Omega)}^{1-\gamma}+C_2k+\lambda C_3\|u_n\|_{L^{2_s^*}(\Omega)}^{2_s^*-1}\nonumber\\&\leq Ck.
\end{align}
Thus, $(T_k(u_n))$ is bounded in $H_0^s(\Omega)$. Consider the sets $I=\{x\in \Omega: |(-\Delta)^{s/2}u_n|\geq t \}$, $I_1=\{x\in \Omega: |(-\Delta)^{s/2}u_n|\geq t, u_n\leq k \}$ and $I_2=\{x\in \Omega: u_n>k\}$. Then, $I\subset I_1\cup I_2$, which implies $m(I)\leq m(I_1)+m(I_2)$, where $m$ is the Lebesgue measure. Using the Sobolev inequality, stated in Theorem $\ref{constant}$, we establish
\begin{align}\label{4p7}
\left(\int_{\Omega}|T_k(u_n)|^{2_s^*}\right)^{\frac{2}{2_s^*}}&\leq C^\prime \int_{\mathbb{R}^N}|(-\Delta)^{s/2}T_k(u_n)|^2\nonumber\\&\leq Ck.
\end{align}
Then on $I_2$, the equation $\eqref{4p7}$ becomes
\begin{align}\label{4p8}
k^2 m(I_2)^{\frac{2}{2_s^*}}&\leq Ck\nonumber\\ m(I_2)& \leq \frac{ C}{k^{\frac{N}{N-2s}}}, ~\forall k\geq 1.
\end{align}
This proves $(u_n)$ is bounded in ${M}^{\frac{N}{N-2s}}(\Omega)$. Similarly on $I_1$, the equation $\eqref{4p6}$ becomes
\begin{align}\label{4p9}
t^2 m(I_1)&\leq Ck\nonumber\\ m(I_1)&\leq  \frac{Ck}{t^2}, ~\forall k>1.
\end{align}
\noindent On combining $\eqref{4p8}$ and $\eqref{4p9}$ we have
\begin{align}
m(I)\leq \frac{Ck}{t^2}+\frac{C}{k^{\frac{N}{N-2s}}}, ~\forall k>1.\nonumber
\end{align}
Choose $k=t^{\frac{N-2s}{N-s}}$. Thus, we obtain $$m(I)\leq \frac{C}{t^{\frac{N}{N-s}}},~\forall t\geq 1.$$
Therefore, the sequence $((-\Delta)^{s/2}u_n)$ is bounded in $M^{\frac{N}{N-s}}(\Omega)$. By the continuous embedding $\eqref{continuous}$, the sequences $(u_n)$ and $((-\Delta)^{s/2}u_n)$ are bounded in $L^q(\Omega)$ for every $q<\frac{N}{N-s}$. From Definition $\ref{Bessel}$ and Theorem $\ref{properties}$, we conclude that $(u_n)$ is uniformly bounded in $L^{s,q}(\mathbb{R}^N)$  and hence bounded in $W_0^{s_1,q}(\Omega)$, for every $s_1<s$ and $q<\frac{N}{N-s}$.
	\end{proof}
\end{lemma}
	\noindent We now pass the limit $n\rightarrow\infty$ in the weak formulation $\eqref{approximating weak}$ to prove the existence of a SOLA to $(P_\lambda)$.
