Harry Feiner's set for act one of "The Importance of Being Earnest" at the Worcester Foothills Theatre is all lush Moorish marzipan, with rugs and cushions and hangings. His set for act two is a spun-sugar filigree garden. And, since the division between them has been blurred, no matter what the program says there really is no act three. Even so, with a play in which every line is a laugh-line and nothing is ever taken seriously, the danger of dining on nothing but confectioners sugar is that one may happily laugh oneself sick. This is a totally professional production, in one of the best new theatres in New England.
Everyone in Oscar Wilde's play is of that posh, effete set who will never work an hour in their lives, who will fly to country estates when their London town-houses become a bore, whose aunts will plot engagements and marriages consulting a Stud Book of pedigree and a cold eye toward investments and income, and who can decide at age eighteen that no one not bearing the name "Earnest" can ever be a suitable husband. If this is not a world made of spun sugar and marzipan, what is?
The problem here is not to make the audience laugh, but to choose which of the myriad possibilities to point them toward. Director Peter Bennett has kept everything as light and frothy as a meringue, so that a slight pause, a drop in tone, or a snarled line-end is all it takes to lay the audience in the aisles. Deftest at slight pauses is Kathleen Huber as Lady Bracknell, whose seconds of silence can draw blood by their implication. Jennifer DiBella and Charlotte Anne Dore as Gwendolen and Cecily (who, arm in arm, might well have "Tweedle-" embroidered round the backs of their collars) are past-mistresses of the dropped tone, while Margery Shaw's Miss Prism excels in snarls. The variety of approach in these ladies' delivery lends a subtlety to the surface of the play that scholars could study at length without ever comprehending.
As Algy and Jack, who delight in playing at aliases, Matthew Amory and Steve Dane start the show trading epigrams and dryly toppling conventions on all sides. They are the sort whose soft breath can significantly alter the sheen on their fingernails, and --- until love galvanizes them into extraordinary action --- to whom the alteration in sheen is uncommonly significant.
Bill Taylor as Algy's Butler and Steven Barkheimer as Jack's are dutiful automatons obeying immediately and unquestioningly the every whim of their masters so crisply and devotedly as to call double-takes from the audience. And Michael G. Dell'Orto as a smitten and therefore tongue-tangled Reverend Chausuble rounds out the trio of eventually mated spouses. These are supposedly small roles perfectly delineated, with their own sorts of significant detail adding to the whole.
It would be easy for a mass audience to miss the subtleties or ignore the delicate detail in so frothy a confection, to denigrate the simple stand-and-deliver accuracy of these line-readings. Americans are ill-equipped to respond to the profound surfaces of English Edwardian fops. But a few hours spent at the Worcester Foothills Theatre studying Peter Bennett's excellently well-drilled cast should open their eyes and unlace their guffaws. This is a totally professional production, and funny as all hell.


an art exhibition curated by Liliana Gallagher and Anna Fabres.
'Here and Now' showcases new artwork from a group of national and international Artists who offer an intriguing glimpse of their work in a variety of media reflecting their personal styles and approaches to contemporary and modern art.
Using an eclectic mix of different media including drawing, painting, print, installation, photography, textiles and sculpture, this show promises to be a fascinating journey exploring many pathways.
The exhibition will be on view from June 6 through June 11, 2017 at The Tabernacle gallery in Notting Hill.
The private view is on Tuesday, the 6th of June from 6pm to 9pm and it is a guest listed event.
There will be a band playing jazz on location from 9pm to 10:30pm.
Don't miss the opportunity to view the work of our talented Artists!


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Two Workshops with Peter S. Adler, PhD
by Ku'ikahi Mediation Center
Fri, Jun 4, 2021, 8:00 AM –
Fri, Jun 18, 2021, 12:00 PM HST
Refunds up to 1 day before event
"Collaborative Planning & Problem-Solving" Training Series
Sign up for 1 or both!
THE MULTI-PARTY POLICY PROBLEM
Friday, June 4, 2021 from 8:00am to 12:00pm
"This workshop focuses on politically contentious planning and regulatory conflicts that often have communities, government agencies, and businesses at loggerheads and too often result in political gridlock or litigation," says Dr. Adler. "The focus of this workshop is on organizing, designing, and managing complex problem solving processes; engaging in constructive interchange; and producing policy 'impacts.'" Recommended for individuals who work in government, business, and community affairs and must interact with others to prevent, manage, or resolve disagreements.
THE FACT INTENSIVE CONTROVERSY
Friday, June 18, 2021 from 8:00am to 12:00pm
"Most of us were taught that a 'fact' is something proven through scientific, legal, or rules-of-logic reasoning. Once established, facts are supposed to be incontrovertible until something new is shown to be valid," says Dr. Adler. "In some ways, nothing is a fact until we agree it's a fact or a credible authority says so." This workshop will offer approaches to working with different assertions of 'the truth' for purpose of negotiation and problem solving. Recommended for legal, technical, and scientific experts—or people who rely on them—and who must engage counterparts on difficult problems.
Peter S. Adler, PhD is a planner and conflict management expert with a long track record of helping public, private, and civil sector organizations achieve alignment and solve tough problems. He is affiliated with The ACCORD3.0 Network and GUILD Consulting and is sought after for his expertise in resolving complex and consequential issues. Prior executive experience includes serving as President and CEO of The Keystone Center, Executive Director of the Hawai'i Justice Foundation, and founding Director of the Hawai'i State Judiciary's Center for Alternative Dispute Resolution. He has helped groups have productive discussions about business strategies, environmental problems, and public health controversies. In addition to "Stories from Ground Zero" some of his other major projects include a negotiated "Joint Fact Finding" agreement on the pesticides used by GMO corn seed producers, a new strategic plan for a major television station, and the acceleration of cooperation strategies for the ozone treaty under the UN Montreal Protocol. Dr. Adler is the author of four books and numerous academic and feature articles.
Tuition is $55 for 1 workshop or $99 for both (individual needs-based partial scholarships available). Contact Majidah at 935-7844 ext. 3 or majidah@hawaiimediation.org to pay with cash, check, or purchase order/invoice (for businesses only) or to request a scholarship application.
This "Collaborative Planning & Problem-Solving" Training Series is made possible thanks in part to funding from the County of Hawai'i, County of Hawai'i Department of Research and Development, and Hawai'i Island United Way.
Organizer Ku'ikahi Mediation Center
Organizer of Two Workshops with Peter S. Adler, PhD
We empower people to come together—to talk and to listen, to explore options, and to find their own best solutions.
To achieve this mission, we offer mediation, facilitation, and training to strengthen the ability of diverse individuals and groups to resolve interpersonal conflicts and community issues.
Ku'ikahi Mediation Center
101 Aupuni Street, Suite PH 1014 B-2
Hilo, HI 96720
Email: info@hawaiimediation.org
Follow up PhD viva workshop: Mock viva
Grant Writing Workshop for Emerging Curators with Peter Kingstone
Tue, Feb 1 6:30 PM
Unfinished business: two creative workshops with Liv Wynter
VIRTUAL Mental Wellness for Business with Lynn Hynes, PhD
The Poetic Voice: Persian Singing Workshops with Mahsa Vahdat
#Music #Class
Tue, Feb 1 6:30 AM
ActionCLUB – Business Skills Workshops – Sales with Derek Lancaster
Redefining the Warrior Two-Day Workshop/Orientation
Data Protection Officer (DPO) Immersion and Competency Workshop (Two Days)
Grow With Google | FREE Digital Skills Workshops to Grow Your Biz & Brand
Mastering the challenges of online meetings & workshops with Collaboard

redpajama_set_name: RedPajamaCommonCrawl


Go Local was created to offer visitors the chance to feel like a local and experience authentic Basque life in San Sebastián and its surrounding areas. Our young and energetic team has been described as 'brilliantly informal'. We invite you to discover San Sebastián and feel like a real donostiarra with GO LOCAL!


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The Theory Behind That Charlottesville Slogan
TOPICS:Abraham H. Milleranti-SemitismCharlottesvilleHaym Salomon CenterKevin B. MacDonaldwhite supremacists
MacDonald rails against Jews at a White Supremacist event.
Posted By: Abraham H. Miller April 3, 2018
'The Jews will not replace us,' they chanted. What do they mean, 'replace us'?
By Abraham H. Miller – Originally published in the Wall Street Journal
When white nationalists and supremacists gathered in Charlottesville, Va., last summer, they marched with tiki torches and chanted: "The Jews will not replace us." Obviously they don't care for Jews. But what do they mean, "replace us"?
Understanding the chant requires an examination of the work of Kevin B. MacDonald, a 74-year-old psychologist and retired professor at California State University, Long Beach. Mr. MacDonald's theories about Jews have become the philosophical and theoretical inspiration for white supremacist and nationalist movements.
Mr. MacDonald characterizes Jewish behavior in terms of the theory of group evolutionary psychology, based on competition among groups for resources and survival. Most scholars of evolutionary psychology reject Mr. MacDonald's methods and conclusions. White nationalists and supremacists embrace him, and he returns their affection. He edits a website called Occidental Observer that focuses on "White Identity, Interests and Culture."
Mr. MacDonald claims that Jewish traits, such as high verbal intelligence and ethnocentrism, have evolved to the point that Jews, as a group, outcompete non-Jews at the expense of Christian majorities.
He further argues that Jews are genetically programmed to undermine Christian civilization. Intellectual movements such as multiculturalism and liberalism serve, in his view, to heighten Jewish advantage because a Christian majority mired in a multicultural society is less likely to foster anti-Semitism.
Which brings us to his explanation of the Immigration and Naturalization Act of 1965. Mr. MacDonald claims that Jewish communal organizations led the effort to enact the law, which eliminated preferential quotas for Western European immigration and increased the total number of immigrants. That led to an increase in non-European immigration. In Mr. MacDonald's view, the act started the "replacement" of white Christians by a more ethnically diverse population.
It is true that Jewish communal organizations are major supporters of multiculturalism. Then again, so are most mainstream churches, on both sides of the papal divide. Christian communal groups loudly extol their commitment to inclusion and diversity.
But Jewish leadership was neither a necessary nor a sufficient condition for the passage of the 1965 immigration law. Rather, a liberalizing wave had already swept the West in the decades after World War II, bringing an end to colonialism and informing U.S. competition with the Soviet Union. In this context, America's 1924 immigration law, which favored Western European immigration, had become an international embarrassment.
As early as 1952, President Harry Truman appointed a Commission on Immigration and Naturalization, whose report served as the foundation for the 1965 act. The report affirmed the country's need for workers, but and it was highly critical of the foreign-policy problems caused by the then-current immigration policy. The report specifically cited Radio Moscow's use of America's exclusionary immigration policy to spread anti-Americanism in Asia, especially Korea.
The 1965 law's chief sponsors were Sen. Philip Hart of Michigan and Rep. Emanuel Celler. Hart was Catholic; Celler had three Jewish grandparents and one Catholic one. Would Mr. MacDonald say Celler's Jewish and Catholic genes were in conflict over immigration policy?
One of the most significant provisions of the act allowed chain migration, in which immigrants sponsor family members to come to America. Rep. Michael A. Feighan of Ohio, a conservative Democrat and Catholic, was responsible for this measure. He saw chain migration as promoting family values.
The bill passed overwhelmingly, with support from 85% of Republicans and 74% of Democrats. Sen. Edward Kennedy, a Catholic, was a major proponent and legislative orchestrator of the bill. President Lyndon Johnson, who belonged to the Disciples of Christ, signed the bill at the Statue of Liberty. Jewish support was not the key to its passage. The bill was widely popular—and more so with Republicans, even though most Jews, then as now, were Democrats.
Mr. Miller is an emeritus professor of political science at the University of Cincinnati and a distinguished fellow with the Haym Salomon Center.
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The Current Democratic Party Has No Use For Jews
Be the first to comment on "The Theory Behind That Charlottesville Slogan"
Islamophobia and the justification of violence
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UNION FOR REFORM JUDAISM PRACTICES INTOLERANCE IN THE PURSUIT OF SOCIAL JUSTICE


Long-distance runners are water-obsessed by necessity, but Mina Guli is on another level. She's trying to run more than 1,000 miles through seven deserts on seven continents in seven weeks to raise awareness of worldwide water shortages. She has completed six of the runs, having jogged an average of 150 miles each through harsh conditions in Spain, Jordan, Antarctica, Australia, South Africa and Chile.
Now, the Australian is in drought-stricken California to run her final leg through Death Valley and into Nevada. She hopes to finish Tuesday, which the United Nations has designated World Water Day. If she does, she will have run the equivalent of 40 marathons. Guli said she was feeling tired but inspired as she spoke to The Associated Press by phone in Santiago, Chile, where she was boarding a flight to Los Angeles on Thursday.