	\begin{proof}[Proof of Theorem $\ref{main theorem}$]
Let $\mu\in \mathcal{M}(\Omega)$, $0<\gamma< 1$ and $u_n\in H$ be a weak solution of $(P_{\lambda,n})$ for $\lambda\in (0,\Lambda)$. According to Lemma $\ref{lemma1}$, $(u_n)$ is bounded in $W_0^{s_1,q}(\Omega)$,  for every $0<s_1<s$ and $ q<\frac{N}{N-s}$. Hence, there exists $u\in W_0^{s_1,q}(\Omega)$ such that $u_n\rightarrow u$ weakly in $W_0^{s_1,q}(\Omega)$. Thus, $u_n\rightarrow u$ a.e. in $\mathbb{R}^N$ and $u\equiv 0$ in $\mathbb{R}^N\setminus\Omega$. \\
Denote $$\Phi(x,y)=\phi(x)-\phi(y),~ \bar{U}_n(x,y)=u_n(x)-u_n(y),$$ $$\bar{U}(x,y)=u(x)-u(y) ~\text{and}~ d\nu=\frac{dxdy}{|x-y|^{N+2s}}.$$
For every $\phi\in C_c^\infty(\Omega)$, from the weak formulation of $(P_{\lambda,n})$ we have
\begin{align}
\int_{\mathbb{R}^{2N}} \bar{U}_n(x,y)\Phi(x,y)d\nu&=\int_{\Omega}\frac{1}{(u_n+\frac{1}{n})^\gamma}\phi+\int_{\Omega}\lambda u_n^{2_s^*-1}\phi+\int_{\Omega}\mu_n \phi.\nonumber
\end{align}
We can rewrite the above equation as
\begin{align}\label{4p10}
 \int_{\mathbb{R}^{2N}} \bar{U}(x,y)\Phi(x,y)d\nu+\int_{\mathbb{R}^{2N}} (\bar{U}_n(x,y)-\bar{U}(x,y))\Phi(x,y)d\nu= \int_{\Omega}\frac{1}{(u_n+\frac{1}{n})^\gamma}\phi+\int_{\Omega}\lambda u_n^{2_s^*-1}\phi+\int_{\Omega}\mu_n \phi.
\end{align}
Clearly, $$\lim\limits_{n\rightarrow\infty}\int_{\Omega}\mu_n\phi=\int_{\Omega}\phi d\mu,$$
$$\lim\limits_{n\rightarrow\infty}\lambda\int_{\Omega}u_n^{2_s^*-1}\phi=\lambda\int_{\Omega}u^{2_s^*-1}\phi.$$
On using the Dominated Convergence Theorem we get
$$\lim\limits_{n\rightarrow\infty}\int_{\Omega}\frac{1}{(u_n+\frac{1}{n})^\gamma}\phi=\int_{\Omega}\frac{1}{u^\gamma}\phi.$$
Now the integral on the left hand side of $\eqref{4p10}$ can be expressed as follows.
\begin{align}
\int_{\mathbb{R}^{2N}} (\bar{U}_n(x,y)-\bar{U}(x,y))\Phi(x,y)d\nu&=\int_{\Omega\times\Omega} (\bar{U}_n(x,y)-\bar{U}(x,y))\Phi(x,y)d\nu\nonumber\\&~~+\int_{\Omega\times(\mathbb{R}^{N}\setminus\Omega)} (\bar{U}_n(x,y)-\bar{U}(x,y))\Phi(x,y)d\nu\nonumber\\&~~+\int_{(\mathbb{R}^{N}\setminus\Omega)\times\Omega} (\bar{U}_n(x,y)-\bar{U}(x,y))\Phi(x,y)d\nu\nonumber\\&=J_{1,n}+J_{2,n}+J_{3,n}.\nonumber
\end{align}
Observe $\bar{U}_n\rightarrow \bar{U}$ a.e. in $\mathbb{R}^N$. Since $\Omega$ is bounded, using Lemma $\ref{lemma1}$ and Vitali's lemma we have $\bar{U}_n\rightarrow \bar{U}$ strongly in $L^1(\Omega\times \Omega, d\nu)$. Hence, $J_{1,n}\rightarrow 0$ as $n\rightarrow\infty$. \\
Let $(x,y)\in \Omega\times(B_R\setminus\Omega)$, then $$\underset{(x,y)\in \Omega\times(B_R\setminus\Omega)}{\sup}\frac{1}{|x-y|^{N+2s}}\leq C<\infty.$$ Hence, by the Dominated Convergence Theorem, $J_{2,n}\rightarrow 0$ as  $n\rightarrow\infty$ and similarly $J_{3,n}\rightarrow 0$ as $n\rightarrow \infty$. Thus on passing the limit $n\rightarrow \infty$ in $\eqref{4p10}$, we obtain
\begin{align}
\int_{\mathbb{R}^{2N}} \bar{U}(x,y)\Phi(x,y)d\nu= \int_{\Omega}\frac{1}{u^\gamma}\phi+\int_{\Omega}\lambda u^{2_s^*-1}\phi+\int_{\Omega} \phi d\mu.