You might think it a little odd to cruise around tropical South Africa on a luxury cruise ship built to withstand the harshest of polar weather. Yet, despite the lack of blizzard conditions, Andy Mossack goes aboard Le Lyrial to experience Ponant's distinctly French approach to cruising.
It's early evening in Richards Bay and I'm on the bridge of Le Lyrial sitting in Captain Etienne Garcia's chair. I'm tempted to use the opportunity to make a passenger announcement but, luckily for Ponant, my French is not quite good enough for that and Captain Garcia is close enough to clap me in irons. Still, I get enough of a frisson simply sitting in the skipper's hot seat.
Ponant has an open bridge policy so guests are welcome to have a peek up here if they ask nicely. It really is quite an experience; perhaps better resembling the Starship Enterprise than a ship's bridge. Not so surprising when you consider Ponant ships provide luxury expedition cruises to the world's remotest locations. Ready and equipped for all eventualities.
Rewind a few days and Le Lyrial looked none the worse for wear after her latest adventure in Antarctica as I joined her in Cape Town. Sleek and elegant, looking more like a billionaire's yacht than a cruise liner, she sat there silently as I came onboard. An eight-night cruise around South Africa's coast to Durban lay ahead with a few stops to visit South Africa's game reserves along the way. A piece of cake compared to polar ice caps.
With just 122 cabins and a couple of hundred passengers, this is light years away from the ever-expanding floating behemoths that graze the high seas. It's personal, informal and distinctly French. It's like waking up in the south of France each morning to blue skies, freshly baked pain au chocolat and great coffee.
My stateroom was on the Prestige deck, one down from the top. It was a little bijou, as the French say, but very comfortable just the same. Designed by Jean-Phillipe Nuel in cream and turquoise pastels with high thread-count bed linen, luxury French toiletries in the shower, flat screen TV, a writing desk and patio doors leading out to a balcony.
Back down in the main lobby, I decided to be smart and reserve the gastronomic restaurant for a couple of nights ahead of everyone else.
An awkward moment ensued while I attempted to act like I knew this all along. But In my head, I was berating myself. It's a French ship, of course, the main restaurant is gastronomic. What else would it be?
In the end, I found myself hopping between both; enjoying the sumptuous buffet breakfasts and lunches at an outside table on the pool deck and then having à la carte dinners down in the gastronomic La Celeste restaurant on Deck 2. It was all deliciously French. The wines, the cheeses, the charcuterie, the salads and of course the fish, seafood and meats. As with most cruise ships, the bulk of the serving staff are Filipino and I can't speak highly enough for the clutch of servers who looked after me during my meal times. They remembered everything. How I liked my eggs, what coffee I preferred and even my penchant for freshly sliced avocado and kiwi every day. The plate materialized almost before I sat down.
Ponant has an all-inclusive policy on meals and non-premium bar drinks, wines and cocktails with enough quality brands in stock to satisfy most preferences. There are, of course, a full complement of premium brands for those connoisseurs who prefer to only drink their favourite tipple.
In spite of its smaller size, Le Lyrial has plenty of spaces to hide away if you want some peace and quiet. Particularly for me, up on the top deck or in the library or the panoramic lounge. Each of them light and airy, contemporary and elegant. The ship also has a theatre for shows and guest speakers a spa, hammam, hairdresser and an impressively modern gym.
We stayed overnight in Cape Town and I spent my first dinner in the company of a couple of highly entertaining South Africans. I bumped into them at the welcome safety briefing when we discovered we would be sharing the same lifeboat if disaster struck. It's worth pointing out here that although Ponant is French, all announcements are in English and French and the crew bilingual.
Over an excellent dinner (where an adjacent French table of diners debated the merits of the white over the red with the wine waiter for a good ten minutes. True story) we compared notes on their home city and I got a few local tips on the upcoming stops at Port Elizabeth, Richards Bay and Durban.
After dinner, I found myself in the main lounge listening to a talented French singer who I swear not only looked like Celine Dion but almost sounded like her too. After my second Cognac, I had convinced myself she was no doppelganger. Celine was most definitely in the auditorium.
Leaving Cape Town behind the next day, we spent it at sea following the famed Garden Route along the coast. It was a perfect opportunity to relax and soak up the South African sunshine despite having to endure my French shipmates somehow managing to still look chic and elegant just wearing swimming costumes.
South Africa's famous wildlife offers up exciting game drive excursions and this cruise takes full advantage with numerous trips ashore (excursion fees applicable). Out first stop at Port Elizabeth provided the prospect of a trip to the Addo National Park and its 500 elephants. Another stop at Richards Bay, bordering the Indian Ocean, offered a safari drive around the Hluhluwe Game Reserve, the second biggest in South Africa after Kruger. Other guests opted for a chance to experience Zulu culture and tradition at Shakaland.
Back up on the bridge, Captain Garcia skilfully avoids my offer of assistance and waves me adieu as I leave him to guide us safely out of port and on to Durban. Tonight is the second and final formal night of the cruise and I inwardly groan knowing my fellow guests will no doubt all materialise looking fashionable chic. I, on the other hand, will be wearing a repeat of my previous outfit. Distinctly non-chic. C'est la vie.
Prices start from £3,253.67 per person – excluding flights. Excursion fees are applicable and start from €45 per person.
British Airways flies a daily service from London Heathrow to Cape Town from £470 return.


April 29th-April 30th, 2016 at Camp Thunderbird in Olympia. The course starts at 6:00pm on Friday and concludes at 6:00pm on Saturday.
required to obtain the First Class rank. Along with Scoutmaster Specifics, this course is required of all direct contact leaders registered in Boy Scout troops and Varsity teams, in order to be considered "trained".
The course fee is $35 until April 15th and $50 after. Online registration closes on April 22nd so that materials can be procured. Online registration is mandatory.
Contact Lee Perez at wiffshusband@yahoo.com with questions.


The Governance SIG's latest evening event saw the SIG collaborate once more with APM Corporate Member Pricewaterhouse Coopers (PwC).
What is the Board's role in governance and do they follow good practice?
Are there good practice guidelines available?
And in particular how can you influence your board to adopt good practice in governance of project management?
These are some of the questions that were answered at this session.
This was one of a series of sessions that the Governance SIG is presenting to enable good practice to be shared. The next event in the series being our autumn conference being held on 1st October in London "Achieving change successfully - why good governance matters"
Poor governance of projects and project management is a major cause of project failure. Recent research (by both PwC and APM) has shown a direct correlation between organisational and project success and good governance. So good governance is the key success factor in delivering successful project outcomes. Shouldn't Boards take note and make improvement of governance a strategic objective?


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Style Guide: How to write appallingly about Gran Canaria (or anywhere else)
We read dozens of articles about Gran Canaria every week. Most of the time it hurts. In fact, you've got more chance of finding an original piece about the island as you do of finding a Canarian in the Puerto Rico Shopping Centre.
We've decided to kill the cliché and murder the tired metaphor so we've put our heads together and set the bar high:
Here's our style-guide for Gran Canaria Info articles.
By the way, if we break our own rules, please feel free to shout at us.
The essential clichés
No article can be published without one of the following insightful themes:
Starting with a pun on the island's roundness
Saying it's like Spain but different
Setting out to explore the other, real, Gran Canaria
Revealing that it's an island of contrasts
Mentioning the tourist bight, sprawl, plague, etc, within two sentences
Referring to Gran Canaria as la Isla Bonita (that's La Palma)
Discover Stuff: You're an explorer
To write in an original way you must discover a secret beach, explore a hidden valley or scale a rugged peak. Seeing the resorts and popular beaches with a fresh eye isn't worth considering. Do not under any circumstances focus on details but rather always try to cram the entire island into 400 words.
Do stay in a resort and explore the rest of the island in a day on a Jeep Safari. You'll see it all. In your article only mention the day out.
Scrape the surface: Make as little effort as possible to find anything interesting during your visit. If you must delve, delve into Google.
Unique and wonderful vocabulary
Go heavy on the word unique. Use it at least every sentence and don't both to explain why anything is unique: The readers already know. If you think you're using it too often just switch to 'perfect'.
Fascinating, gorgeous, wonderful, incredible, amazing, picture postcard perfect, etc are also perfectly viable alternatives. All other describing words are redundant.
Use the words traditional and sustainable at least once per article. It doesn't matter where: Just get them in. Combined with "perfect example of" they get you double points.
The word nice is due for a revival: Use it often as it makes your prose incisive.
The word literally is free. Slap it on the page.
Use words like impart, integrated, intense: Any word that you don't use in everyday conversation makes you sound clever. Better still, combine several beginning with the same letter in one phrase: Alliteration is always fun.
Help us get the word amazeballs off the Twittersphere and into quality prose. The world needs it.
English is too short: Make sure you puff out your prose with lots of wills, woulds, shoulds, coulds, can.
The active voice is nasty. Passive is always the way to go.
Mince the metaphor, crunch the cliché
In the lively resorts, hotels perch, pools are situated, you hit the sun-drenched beaches along with hordes of tourists, cash is splashed, the Yumbo is nudge, nudge, wink,wink (insert funny joke).
Off the beaten track the mountains float in the air above you, remote villages nestle quaintly, views are panoramic and breathtaking, markets and fiestas are colourful and vibrant and full of friendly locals.
Food is always succulent, mouth-watering and delectable. You stumble upon charming local restaurants that are always hidden gems.
Facts get in the way
Facts are so yesterday. Only check yours by referencing other articles found on Google. Get at least one basic fact per article so wrong that it makes the reader's teeth grind. It makes your words memorable.
Locals, what locals
There are no local people in Gran Canaria except friendly ones dressed up in traditional costume dancing charmingly at authentic local fiestas. If you must mention a local in any other context, make sure it's a barman with a dodgy accent.
Narrate to irritate
Refer to yourself in the third person. It's what the queen does and if its good enough for German pensioners, then it's good enough for you.
You need a nickname. Pick one that nobody has ever called you for real authenticity.
Tell the reader all about yourself. They care more about you than about the destination. Include lots of personal details that are unrelated to the content. Blurry photos of your family get you noticd on social media.
Engage with the audience
Beg for likes on social media: People will admire your determination
Post a link repeatedly just in case somebody missed it the first 23 times.
Join a web ring, blog ring, book club or any other network dedicated to sharing each other's content. Promote your buddy's irrelevant content to your audience.
Change the title of your old stuff and repost it as new. Do this often and without warning.
Photos are essential: Include blurry, cloudy and downright terrible ones to boost your reach. If your camera is broken use one of the 10 free photos available on Flickr or Creative Commons. Or steal from the internet: Never credit the photographer.
Want to submit a guest post to Gran Canaria Info? Please use the above as a guide and include plenty of links to your online casino or Viagra shop. We'll definitely publish it on the front page.
Published in Frontpage Blog
Arucas: Bananas, Rum and a Giant Church
Arucas town is famous for its colossal Gothic church built with bug money (see below). Oh, its also got a rum distillery and one of the best preserved town centres on the island.
Published in Resorts & Places
From Magma to Spaniards: A Brief History of Gran Canaria
Gran Canaria startede under the ocean and that's the way it'll end up. In the m,eantime, it's had a pretty eventful history.
Published in History
The Economy: Boom Bust and Sunshine
From prehistoric times Gran Canaria's economy has depended on turning the sunshine into things to sell to passing ships. As demand went up and down for what the island grew, it experienced booms and busts: It's why the cathedral in Las Palmas took hundreds of years to finish.
Even before the Spanish it's likely that the island's original inhabitats traded with passing roman and Phoenecian ships until these disappeared. The Canarii then went through a few lean centuries when they were the main export of the islands: Slavers from Europe and North Africa raided regularly.
At different times large areas of the island were planted with sugar cane, coffee, grape vines, and cactus for cochineal. The Caribbean proved better for sugar production (although rum is still made on the islands), disease put paid to the wine (mentioned by Shakespeare), and artificial dyes destroyed the cochineal trade. Bananas are still a faithful crop but have to compete with cheaper Caribbean imports.
The lean periods saw thousands of Canarians migrate to South America, and especially Cuba and Venezuela. The modern economy of Gran Canaria is based around tourism, and Las Palmas' huge container port. The island grows bananas, tomatoes, cucumbers and aubergines for the European market and olive oil and coffee are comig back.
Canarian wines and cheese are earning good reputations abroad. Gran Canarian cheese, particularly goat and sheep cheese from Guia in the north, regularly wins international awards. Gran Canarian wine is a mixed bag: The whites are excellent but the reds still lag behind vintages from Tenerife and Lanzarote.
Catering to tourists dwarfs all other economic activities in Gran Canaria: Over three million visit Gran Canaria every year. The huge majority stay in the tourist resorts along the south coast.
Tourism is now the islands largest industry and its biggest employer.
Published in Guide
What To Do When It's Raining In Gran Canaria
Gran Canaria sells itself to tourists as a sunshine destination where good weather is guaranteed. However, it does have to be wet every now and then or we'd be the Sahara desert. The good news is that there is plenty to do in Gran Canaria on the odd rainy day.
"Horrible" Gofio: The Canary Island Soul Food
Gofio is soul food in the Canary Islands. Most Canarians were weaned on bananas mashed up with gofio and many still eat it every day. It is so tightly entwined with local identity that it is best not not to tell anyone on the islands if you don't like it. At the same time Canarians aren't stupid. They enjoy cake and chocolate and ice cream just like everyone else. They wouldn't eat gofio if it was horrible. Wwould they?
Published in Eat Drink & Have Fun in Gran Canaria
Gran Canaria Oilve Oil: Gold from Sunshine
Gran Canaria is lucky that olive trees live a long time and don't bear grudges. For centuries the island's olivos were used as windbreaks and goat shade, their fruit pickled in mojo sauce or left to rot. Then, about 10 years ago, someone decided to crush a few olives and taste the oil.
Gran Canaria olive oil turns out to be delicious and farmers lucky enough to have olive trees on their land are relearning what their grandparents forgot. Our olives have gone from aperitif to the stars of the table.
Old Trees: New Tricks
There were presses in Gran Canaria in the 16th century so the Spanish must have planted olives as soon as they arrived. Most Canarian trees are old varieties that are rare elsewhere in Spain while one, the Verdial de Canarias, is unique to the islands.
These old cultivars give Gran Canaria olive oil a distinctive flavour: Fruity with a herbal edge like the smell or ripe tomatoes. It also has that peppery after taste that tickles the back of your throat: A classic sign of great olive oil.
Gran Canaria's olive trees grow on terrain too steep for mechanical harvesting so the whole crop is hand-picked. Since 100 kilos of island olives yields only 16.5 litres of oil all production is cold-pressed extra virgin olive oil.
The best Gran Canaria olive oil is sold unfiltered (therefore slightly cloudy) in local markets at 5 euros per 250 ml. You can also buy it in shops in the resorts but expect to pay more. Drizzle your oil on a simple salad or eat with crusty bread to appreciate the flavour. Don't combine it with balsamic vinegar (it's too strong and everyone is sick of the stuff).
Olive Country
There are olive trees all over the south east of Gran Canaria from Telde to San Bartolome but Temisas, between Santa Lucia and Agüimes, is the olive's heartland. The island's oldest olive press is in the village and it was here that the industry restarted. You can't miss the olivos fringing the terraced fields as you come into Temisas, especially if the trees are wearing their white hats in the wind. Bars and shops in the village sell the oil.
More To Olives Than Oil
So far the olive boom hasn't affected the supply of fresh olives in the markets. They taste horrible unless you cure them but the process is simple and worthwhile. It takes too long to do on holiday but if you live in Gran Canaria take advantage (and congratulations on a fine choice of island).
Look out for fat olives in the markets in November and December, or head to Temisas for the pick of the crop. The best ones are deep purple with a pointy end and soft, creamy flesh. To cure, nick each one with a knife and put them in an empty garafa (big plastic water bottle). Fill with water salty enough to float an egg and leave for two weeks. Then change the water and wait another two weeks.
To check if your olives are cured taste one: If your mouth puckers up wait another two weeks. When they are ready they are as good as the kalamata olives from Greece sold in little tubs in posh London delis. Bung them in mojo if you must, or store in olive oil or salty water before serving them smugly to your friends.
Gran Canaria's ancient olive trees, freshly pruned and fertilized, are standing proud again: The island has found yet another way of turning sunshine into gold.
The Historic Maspalomas Space Station Has Saved 10,000 Lives
The Maspalomas Space Station has saved 10,000 lives in 20 years by detecting distress signals from ships and planes. This is on top of its historic part in the Apollo moon landings.
The Maspalomas Space Station, also called the Canary Islands Space Centre, detects signals from distress beacons on ships and planes and from expeditions in remote areas. It then transmits their locations to the relevant emergency services. It is believed to have saved 10,000 lives in the past 20 years. That's more than one person per day!
In December 2012 it picked up a distress signal from a sinking yacht over 500km from the Canary Islands. A Superpuma helicopter saved the crew before the yacht sank, and broke the world record for a long distance maritime rescue at the same time.
Maspalomas and the moon
Started in 1960 as a NASA satellite tracking station it was extended in the 1960s to act as a receiving station for transmissions from the Apollo spacecraft. It was the first place on Earth to receive Neil Armstrong's famous words from the moon in 1969:
"That's one small step for (a) man, one giant leap for mankind"
Maspalomas' role in the moon landings was so important that Neil Armstrong himself visited the island just three months after returning from the moon. He flew to Gran Canaria aboard Air Force One along with fellow moonlanders Michael Collins y Edwin Aldrin and stayed at the Maspalomas-Oasis hotel for three nights.
By all accounts the astronauts had a good time in Gran Canaria even though it rained on their last day. They took a boat trip along the south coast, got on a camel, partied late into the night and enjoyed the food. Armstrong's wife liked mojo sauce so much that she took the recipe home with her.
While in Gran Canaria, Armstrong said that the moon was "a wonderful place, just like Maspalomas. The whole island is amazing".
You can see the receiving antennas of the space station from the roads between Maspalomas and Arguineguin. It is just up the hill from Pasito Blanco harbour. At this time visits are not possible.
Come To Las Palmas For Culture, Stay For Food And Fun
In brochure-land Las Palmas is full or tourists wandering around cobbled streets between museums and galleries. They eat authentic Canarian food, buy arts and crafts, and enjoy themselves in a demure, cultured sort of way without breaking a sweat.
You can do Las Palmas this way if you want: As the first Spanish city founded in the New World, it's gravid with history. It was attacked by cavemen, visited by Columbus, fought off an attack by Sir Francis Drake and got sacked by Dutch privateers. Vegueta is the perfect cultural destination: Compact, well-preserved and full of small, interesting museums. However, there is also a spicier side to the city that you miss if you focus on the past.
Cocktails at sunset
Down at the modern end by the beach, tourism is about catching waves, learning Spanish, scoffing tapas and dancing salsa until you drop. Visitors are discovering Las Palmas as an exciting, modern city with Latino flair. They may wander over to Vegueta to visit Columbus' old haunts but they spend most of their time having fun in the sun.
This alternative scene isn't as new as it sounds. Until the late 1960s Las Palmas was the main tourist destination on the island and the beach was covered in Scandinavians and Brits. After the resorts in the south took off the city went to sleep, resting on its ancient laurels.
Now not then
With millions of cruise ship passengers spending a day in the city, and three million tourist a year on the island for a week or two, Las Palmas has an opportunity to remind people that it is a destination in its own right. But it's not history and architecture that will bring the visitors back, it's today's city with its beach culture, great Spanish food and laid back cool.
We believe that modern Las Palmas is missing a trick by focusing on the city it was 500 years ago rather than the city it is now. Nobody leaps out of bed in the morning because they are excited about visiting a cathedral. But sunset cocktails, tasty tapas, warm sand and Latino rhythm? That sounds like fun!
There are bars by Las Canteras that do the best caipirinhas this side of Copacabana and the streets behind the beach are full of restaurants knocking out quality Spanish tapas. And then there's live music on the beachfront every Saturday, tapas night in Vegueta on Thursdays, carnival, the surf competitions, the marathon, the food and film weekend ... Las Palmas is already happening, it's just that nobody knows about it yet!
It is time for Las Palmas to get its mojo back and remind the rest of Europe that there is a Habana on this side of the Atlantic: That's the sexy Habana with mojitos and rumba by the way, not the Habana of crumbling buildings and long dead heroes.
Published in Las Palmas
Overheard in Gran Canaria: Tourist Complaints
Three million tourists come to Gran Canaria every year so we get our share of complaints. While some are legitimate many are caused by people with ridiculous expectations and no knowledge of local culture. Gran Canaria makes a huge effort to please all its guests and most go home tanned and happy. That said, there's no pleasing some people:
The Best Nudist Beaches In Gran Canaria
A useful guide to the best nudist beaches in Gran Canaria; All the info you need to find them and enjoy them with no hassle.
Published in Guide to all the beaches on Gran Canaria
Gran Canaria Versus Tenerife: Which Island Is Better for Your Holiday
Gran Canaria and Tenerife are the two biggest tourist destinations in the Canary Islands. Since most visitors choose one island for their holidays, which is better, Gran Canaria or Tenerife?
Both islands get great weather all year round and are the same distance from Europe. We live in Gran Canaria and love the island, but Tenerife is a cracking destination as well. Here's what we think:
The beaches are why you guys head to the Canary Islands so let's have a look at what the two islands offer:
We think Gran Canaria has the edge because of its natural sandy beaches: Between the golden beach and dunes at Maspalomas and Playa del Inglés, and the stunning Las Canteras Beach in Las Palmas, Gran Canaria has two of Europe's best beaches. Tenerife has beautiful natural stretches of sand, such as El Medano, La Tejita and Playa Jardín, but they are not on the same scale.
When it comes to artificial beaches both Gran Canaria and Tenerife have some crackers. Las Teresitas in Tenerife is beautiful, as are Playa de las Americas, Playa del Duque and Fañabé. On Gran Canaria you have Anfi, Amadores and Mogan, all pretty beaches close to the resorts. Again, Gran Canaria has the edge, in our slightly biased opinion, due to its number and range of beaches.
Both Tenerife and Gran Canaria have excellent little beaches that rarely get any tourists. The best beaches on both islands are often the little ones at the mouth of quiet barrancos. You get the sun and the sea all to yourself! Overall, we give the beach crown to Gran Canaria.
Tenerife just seems to do food better than Gran Canaria. From its little Guachinche restaurants in the hills to its quality resort restaurants, Tenerife is a cut above. There are plenty of good places to eat in Gran Canaria, especially in small towns and in the capital Las Palmas, but Tenerife still has the edge: We think it is because is more popular with French and Italian visitors and residents.
Tenerife's Dorada beer is slightly nicer than Gran Canaria's Tropical (hope nobody from Gran Canaria is reading this ;-D), even though both brews are made by South Africa's SAB Miller brewery. Gran Canaria's Arehucas rum is the Canary Islands' most popular spirit.
With Tenerife's beer topping Tropical and Arehucas carrying the spirits category we have to give the overall food and drink prize to Tenerife.
Tenerife has a 3,718 metre volcano and you can't argue with that: The Las Cañadas national park is unique in Europe and Teide is a stunning peak. The far north of Tenerife is rugged and spectacular and its laurel and pine forests are pristine.
Gran Canaria on the other hand has its spectacular ravines or barrancos and gorgeous pine forests, as well as the sheers cliffs of the west coast. Both islands have a huge range of outdoor sporting options such as cycling, diving and hiking. In the battle of landscapes, we think both islands come out equal.
Nightlife and Fun
Tenerife's Playa de Las Americas is the liveliest resort in the Canary Islands and beats Playa del Inglés in Gran Canaria. The famous Veronica Strip is where the young-uns head for fun after sunset. Where Gran Canaria wins out is in its local night-life: Las Palmas' bars and clubs are full most nights of the week and Santa Cruz just doesn't have the same energy. While La Laguna has a fantastic university scene, we think that Las Palmas' is better thanks to its size and diversity. With Tenerife taking the resort crown and Gran Canaria the local one, its another draw.
So which island is better, Gran Canaria or Tenerife? It's a tough call as both islands have world-class highlights, but we have to give the overall prize to Gran Canaria. But then we would: If we liked Tenerife more we would move there!
What do you think? Have we been fair? Or have we missed something that pushes Tenerife into the top spot? Let us know in a comment, or visit our Gran Canaria Facebook page.
Sabotage: The Real Reason Columbus Stopped In Gran Canaria
Christopher Columbus stopped in Las Palmas de Gran Canaria in 1492 on his way to discovering the Americas not because he wanted to, but because his crew sabotaged his ship within three days of setting sail from Spain.
Is Gran Canaria A Safe Holiday Destination?
You might expect a volcanic island close to the Sahara desert to have its fair share of dangerous animals and freak weather.
Published in Body & Soul
Gran Canaria: Restaurant Daily Menus
You never see a Canarian rushing down the high street juggling a half-eaten bocadilllo and a plastic cup of takeaway coffee. That's because of the venerable tradition of the Menu del Dia.
Las Palmas de Gran Canaria: Europe's Latino City
A city destination with miles of golden sand, bars serving local rum, a raucous month-long carnival and Christopher Columbus' house. The legendary Habana perhaps, or Colombia's exotic Cartagena de Indias? You might be surprised to find out that this exotic destination is much closer to home. Las Palmas, Europe's secret Latino city, is just four hours flight from Britain and Germany.
Las Palmas is one of the biggest cities in Spain, boasts the best city beach in Europe and still parties like it's 1999. Its old town district is on the short-list to become a World Heritage Site and its shops are among the cheapest in Europe. Las Palmas is 'the city with the best climate in the world' according to New York's Syracuse University. Average temperatures range between 20 degrees in the winter and 26 degrees in the summer. It receives six hours of sunshine every day during the winter, ten hours in the summer.
By rights Las Palmas should be overflowing with tourists but somehow it has slipped under the radar. Overshadowed by the huge, package-tourist resorts in the south of Gran Canaria, the city hardly caters to visitors. There are no rep-led pub crawls or time-share salesmen in Las Palmas.
In February, the annual carnival takes over the city for a whole month of parades, parties and concerts. No other European city can match Las Palmas' blend of Spanish and Latino culture. Sometimes, it is like a small chunk of Cuba floated across the ocean and nobody noticed.
Las Canteras Beach Guide: How To Choose A Spot
Las Canteras beach, a 3.5km of golden sand that sweeps down one side of the city, is the jewel of Las Palmas. It's so big that each section has its own name and character. Here're the main areas from north to south.
The Top 20 Best Beaches in Gran Canaria
Gran Canaria has dozens of beaches, from the Saharan splendour of Maspalomas to the tiny patch of sand at Sardina del Norte. Some are packed with sun loungers and parasols, while others are hours away from the nearest Full English Breakfast. Whatever your taste, the island has the perfect beach for you.
Five Cool North Gran Canaria Beaches The Locals Keep to Themselves
There are quiet local beaches dotted all around the north coast of Gran Canaria. If you get sick of long sand beaches covered in sun loungers and parasols then head to one of these secret spots. All of them are great for snorkelling.
Gran Canaria Beaches: The Legendary Güi Güi
If Güi Güi beach was next door to Maspalomas nobody would think twice about it. It's not particularly pretty and at high tide most of it is underwater. The sand is on the dark side and disappears completely during the winter. At times it is covered in driftwood. Nevertheless, Güi Güi's remote location and high cliffs make it Gran Canaria's Shangri La.
Published in Every Gran Canaria Beach
Tip Of The Day: Arucas Old Town Is Worth A Wander
There's more to Arucas old town than the great big church. Walk up the narrow cobbled streets behind the cathedral for great old Canarian houses and local street life, then head down narrow Calle Gourié to the High Street (Calle León y Castillo).