\end{align}
Thus, $u$ is a positive SOLA to $(P_\lambda)$, in the sense of Definition $\ref{SOLA}$.
\end{proof}
	\section*{Appendix}\label{sec 4}
	We now discuss about the multiplicity of solutions to the problem $(P_\lambda)$ for the case $0<\gamma<1$. Let $I_\lambda$ be the corresponding energy functional of $(P_\lambda)$ given by
	$$I_\lambda(u)=\frac{1}{2}\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy-\frac{1}{1-\gamma}\int_{\Omega}u^{1-\gamma}-\frac{\lambda}{2_s^*}\int_{\Omega}u^{2_s^*}-\int_{\Omega}ud\mu.$$
For $0<\gamma<1$, $I_\lambda$ is a $C^1$ functional and $I_\lambda(0)=0.$ From Theorem $\ref{main theorem}$, $u\in W_0^{s_1,q}(\Omega)$ is a  positive SOLA to $(P_\lambda)$ for every $s_1<s$, $q<\frac{N}{N-s}$. Then $u$ is also a Nehari solution of $(P_\lambda)$, i.e. $u\in N_\lambda=\{u\in W_0^{s_1,q}(\Omega):\langle I_\lambda^\prime(u),u\rangle=0 \}$. Here
$$\langle I_\lambda^\prime(u),u\rangle=\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy-\int_{\Omega}u^{1-\gamma}-\lambda\int_{\Omega}u^{2_s^*}-\int_{\Omega}ud\mu.$$
Consider the fibre map $\psi:(0,\infty)\rightarrow\mathbb{R}$ defined by
$$\psi(t)=\frac{t^2}{2}\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy-\frac{t^{1-\gamma}}{1-\gamma}\int_{\Omega}u^{1-\gamma}-\frac{\lambda t^{2_s^*}}{2_s^*}\int_{\Omega}u^{2_s^*}-t\int_{\Omega}ud\mu.$$
Then $$\psi^\prime(t)=At-Bt^{-\gamma}-\lambda C t^{2_s^*-1}-D$$ and $$\psi^{\prime\prime}(t)= A+\gamma Bt^{-\gamma-1}-(2_s^*-1)\lambda Ct^{2_s^*-2}$$
where $A=\int_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy$, $B=\int_{\Omega}u^{1-\gamma}$, $C=\int_{\Omega}u^{2_s^*}$ and $D=\int_{\Omega}ud\mu$. Since $u\in N_\lambda$, $\psi^\prime(1)=A-B-\lambda C-D=0.$ Clearly $\psi^\prime(t)\rightarrow -\infty$ as $t\rightarrow 0$ and $t\rightarrow \infty$.\\
{\it Case 1:} If $\Lambda>\lambda>\frac{A+\gamma B}{(2^*_s-1)C}$, then $\psi^{\prime\prime}(1)<0.$ Hence, there exists at least one Nehari solution to $(P_\lambda)$.\\
{\it Case 2:} If $\lambda<\min \{\frac{A+\gamma B}{(2^*_s-1)C},\Lambda\}$, then $\psi^{\prime\prime}(1)>0$ and we guarantee the existence of at least three nontrivial Nehari solution to $(P_\lambda)$.\\
{\it Case 3:} If $\lambda=\frac{A+\gamma B}{(2^*_s-1)C}$, then $\psi^{\prime\prime}(1)=0$. Thus, we obtain a saddle point and hence there exists at least one Nehari solution to $(P_\lambda)$.
	\section*{Acknowledgement}
The author Debajyoti Choudhuri thanks the grant received from Science and Engineering Research Board (SERB) for the project MATRICS- MTR/2018/000525. Akasmika Panda thanks the financial assistantship received from the Ministry of Human Resource Development (M.H.R.D.), Govt. of India. Ratan Kr. Giri acknowledges the financial support and facilities
received from the Mathematics Department, Technion - Israel Institute of Technology, Haifa. The authors thank the anonymous reviewers for their comments and suggestions.

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