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I'm an artist and comic author, responsible most famously for Incubus Tales, which ran for over 7 years and 700 pages.
Currently I'm doing Demoniac Verse, after a brief time away from the genre.
Be sure to check my Red Curtain gallery too!


Kids are idolising the man who fell from space, but for the wrong reasons.
Felix's impressive feat not only broke records, but the sound barrier on his brief and nerve-wracking drop from where the sky begins. I looked at that, and thought, it's a little like being born.
There you are, a baby, tucked safely into your womb-balloon after months of preparation. You can't go up any more, and there's only one way down- to slip into the world- falling gracefully into the unknown, maybe hitting one or two wobbly patches on the way out, but breaking a sound barrier of your own as the shockingly cold air buffets against your human flesh for the first time.
This is where Felix gets it all wrong: According to News24, now that he's completed his leap to earth, he's retiring: "There's nothing else for me to achieve", he reportedly said. "I have no energy left". He's 43-years-old.
For a baby, after you've left your balloon where everything is designed to protect you from the unknown, your journey is only beginning. It's exhausting, being born, but the whole point of landing on earth is that you have EVERYTHING to achieve.
Do you agree that parents sometimes focus too much on birth rather than actually having a baby?


text:
\section{Introduction}

In this paper, given a bounded domain $\Omega \subset \mathbb{R}^N$, $N\geq 2$, with a Lipschitz boundary $\partial\Omega$, we are concerned with the existence and uniqueness of solutions to the following elliptic system
	\begin{align}
	\label{syst}
	\begin{aligned}
	-\divergenz \l(|\nabla u|^{p_1-2} \nabla u+ \mu_1(x) |\nabla u|^{q_1-2} \nabla u\r)&= f_1(x, u, v, \nabla u, \nabla v) \quad && \text{in } \Omega, \\
	-\divergenz \l(|\nabla v|^{p_2-2} \nabla v+ \mu_2(x) |\nabla v|^{q_2-2} \nabla v\r)&= f_2(x, u, v, \nabla u, \nabla v) \quad && \text{in } \Omega, \\
	u&= v= 0 && \text{on } \partial\Omega,
	\end{aligned}
	\end{align}
where $1< p_i< q_i< N$, $\mu_i\colon \overline{\Omega} \to [0, \infty)$ are Lipschitz continuous and $f_i\colon \Omega \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ are Carath\'eodory functions, $i= 1, 2$, that satisfy suitable structure conditions, see hypotheses (H) in Section \ref{section_3}.

Here, the operator is the so-called double-phase operator, that is
\begin{align*}
        -\divergenz \l(|\nabla u|^{p-2} \nabla u+ \mu(x) |\nabla u|^{q-2} \nabla u\r)\quad \text{for }u\in \Wp{\mathcal{H}},
\end{align*}
where $1<p<q<N$ and with a suitable Sobolev Musielak-Orlicz space $\Wp{\mathcal{H}}$, see its definition in Section \ref{section_2}. Such an operator is the extension of the so-called weighted $(q,p)$-Laplacian when $\inf_{\overline{\Omega}} \mu>0$ and of the $p$-Laplace differential operator when $\mu\equiv 0$.

The novelty of this work is an existence and uniqueness result for problems of the form \eqref{syst} by using the surjectivity result for pseudomonotone operators, see Definition \ref{SplusPM} and Theorem \ref{theorem_pseudomonotone}. To the best of our knowledge, this is the first work dealing with a double phase operator and a convection term (that is, the right-hand side depends on the gradient of the solution) in the context of elliptic systems.

 Zhikov \cite{Zhikov-1986} was the first who studied so-called double phase operators in order to describe models of strongly anisotropic materials by studying the functional
\begin{align}\label{integral_minimizer}
   u \mapsto \int \left(|\nabla  u|^p+\mu(x)|\nabla  u|^q\right)\,dx,
\end{align}
where $1<p<q<N$, see also Zhikov \cite{Zhikov-1995}, \cite{Zhikov-1997} and the monograph of Zhikov-Kozlov-Oleinik \cite{Zhikov-Kozlov-Oleinik-1994}. Functionals of the expression \eqref{integral_minimizer} have been studied by several
 authors with respect to regularity results and nonstandard growth, see for example,
 Baroni-Colombo-Mingione \cite{Baroni-Colombo-Mingione-2015}, \cite{Baroni-Colombo-Mingione-2016}, \cite{Baroni-Colombo-Mingione-2018},
 Baroni-Kuusi-Mingione \cite{Baroni-Kuusi-Mingione-2015},
 Cupini-Marcellini-Mascolo \cite{Cupini-Marcellini-Mascolo-2015},
 Co\-lom\-bo-Mingione \cite{Colombo-Mingione-2015a}, \cite{Colombo-Mingione-2015b},
 Marcellini \cite{Marcellini-1989},
 \cite{Marcellini-1991} and the references therein.

 The motivation of this work was on the one hand the work of Gasi\'nski-Winkert \cite{Gasinski-Winkert-2020b} who proved existence and uniqueness for the problem
 \begin{equation}\label{problem}
    \begin{aligned}
	-\divergenz\left(|\nabla u|^{p-2}\nabla u+\mu(x) |\nabla u|^{q-2}\nabla u\right) & =f(x,u,\nabla u)\quad && \text{in } \Omega,\\
	u & = 0 &&\text{on } \partial \Omega,
    \end{aligned}
\end{equation}
following the paper of Averna-Motreanu-Tornatore \cite{Averna-Motreanu-Tornatore-2016}. On the other side, we were also motivated by the paper of Motreanu-Vetro-Vetro \cite{Motreanu-Vetro-Vetro-2016} who treated elliptic systems for $(p_i,q_i)$-Laplace operators of the form
\begin{equation}\label{problem2}
        \begin{aligned}
                -\Delta_{p_1}u_1-\mu_1 \Delta_{q_1} u_1 & =f_1(x,u_1,u_2,\nabla u_1, \nabla u_2)\quad && \text{in } \Omega,\\
                -\Delta_{p_2}u_2-\mu_2 \Delta_{q_2} u_2 & =f_2(x,u_1,u_2,\nabla u_1, \nabla u_2)\quad && \text{in } \Omega,\\
                u_1=u_2 & = 0 &&\text{on } \partial \Omega.
    \end{aligned}
\end{equation}
The idea in the current paper is to combine both problems \eqref{problem} and \eqref{problem2} which gives our model problem \eqref{syst}. Such new class of problems brings lots of difficulties to be overcome like the Orlicz space in order to deal with the double phase operator, the gradient dependence of the right-hand side which implies that we cannot use variational tools and the fact that we treat this for elliptic systems. Our results extend those in Gasi\'nski-Winkert \cite{Gasinski-Winkert-2020b} and Motreanu-Vetro-Vetro \cite{Motreanu-Vetro-Vetro-2016}.

In the case of single-valued equations like \eqref{problem} without convection we refer to the works of Colasuonno-Squassina \cite{Colasuonno-Squassina-2016}, Gasi\'nski-Papageorgiou \cite{Gasinski-Papageorgiou-2019}, Gasi\'nski-Winkert \cite{Gasinski-Winkert-2020a},  Liu-Dai \cite{Liu-Dai-2018}, Perera-Squassina \cite{Perera-Squassina-2019} concerning existence and multiplicity results.

Elliptic systems with the shape as in \eqref{problem2} have been considered by a number of authors. Existence results can be found, for examples, in Boccardo-de Figueiredo \cite{Boccardo-de-Figueiredo-2002}, Carl-Motreanu \cite{Carl-Motreanu-2017}, \cite{Carl-Motreanu-2015}, Dr\'{a}bek-Stavrakakis-Zographopoulos \cite{Drabek-Stavrakakis-Zographopoulos-2003}, Motreanu-Vetro-Vetro \cite{Motreanu-Vetro-Vetro-2018} and the references therein.

 Works which are closely related to our paper dealing with certain types
 of double phase problems, convection terms or elliptic systems can be found in Bahrouni-R\u{a}dulescu-Repov\v{s} \cite{Bahrouni-Radulescu-Repovs-2019}, Bahrouni-R\u{a}dulescu-Winkert \cite{Bahrouni-Radulescu-Winkert-2019}, Cencelj-R\u{a}dulescu-Repov\v{s} \cite{Cencelj-Radulescu-Repovs-2018}, Marano-Marino-Moussaoui \cite{Marano-Marino-Moussaoui-2019}, Marano-Winkert \cite{Marano-Winkert-2019}, Marino-Winkert \cite{Marino-Winkert-2020}, Motreanu-Winkert \cite{Motreanu-Winkert-2019}, Papageorgiou-R\u{a}dulescu-Repov\v{s} \cite{Papageorgiou-Radulescu-Repovs-2019a}, \cite{Papageorgiou-Radulescu-Repovs-2019b}, \cite{Papageorgiou-Radulescu-Repovs-2020}, R\u{a}dulescu \cite{Radulescu-2019}, Zhang-R\u{a}dulescu \cite{Zhang-Radulescu-2018}, Zheng-Gasi\'nski-Winkert-Bai \cite{Zheng-Gasinski-Winkert-Bai-2020} and the references therein.

 The paper is organized as follows. In Section \ref{section_2}  we recall the definition of the Musielak-Orlicz spaces $\Lp{\mathcal{H}}$ and its corresponding Sobolev spaces $\Wp{\mathcal{H}}$ and we recall the surjectivity result for pseudomonotone operators. In Section \ref{section_3} we present the full assumptions on the data of problem \eqref{syst},
 give the definition of the weak solution and state and prove our main existence result, see Theorem \ref{main}. In the last part, namely Section \ref{section_4}, we state some conditions on $f_i$, $i=1,2$, in order to prove the uniqueness of weak solutions of \eqref{syst}, see Theorem \ref{theorem_uniqueness}.

\section{Preliminaries}\label{section_2}

For every $1 \le r< \infty$ we consider the usual Lebesgue spaces $L^r(\Omega)$ and $L^r(\Omega; \mathbb{R}^N)$ equipped with the norm $\|\cdot \|_r$. When $1< r< \infty$ we denote by $W^{1, r}(\Omega)$ and $W^{1,r}_0(\Omega)$ the corresponding Sobolev spaces  equipped with the norms $\|\cdot \|_{1,r}$ and $\|\cdot \|_{1,r,0}$, respectively. By $r'$, we denote the conjugate of $r \in (1,\infty)$, that is, $\frac{1}{r}+\frac{1}{r'}=1$.

For $i= 1, 2$ we define functions $\mathcal H_i\colon \Omega \times [0, \infty) \to [0, \infty)$ by
\begin{align*}
        \mathcal H_i(x, t)= t^{p_i}+ \mu_i(x) t^{q_i},
\end{align*}
where $1< p_i< q_i< N$ and
\begin{align}\label{1}
        \frac{q_i}{p_i}< 1+ \frac{1}{N}, \qquad \mu_i\colon \overline{\Omega} \to [0, \infty) \text{ is Lipschitz continuous.}
\end{align}

\begin{remark
        From the condition above we easily see that
        \begin{align*
                q_i< p_i^*, \quad i= 1, 2,
        \end{align*}
		where $p_i^*$ is the critical Sobolev exponent of $p_i$ given by
		\begin{align*
			p^*_i:= \frac{N p_i }{N- p_i}.
		\end{align*}
        Indeed, for fixed $ i \in \{1, 2\}$, we have to show that $ \displaystyle q_i< \frac{N p_i }{N- p_i}$, that is $Nq_i - p_i q_i< Np_i $. From condition \eqref{1} we have $ Nq_i - p_i< Np_i $. Moreover, since $q_i> 1$, we have
        \begin{align*}
                N q_i - p_i q_i< N q_i - p_i< Np_i ,
        \end{align*}
        which shows the assertion.
\end{remark}

Let  $\displaystyle \rho_{\mathcal H_i}(u):= \int_{\Omega} \mathcal H_i(x, |u|) dx$. Then the Musielak-Orlicz space $L^{\mathcal H_i}(\Omega)$ is given by
\begin{align*}
        L^{\mathcal{H}_i}(\Omega)=\left \{u ~ \Big | ~ u: \Omega \to \mathbb{R} \text{ is measurable and } \rho_{\mathcal{H}_i}(u)< +\infty \right \}
\end{align*}
equipped with the Luxemburg norm
\begin{align*}
        \|u\|_{\mathcal H_i}:= \inf \l\{ \tau>0 : \, \rho_{\mathcal H_i}\l(\frac{u}{\tau}\r) \le 1 \r\}.
\end{align*}
Then $L^{\mathcal H_i}(\Omega)$  becomes uniformly convex, and so a reflexive Banach space. Moreover we define the space
\begin{align*}
    L^{q_i}_{\mu_i}(\Omega)=\left \{u ~ \Big | ~ u: \Omega \to \mathbb{R} \text{ is measurable and } \int_{\Omega} \mu_i(x) | u|^{q_i} \,dx< +\infty \right \}
\end{align*}
endowed with the seminorm
\begin{align*}
        \|u\|_{q_i, \mu_i}:= \l(\int_{\Omega} \mu_i(x) |u|^{q_i} dx \r)^{\frac{1}{q_i}}.
\end{align*}
From Colasuonno-Squassina \cite[Proposition 2.15]{Colasuonno-Squassina-2016} we have the following continuous embeddings
\begin{align*}
        L^{q_i}(\Omega) \hookrightarrow L^{\mathcal H_i}(\Omega) \hookrightarrow L^{p_i}(\Omega) \cap L_{\mu_i}^{q_i}(\Omega).
\end{align*}
For $u \ne 0$ we have $\displaystyle \rho_{\mathcal H_i}\l(\frac{u}{\|u\|_{\mathcal H_i}}\r)= 1$, so it is easy to see that
\begin{align}	\label{2}
        \min \left\{\|u\|_{\mathcal H_i}^{p_i}, \|u\|_{\mathcal H_i}^{q_i}\right\}
        \leq \|u\|_{p_i}^{p_i}+ \|u\|_{q_i, \mu_i}^{q_i}
        \leq \max \left\{\|u\|_{\mathcal H_i}^{p_i}, \|u\|_{\mathcal H_i}^{q_i}\right\}
\end{align}
for every $u \in L^{\mathcal H_i}(\Omega)$. Then we can introduce the corresponding Sobolev space $W^{1, \mathcal H_i}(\Omega)$ defined by
\begin{align*
        W^{1, \mathcal H_i}(\Omega):= \left\{u \in L^{\mathcal H_i}(\Omega) : \, |\nabla u| \in L^{\mathcal H_i}(\Omega) \right\}
\end{align*}
with the norm
\begin{align*}
        \|u\|_{1, \mathcal H_i}:= \|\nabla u \|_{\mathcal H_i}+ \|u\|_{\mathcal H_i},
\end{align*}
where $\|\nabla u\|_{\mathcal H_i}= \|\, |\nabla u| \, \|_{\mathcal H_i}$.

Moreover, we write $W^{1, \mathcal H_i}_0(\Omega):= \overline{C^{\infty}_0(\Omega)}^{\|\cdot \|_{1, \mathcal H_i}}$ being the completion of $C^\infty_0(\Omega)$ in $W^{1, \mathcal H_i}(\Omega)$. Taking \eqref{1} into account  we can refer to Colasuonno-Squassina \cite[Proposition 2.18]{Colasuonno-Squassina-2016} in order to consider an equivalent norm on $W^{1, \mathcal H_i}_0(\Omega)$ given by
	\begin{align*}
	\|u\|_{1, \mathcal H_i, 0}= \|\nabla u\|_{\mathcal H_i}.
	\end{align*}
Note that $W^{1, \mathcal H_i}(\Omega)$ as well as $W^{1, \mathcal H_i}_0(\Omega)$ are uniformly convex, and so reflexive Banach spaces.

Since $1< p_i< N$, we know that the embedding
\begin{align}\label{embedding_sobolev}
            W^{1,\mathcal{H}_i}_0(\Omega) \hookrightarrow \Lp{r_i}
\end{align}
is compact whenever $r_i<p^*_i$, see Colasuonno-Squassina \cite[Proposition 2.15]{Colasuonno-Squassina-2016}.

From equation \eqref{2} we directly obtain
\begin{align}\label{5}
    \begin{split}
        \min \l\{\|u\|_{1, \mathcal H_i, 0}^{p_i}, \|u\|_{1, \mathcal H_i, 0}^{q_i}\r\}
        & \leq \|\nabla u\|_{p_i}^{p_i}+ \|\nabla u\|_{q_i, \mu_i}^{q_i}\\
        & \leq \max \l\{\|u\|_{1, \mathcal H_i, 0}^{p_i}, \|u\|_{1, \mathcal H_i, 0}^{q_i}\r\}
    \end{split}
\end{align}
for every $u \in W^{1, \mathcal H_i}_0(\Omega)$.

For $1< r< \infty$ we consider now the eigenvalue problem for the $r$-Laplacian with homogeneous Dirichlet boundary condition given by
	\begin{align}
	\label{6}
	\begin{aligned}
	-\Delta_r u&= \lambda |u|^{r-2} u \quad && \text{in } \Omega, \\
	u&= 0 && \text{on } \partial\Omega.
	\end{aligned}
	\end{align}
Let us denote by $\lambda_{1, r}$ the first eigenvalue of \eqref{6}. It is well known that $\lambda_{1, r}$ is positive, simple, and isolated, see L{\^e} \cite{Le-2006}. Moreover, we have the following variational characterization
	\begin{align}
	\label{7}
	\lambda_{1, r}= \inf_{u \in W^{1,r}_0(\Omega)} \l\{\int_{\Omega} |\nabla u|^r dx: \, \int_{\Omega} |u|^r dx= 1\r\}.
	\end{align}

We now recall some definitions that we will use in the sequel.

\begin{definition}\label{SplusPM}
    Let $X$ be a reflexive Banach space, $X^*$ its dual space and denote by $\langle \cdot \,, \cdot\rangle$ its duality pairing. Let $A\colon X\to X^*$, then $A$ is called
    \begin{enumerate}[leftmargin=1cm]
	\item[(a)]
	    to satisfy the $(\Ss_+$)-property if $u_n \rightharpoonup u$ in $X$ and $\limsup_{n\to \infty} \langle Au_n,u_n-u\rangle \leq 0$ imply $u_n\to u$ in $X$;
	 \item[(b)]
	    pseudomonotone if $u_n \rightharpoonup u$ in $X$ and $\limsup_{n\to \infty} \langle Au_n,u_n-u\rangle \leq 0$ imply $Au_n \rightharpoonup u$ and $\langle Au_n,u_n\rangle \to \langle Au,u\rangle$;
    \item[(c)]
        coercive if
            \begin{align}
            \label{coercivity}
            \lim_{\|u\|_X \to \infty} \frac{\langle Au, u \rangle}{\|u\|_X}= \infty.
            \end{align}
    \end{enumerate}
\end{definition}

Our existence result is based on the following surjectivity result for pseudomonotone operators, see, e.\,g., Carl-Le-Motreanu \cite[Theorem 2.99]{Carl-Le-Motreanu-2007} or Papageorgiou-Winkert \cite[Theorem 6.1.57]{Papageorgiou-Winkert-2018}.

\begin{theorem}\label{theorem_pseudomonotone}
    Let $X$ be a real, reflexive Banach space, let $A\colon X\to X^*$ be a pseudomonotone, bounded, and coercive operator, and $b\in X^*$. Then, a solution of the equation $Au=b$ exists.
\end{theorem}

We consider the space $\mathcal{W}:=W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega)$ endowed with the norm
	\begin{align*
	\|(u, v)\|_{\mathcal{W}}:= \|u\|_{1, \mathcal H_1, 0}+ \| v \|_{1, \mathcal H_2, 0},
	\end{align*}
for every $(u, v) \in W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega)$.

Then we consider the operator $A\colon W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) \to (W^{1, \mathcal H_1}_0(\Omega))^* \times (W^{1, \mathcal H_2}_0(\Omega))^* $ defined by
        \begin{align}\label{A}
            \begin{split}
                    \langle A(u, v), (\varphi, \psi)\rangle_{\mathcal H_1 \times \mathcal H_2}&:= \int_{\Omega} \l(|\nabla u|^{p_1-2} \nabla u+ \mu_1(x) |\nabla u|^{q_1-2} \nabla u \r) \cdot \nabla \varphi dx \\
                    & \quad +  \int_{\Omega} \l(|\nabla v|^{p_2-2} \nabla v+ \mu_2(x) |\nabla v|^{q_2-2} \nabla v \r) \cdot \nabla \psi dx,
            \end{split}
        \end{align}
        where $ \langle \cdot \, , \cdot \rangle_{\mathcal H_1 \times \mathcal H_2}$ is the duality pairing between $W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega)$ and its dual space $ (W^{1, \mathcal H_1}_0(\Omega))^* \times (W^{1, \mathcal H_2}_0(\Omega))^* $. The next result summarizes the  properties of the operator $A$.

\begin{lemma}\label{lemma1}
        Let $A\colon W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) \to (W^{1, \mathcal H_1}_0(\Omega))^* \times (W^{1, \mathcal H_2}_0(\Omega))^* $ be the operator defined by \eqref{A}. Then, $A$ is bounded, continuous, monotone (hence maximal monotone), and of type $(S_+)$.
\end{lemma}

\begin{proof}
    The proof is similar to the one in Liu-Dai \cite[Proposition 3.1]{Liu-Dai-2018}.
\end{proof}

\section{Main result}\label{section_3}
We assume the following hypotheses on the nonlinearities $f_1, f_2$.

\begin{enumerate}
    \item[(H)]
            $f_1, f_2 \colon \Omega \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}  $ are Carath\'eodory functions such that
            \begin{enumerate}
                    \item[(i)]
                            There exist $\alpha_i \in L^{\frac{r_i}{r_i-1}}(\Omega) $ ($i=1,2$) such that
                            \begin{align*}
                                \begin{split}
                                    |f_1(x, s, t, \xi, \zeta)|
                                    &\le A_1 |s|^{a_1}+ A_2 |t|^{a_2}+ A_3 |s|^{a_3} |t|^{a_4}+ A_4 |\xi|^{a_5} \\
                                    & \quad + A_5 |\zeta|^{a_6}+ A_6 |\xi|^{a_7} |\zeta|^{a_8}+  |\alpha_1(x)|, \\
                                    |f_2(x, s, t, \xi, \zeta)|
                                    &\le B_1 |s|^{b_1}+ B_2 |t|^{b_2}+ B_3 |s|^{b_3} |t|^{b_4}+ B_4 |\xi|^{b_5} \\
                                    & \quad + B_5 |\zeta|^{b_6}+ B_6 |\xi|^{b_7} |\zeta|^{b_8}+  |\alpha_2(x)|,
                                \end{split}
                            \end{align*}
                            for a.\,a.\,$x \in \Omega$, for all $s, t \in \mathbb{R}$ and for all $ \xi, \zeta \in \mathbb{R}^N$, where $A_j, B_j$, $j=1, \dots 6$, are nonnegative constants and with $1< r_i< p_i^*$, $i= 1, 2$. Moreover,  the exponents $a_\ell, b_\ell $, $\ell= 1, \dots, 8$, are nonnegative and satisfy the following conditions
                            \begin{align*}
                                    \hspace*{1cm}
                                    \begin{array}[]{lll}
                                            \text{(E1)} \quad a_1 \le r_1- 1,
                                            & \text{(E2)} \quad \displaystyle a_2 \le \frac{r_1- 1}{r_1} r_2,\\[3ex]
                                            \text{(E3)} \quad  \displaystyle \frac{a_3}{r_1}+ \frac{a_4}{r_2} \le  \frac{r_1- 1}{r_1},  & \text{(E4)} \quad \displaystyle a_5 \le \frac{r_1- 1}{r_1} p_1,\\[3ex]
                                            \text{(E5)} \quad \displaystyle a_6 \le \frac{r_1- 1}{r_1} p_2,
                                            & \text{(E6)} \quad  \displaystyle \frac{a_7}{p_1}+ \frac{a_8}{p_2} \le  \frac{r_1- 1}{r_1},   \\[3ex]
                                            \text{(E7)} \quad b_1 \le \displaystyle\frac{r_2- 1}{r_2} r_1,
                                            & \text{(E8)} \quad \displaystyle b_2 \le r_2- 1,\\[3ex]
                                             \text{(E9)} \quad \displaystyle \frac{b_3}{r_1}+ \frac{b_4}{r_2} \le \frac{r_2- 1}{r_2},
                                            &\text{(E10)} \quad b_5 \le \displaystyle\frac{r_2- 1}{r_2} p_1,\\[3ex]
                                             \text{(E11)} \quad \displaystyle b_6 \le \frac{r_2- 1}{r_2} p_2,
                                            & \text{(E12)} \quad  \displaystyle \frac{b_7}{p_1}+ \frac{b_8}{p_2} \le  \frac{r_2- 1}{r_2}.
                                    \end{array}
                            \end{align*}
                    \item[(ii)]
                            There exist $\omega \in L^1(\Omega)$ and $\Lambda, \Gamma \ge 0$ such that
                            \begin{align}\label{cond}
                                    \begin{split}
                                            & f_1(x, s, t, \xi, \zeta) s+f_2(x, s, t, \xi, \zeta) t\\
                                            & \le \Lambda \left(|\xi|^{p_1}+|\zeta|^{p_2}\right)+ \Gamma \left( |s|^{p_1}+|t|^{p_2}\right)+ \omega(x),
                                    \end{split}
                            \end{align}
                            for a.\,a.\,$x \in \Omega$, for all $s, t \in \mathbb{R}$ and for all $\xi, \zeta \in \mathbb{R}^N$ and with
                            \begin{align}\label{cond3}
                                    \Lambda +\Gamma \max\left\{\lambda_{1, p_1}^{-1},\lambda_{1, p_2}^{-1}\right\}< 1,
                            \end{align}
                            where $\lambda_{1, p_i}$ is the first eigenvalue of the $p_i$-Laplacian, see \eqref{6}.
            \end{enumerate}
\end{enumerate}

We say that $(u, v) \in W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) $ is a weak solution of problem \eqref{syst} if
\begin{align}\label{weak}
        \begin{split}
                \int_{\Omega} \l(|\nabla u|^{p_1-2} \nabla u+ \mu_1(x) |\nabla u|^{q_1-2} \nabla u \r) \cdot \nabla \varphi \,dx&= \int_{\Omega} f_1(x, u, v, \nabla u, \nabla v) \varphi \,dx, \\
                \int_{\Omega} \l(|\nabla v|^{p_2-2} \nabla v+ \mu_2(x) |\nabla v|^{q_2-2} \nabla v \r) \cdot \nabla \psi \,dx&= \int_{\Omega} f_2(x, u, v, \nabla u, \nabla v) \psi \,dx,
        \end{split}
\end{align}
is satisfied for all test functions $(\varphi, \psi) \in W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega)$. Taking the embedding \eqref{embedding_sobolev} into account, along with the growth conditions in (H)(i), we see that the definition of a weak solution is well-defined. Indeed, if we estimate the integral concerning the function $f_1\colon \Omega \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$  by using condition (H)(i) we obtain several mixed terms. Let us consider, for example, the third term on the right-hand side of the growth of $f_1$. Applying H\"older's inequality we get
\begin{align*
    \begin{split}
        &A_3\int_{\Omega} |u|^{a_3}|v|^{a_4} \varphi \,dx\\
        &\leq A_3\left(\int_{\Omega}|u|^{a_3 s_1}\,dx\right)^{\frac{1}{s_1}}\left(\int_{\Omega}|v|^{a_4 s_2}\,dx\right)^{\frac{1}{s_2}} \left(\int_{\Omega}|\varphi|^{ s_3}\,dx\right)^{\frac{1}{s_3}},
    \end{split}
\end{align*}
where $ (u,v) \in W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) $, $\varphi \in W^{1, \mathcal H_1}_0(\Omega)$ and
\begin{align*}
        \frac{1}{s_1}+\frac{1}{s_2}+\frac{1}{s_3}=1.
\end{align*}
Taking $s_3=r_1$ with $1<r_1<p_1^*$ and using $s_1 \leq \frac{r_1}{a_3}$ as well as $s_2 \leq \frac{r_2}{a_4}$  leads to
\begin{align*
        \frac{a_3}{r_1}+ \frac{a_4}{r_2}\leq \frac{r_1-1}{r_1},
\end{align*}
which is exactly condition (E3). Note that the conditions in (H)(i) are chosen in order to prove our main results by applying the compact embedding \eqref{embedding_sobolev}. Of course, for the finiteness of the integrals in the weak formulation \eqref{weak}, we can also allow critical growth to have a well-defined weak formulation.

Now we are ready to formulate and prove our main result in this section.

\begin{theorem}\label{main}
        Let $1< p_i< q_i< N$, $i= 1, 2$, and let hypotheses \eqref{1} and (H) be satisfied. Then, there exists a weak solution $(u, v) \in W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega)$ of problem \eqref{syst}.
\end{theorem}

\begin{proof}
    Let
    \begin{align*}
        \hat N_{f_i}\colon W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) \subset L^{r_1}(\Omega) \times L^{r_2}(\Omega) \to L^{r_1'}(\Omega) \times L^{r_2'}(\Omega)
    \end{align*}
    be the Nemytskij operator associated to $f_i$. Moreover, let
    \begin{align*}
        j_i^*\colon L^{r_1'}(\Omega) \times L^{r_2'}(\Omega) \to (W^{1, \mathcal H_1}_0(\Omega))^* \times (W^{1, \mathcal H_2}_0(\Omega))^*
    \end{align*}
    be the adjoint operator for the embedding
    \begin{align*}
        j_i\colon W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) \to L^{r_1}(\Omega) \times L^{r_2}(\Omega).
    \end{align*}
    We then define
    \begin{align*}
        N_{f_i}:= j_i^* \circ \hat N_{f_i}\colon W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) \to (W^{1, \mathcal H_1}_0(\Omega))^* \times (W^{1, \mathcal H_2}_0(\Omega))^*,
    \end{align*}
    which is well-defined by hypotheses (H)(i). We set
    \begin{align}\label{mathcal-A}
        \mathcal A(u, v):= A(u, v)- N_{f_1}(u, v)- N_{f_2}(u, v).
    \end{align}
    Our aim is to apply Theorem \ref{theorem_pseudomonotone}. So, we need to show that $\mathcal A$ is bounded, pseudomonotone and coercive.

    {\bf Claim 1:} $\mathcal{A}$ is bounded.

    The boundedness of $\mathcal{A}$ follows directly from the boundedness of $A$ and the growth conditions on $f_1$ and $f_2$ stated in (H)(i).

    {\bf Claim 2:} $\mathcal{A}$ is pseudomonotone.

    To this end, let $\{(u_n, v_n)\}_{n \in \mathbb{N}} \subset W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega) $ be a sequence such that
    \begin{align}\label{weak lim}
        (u_n, v_n) \rightharpoonup (u, v) \quad \text{in } W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega)
    \end{align}
    and
	\begin{align}\label{limsup}
	    \limsup_{n \to \infty} \langle \mathcal A(u_n, v_n), (u_n- u, v_n- v) \rangle_{\mathcal H_1 \times \mathcal H_2} \le 0.
	\end{align}
    Taking the compact embedding \eqref{embedding_sobolev} into account yields
	\begin{align}\label{strong}
	    u_n \to u \quad \text{in } L^{r_1}(\Omega) \quad \text{and} \quad v_n \to v \quad \text{in } L^{r_2}(\Omega),
	\end{align}
	since $r_1< p_1^*$ and $r_2< p_2^*$, respectively.
	We want to show that
	\begin{align}\label{lim-f}
	    \begin{split}
	        \lim_{n \to \infty} \int_{\Omega} f_1(x, u_n, v_n, \nabla u_n, \nabla v_n)(u_n- u)\, dx&= 0, \\
	        \lim_{n \to \infty} \int_{\Omega} f_2(x, u_n, v_n, \nabla u_n, \nabla v_n)(v_n- v) \,dx&= 0.
	    \end{split}
	\end{align}
    Let us consider the first expression in \eqref{lim-f}. By the growth condition (H)(i) it follows
	\begin{align}\label{f1}
	    \begin{split}
	        & \int_{\Omega} f_1(x, u_n, v_n, \nabla u_n, \nabla v_n) (u_n-u) \,dx \\
	        & \leq \int_{\Omega} \bigg(A_1 |u_n|^{a_1}+ A_2 |v_n|^{a_2}+ A_3 |u_n|^{a_3} |v_n|^{a_4} \\
	        & \quad+ A_4 |\nabla u_n|^{a_5}+ A_5 |\nabla v_n|^{a_6}+ A_6 |\nabla u_n|^{a_7} |\nabla v_n|^{a_8}+ |\alpha_1(x)|\bigg) |u_n- u| \,dx.
	    \end{split}
	\end{align}
    Applying H\"older's inequality, \eqref{strong} and conditions (E1) and (E2), respectively, we obtain
	\begin{align*}
	    \begin{split}
	        A_1 \int_{\Omega} |u_n|^{a_1} |u_n- u| \,dx
	        & \leq A_1 \l(\int_{\Omega} |u_n|^{a_1 r_1'}\,dx \r)^{\frac{1}{r_1'}} \|u_n- u\|_{r_1} \\
	        & \le C_{1} \l(1+ \|u_n\|_{r_1}^{r_1-1}\r) \|u_n- u\|_{r_1} \to 0
	    \end{split}
	\end{align*}
    and
	\begin{align*}
	    \begin{split}
	        A_2 \int_{\Omega} |v_n|^{a_2} |u_n- u| \,dx & \le A_2 \l(\int_{\Omega} v_n^{a_2 r_1'} \,dx \r)^{\frac{1}{r_1'}} \| u_n- u\|_{r_1} \\
	        & \le C_{2} \l(1+ \|v_n\|_{r_2}^{\frac{r_2}{r_1'}} \r) \|u_n- u\|_{r_1} \to 0
	    \end{split}
	\end{align*}
	for some $C_1, C_2>0$.
    Moreover, H\"older's inequality with exponents $x_1, y_1, z_1> 1$ such that
	\begin{align*}
	    x_1 a_3 \le r_1, \quad y_1 a_4 \le r_2, \quad z_1= r_1, \quad \frac{1}{x_1}+ \frac{1}{y_1}+ \frac{1}{z_1}= 1
	\end{align*}
    gives, by hypothesis (E3),
	\begin{align*}
	    A_3 \int_{\Omega} |u_n|^{a_3} |v_n|^{a_4} |u_n- u| \,dx \le A_3 \|u_n\|_{a_3 x_1}^{a_3} \|v_n \|_{a_4 y_1}^{a_4} \|u_n- u\|_{r_1} \to 0.
	\end{align*}
    Next we apply H\"older's inequality with exponents $r_1, r_1'$ and use (E4) and (E5) to get
	\begin{align*}
	    \begin{split}
	        A_4 \int_{\Omega} |\nabla u_n|^{a_5} |u_n- u| \,dx
	        & \leq A_4 \l(\int_{\Omega} |\nabla u_n|^{a_5 r_1'} \,dx \r)^{\frac{1}{r_1'}} \|u_n- u\|_{r_1} \\
	        & \leq C_3 \l(1+ \|\nabla u_n\|_{p_1}^{\frac{p_1}{r_1'}} \r) \|u_n- u\|_{r_1} \to 0
	    \end{split}
	\end{align*}
    and
	\begin{align*}
	    \begin{split}
	        A_5 \int_{\Omega} |\nabla v_n|^{a_6} |u_n- u| \,dx
	        & \leq A_5 \l(\int_{\Omega} |\nabla v_n|^{a_6 r_1'} \,dx \r)^{\frac{1}{r_1'}} \|u_n- u\|_{r_1} \\
	        & \leq C_4 \l(1+ \|\nabla v_n\|_{p_2}^{\frac{p_2}{r_1'}} \r) \|u_n- u\|_{r_1} \to 0
	\end{split}
	\end{align*}
for some $C_3, C_4>0$. Furthermore, condition (E6) allows us to apply H\"older's inequality with exponents $x_2, y_2, z_2> 1$ such that
	\begin{align*}
	    x_2 a_7 \leq p_1, \quad y_2 a_8 \leq p_2, \quad z_2= r_1, \quad \frac{1}{x_2}+ \frac{1}{y_2}+ \frac{1}{z_2}= 1
	\end{align*}
    in order to have
	\begin{align*}
	    A_6 \int_{\Omega} |\nabla u_n|^{a_7} |\nabla v_n|^{a_8} (u_n- u) \,dx
	    \leq A_6 \|\nabla u_n \|_{a_7 x_2}^{a_7} \|\nabla v_n\|_{a_8 y_2}^{a_8} \|u_n- u\|_{r_1} \to 0,
	\end{align*}
    since both $\|\nabla u_n \|_{a_7 x_2}$ and $\|\nabla v_n\|_{a_8 y_2}$ are bounded. Finally, for the last term in \eqref{f1} we have
	\begin{align*}
	    \int_{\Omega} |\alpha_1(x)| (u_n- u) \,dx
	    \leq \|\alpha_1\|_{r_1'} \|u_n- u\|_{r_1} \to 0.
	\end{align*}
    Combining all the calculations above give
	\begin{align*}
	    \lim_{n \to \infty} \int_{\Omega} f_1\l(x, u_n, v_n, \nabla u_n, \nabla v_n\r)(u_n- u)\,dx= 0.
	\end{align*}
    Applying similar arguments proves that
	\begin{align*}
	    \lim_{n \to \infty} \int_{\Omega} f_2\l(x, u_n, v_n, \nabla u_n, \nabla v_n\r) (v_n- v) \,dx= 0.
	\end{align*}
	Hence, \eqref{lim-f} is fulfilled. We now take the weak formulation \eqref{weak}, replace $u$ by $u_n$, $v$ by $v_n$, $\varphi$ by $u_n- u$ and $\psi $ by $v_n- v$ and use \eqref{limsup} as well as \eqref{lim-f}  in order to have
	\begin{align}\label{limsup1}
	    \begin{split}
	        & \limsup_{n \to \infty} \langle A(u_n, v_n), (u_n-u, v_n- v)\rangle_{\mathcal H_1 \times \mathcal H_2}\\
	        & = \limsup_{n \to \infty} \langle \mathcal A(u_n, v_n), (u_n- u, v_n- v)\rangle_{\mathcal H_1 \times \mathcal H_2} \le 0.
        \end{split}
	\end{align}
    Since $A$ satisfies the $(S_+)$-property, see Lemma \ref{lemma1}, we derive from \eqref{weak lim} and \eqref{limsup1} that
	\begin{align*}
	    (u_n, v_n) \to (u, v) \quad \text{in } W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega).
	\end{align*}
    Since $\mathcal A$ is continuous we have $\mathcal A(u_n, v_n) \to \mathcal A(u, v)$ in $(W^{1, \mathcal H_1}_0(\Omega))^* \times (W^{1, \mathcal H_2}_0(\Omega))^*$, which proves that $\mathcal A $ is pseudomonotone.

    {\bf Claim 3:} $\mathcal{A}$ is coercive.

    First of all, taking into account the representation \eqref{7} and replacing $r$ by $p_1$ and $p_2$, respectively, we have
	\begin{align}\label{eigenv}
	    \|u\|_{p_1}^{p_1} \leq \lambda_{1, p_1}^{-1} \|\nabla u \|_{p_1}^{p_1} \quad \text{and} \quad \|v\|_{p_2}^{p_2} \leq \lambda_{1, p_2}^{-1} \|\nabla v \|_{p_2}^{p_2},
	\end{align}
    for all $(u, v) \in W^{1, \mathcal H_1}_0(\Omega) \times W^{1, \mathcal H_2}_0(\Omega)$. Note that $W^{1, \mathcal H_1}_0(\Omega) \subseteq \Wpzero{p_1}$ and $W^{1, \mathcal H_2}_0(\Omega)\subseteq \Wpzero{p_2}$. Applying these facts along with \eqref{eigenv}, \eqref{cond}, and \eqref{5} leads to
	\begin{align*}
	    \begin{split}
	        &\langle \mathcal A(u, v), (u, v)\rangle_{\mathcal H_1 \times \mathcal H_2}\\
	        &= \int_{\Omega} \l(|\nabla u|^{p_1-2} \nabla u+ \mu_1(x) |\nabla u|^{q_1-2} \nabla u\r) \cdot \nabla u \,dx \\
	        & \quad+ \int_{\Omega} \l(|\nabla v|^{p_2-2} \nabla v+ \mu_2(x) |\nabla v|^{q_2-2} \nabla v\r) \cdot \nabla v \,dx \\
	        & \quad- \int_{\Omega} f_1(x, u, v, \nabla u, \nabla v)u \,dx- \int_{\Omega} f_2(x, u, v, \nabla u, \nabla v)v \,dx \\
	        & \geq \|\nabla u\|_{p_1}^{p_1}+ \|\nabla u\|_{q_1, \mu_1}^{q_1}+ \|\nabla v \|_{p_2}^{p_2}+ \|\nabla v\|_{q_2, \mu_2}^{q_2} \\
	        & \quad-\Lambda \left(\|\nabla u\|_{p_1}^{p_1}+\|\nabla v\|_{p_2}^{p_2}\right)- \Gamma \left( \|u\|_{p_1}^{p_1}+\|v\|_{p_2}^{p_2}\right)- \|\omega\|_1\\

        	& \geq \l(1- \Lambda- \Gamma \lambda_{1, p_1}^{-1}\r) \|\nabla u\|_{p_1}^{p_1}
        	+ \|\nabla u\|_{q_1, \mu_1}^{q_1} \\
        	& \quad +\l(1- \Lambda- \Gamma \lambda_{1, p_2}^{-1}\r) \|\nabla v\|_{p_2}^{p_2}+ \|\nabla v \|_{q_2, \mu_2}^{q_2}- \|\omega\|_1 \\

        	& \geq \bigg( 1- \Lambda- \Gamma \max\left\{\lambda_{1, p_1}^{-1},\lambda_{1, p_2}^{-1}\right\}\bigg)\left(\min\left\{\|u \|_{1, \mathcal H_1, 0}^{p_1}, \|u\|_{1, \mathcal H_1, 0}^{q_1}\right\} \right.\\
        	& \quad \left.+ \min\left\{\|v \|_{1, \mathcal H_2, 0}^{p_2}, \|v\|_{1, \mathcal H_2, 0}^{q_2}\right\}\right) -\|\omega\|_{L^1(\Omega)}.
	    \end{split}
	\end{align*}
    Since $1< p_i< q_i$ and  condition \eqref{cond3} holds, it follows that  \eqref{coercivity} is satisfied, and hence $\mathcal A$ is coercive.

    From the Claims 1--3 we see that $\mathcal A$ is bounded, pseudomonotone and coercive. Therefore, by Theorem \ref{theorem_pseudomonotone}, there exists $(u, v) \in W_0^{1, \mathcal H_1}(\Omega) \times W_0^{1, \mathcal H_2}(\Omega)$ such that $\mathcal A(u, v)= 0$. Taking into account the definition of $\mathcal A$, see equation \eqref{mathcal-A}, it follows that $(u, v)$ is a weak solution of problem \eqref{syst}. That finishes the proof.
\end{proof}

\section{A uniqueness result}\label{section_4}

Now we consider the uniqueness of solutions of \eqref{syst}. To this end, let ${\bf f}\colon \Omega \times \mathbb{R}^2 \times (\mathbb{R}^N)^2 \to \mathbb{R}^2$ be the vector field defined by
\begin{align*}
    {\bf f}(x,s,\xi)=(f_1(x,s,\xi),f_2(x,s,\xi))
\end{align*}
for a.\,a.\,$x\in\Omega$, for all $s \in \mathbb{R}^2$ and for all $\xi \in (\mathbb{R}^N)^2$. We suppose the following conditions on $f$:
\begin{enumerate}
    \item[(U1)]
        There exists $c_1 \geq 0$ such that
        \begin{align*}
            ({\bf f}(x,s,\xi)-{\bf f}(x,t,\xi))\cdot (s-t) \leq c_1 |s-t|^2
        \end{align*}
        for a.\,a.\,$x\in\Omega$, for all $s,t \in \mathbb{R}^2$ and for all $\xi \in (\mathbb{R}^N)^2$.
    \item[(U2)]
        There exist $\rho=(\rho_1,\rho_2)$ with $\rho_i \in \Lp{s_i}$, $1<s_i<p_i^*$ and $c_2 \geq 0$ such that
        ${\bf f}(x,s,\cdot)-\rho(x)$ is linear on $(\mathbb{R}^N)^2$ for a.\,a.\,$x\in\Omega$ and for all $s\in \mathbb{R}^2$ and
        \begin{align*}
            |{\bf f}(x,s,\xi)-\rho(x)| \leq c_2 |\xi|
        \end{align*}
        for a.\,a.\,$x\in\Omega$, for all $s \in \mathbb{R}^2$ and for all $\xi \in (\mathbb{R}^N)^2$.
\end{enumerate}

Our main result in this section reads as follows.

\begin{theorem}\label{theorem_uniqueness}
    Let \eqref{1}, (H), (U1), and (U2) be satisfied. If $2= p_i< q_i< N$ for $i=1,2$ and
    \begin{align}\label{condition_uniqueness}
        c_1 \lambda_{1,2}^{-1}+c_2 \left(2\lambda_{1,2}^{-1}\right)^{\frac{1}{2}}<1,
    \end{align}
    then there exists a unique weak solution of problem \eqref{syst}.
\end{theorem}

\begin{proof}
    Let $u=(u_1, u_2), v=(v_1, v_2) \in W_0^{1, \mathcal H_1}(\Omega) \times W_0^{1, \mathcal H_2}(\Omega)$ be two weak solutions of \eqref{syst}. Considering the weak formulation for $u$ and $v$, choosing $\varphi= u_1-v_1$ as well as $\psi= u_2- v_2$ and subtracting the related equations gives
	\begin{align}\label{u1}
	    \begin{split}
	        & \int_{\Omega} |\nabla (u_1- v_1)|^2\,dx
	        + \int_{\Omega} |\nabla(u_2- v_2)|^2\,dx\\
	        & \quad + \int_{\Omega} \mu_1(x) \l(|\nabla u_1|^{q_1-2} \nabla u_1- |\nabla v_1|^{q_1- 2} \nabla v_1 \r) \cdot \nabla(u_1- v_1)\,dx \\
	        & \quad + \int_{\Omega} \mu_2(x) \l(|\nabla u_2|^{q_2-2} \nabla u_2- |\nabla v_2|^{q_2-2} \nabla v_2 \r) \cdot \nabla (u_2- v_2) \,dx \\
	        &= \int_{\Omega} \l({\bf f}(x, u, \nabla u)- {\bf f}(x, v, \nabla u)\r) \cdot (u- v) \,dx \\
	        & \quad+ \int_{\Omega} \l({\bf f}(x, v, \nabla u)-\rho(x)- {\bf f}(x, v, \nabla v)+\rho(x)\r)\cdot (u-v)\, dx.
	    \end{split}
	\end{align}
	By the monotonicity of $\xi \mapsto |\xi|^{q_i-2}\xi$ we see that the third and the fourth integral on the left-hand side of \eqref{u1} are nonnegative, that is,
	\begin{align}\label{u2}
	    \begin{split}
	        & \int_{\Omega} |\nabla (u_1- v_1)|^2\,dx
	        + \int_{\Omega} |\nabla(u_2- v_2)|^2\,dx\\
	        & \quad + \int_{\Omega} \mu_1(x) \l(|\nabla u_1|^{q_1-2} \nabla u_1- |\nabla v_1|^{q_1- 2} \nabla v_1 \r) \cdot \nabla(u_1- v_1)\,dx \\
	        & \quad + \int_{\Omega} \mu_2(x) \l(|\nabla u_2|^{q_2-2} \nabla u_2- |\nabla v_2|^{q_2-2} \nabla v_2 \r) \cdot \nabla (u_2- v_2) \,dx \\
	        & \geq \int_{\Omega} |\nabla (u_1- v_1)|^2\,dx
	        + \int_{\Omega} |\nabla(u_2- v_2)|^2\,dx\\
	        & = \|\nabla (u_1-v_1)\|_2^2+\|\nabla (u_2-v_2)\|_2^2.
	    \end{split}
	\end{align}
	On the other side, by applying (U1) to the first integral on the right-hand side of \eqref{u1} and (U2) to the second we obtain along with H\"older's inequality
	\begin{align}\label{u3}
	    \begin{split}
	        &\int_{\Omega} \l({\bf f}(x, u, \nabla u)- {\bf f}(x, v, \nabla u)\r) \cdot (u- v) \,dx \\
	        & \quad+ \int_{\Omega} \l({\bf f}(x, v, \nabla u)-\rho(x)- {\bf f}(x, v, \nabla v)+\rho(x)\r)\cdot (u-v)\, dx\\
	        & \leq c_1 \left(\|u_1-v_1\|_2^2+\|u_2-v_2\|_2^2 \right) \\
	        & \quad+ \int_{\Omega} (f_1(x,v_1,v_2,(u_1-v_1)\nabla (u_1-v_1),(u_1-v_1)\nabla (u_2-v_2))-\rho_1(x) )\,dx\\
	        & \quad + \int_{\Omega} (f_2(x,v_1,v_2,(u_2-v_2)\nabla (u_1-v_1),(u_2-v_2)\nabla (u_2-v_2))-\rho_2(x) )\,dx\\
	        & \leq c_1 \lambda_{1,2}^{-1}\left(\|\nabla (u_1-v_1)\|_2^2+\|\nabla (u_2-v_2)\|_2^2 \right)\\
	        & \quad +c_2 \int_{\Omega} \left(|u_1-v_1|+|u_2-v_2| \right) \left(|\nabla (u_1-v_2)|^2+|\nabla (u_2-v_2)|^2\right)^{\frac{1}{2}}\,dx\\
	        & \leq \left(c_1 \lambda_{1,2}^{-1}+c_2 \left(2\lambda_{1,2}^{-1}\right)^{\frac{1}{2}}\right)\left(\|\nabla (u_1-v_1)\|_2^2+\|\nabla (u_2-v_2)\|_2^2 \right).
	    \end{split}
	\end{align}
	Combining \eqref{u1}, \eqref{u2} and \eqref{u3} gives
	\begin{align}\label{u4}
	    \begin{split}
	        & \|\nabla (u_1-v_1)\|_2^2+\|\nabla (u_2-v_2)\|_2^2\\
	        & \leq \left(c_1 \lambda_{1,2}^{-1}+c_2 \left(2\lambda_{1,2}^{-1}\right)^{\frac{1}{2}}\right)\left(\|\nabla (u_1-v_1)\|_2^2+\|\nabla (u_2-v_2)\|_2^2 \right).
	    \end{split}
	\end{align}
	Taking \eqref{condition_uniqueness} into account, we see from \eqref{u4} that $u_1=v_1$ and $u_2=v_2$ and so the solution of \eqref{syst} is unique.
\end{proof}

redpajama_set_name: RedPajamaArXiv