On Flightaware there doesnt look like any airport service the islands.but they must have airlines that service them?
The Falklands get the amount of visitors one would expect for a pile of rocks in the middle of the ocean. When I was there I think LAN flew in once a week or so, not sure why they don’t show up on the EGYP page. Must be outside of FlightAware’s service area. However, the majority of visitors (even civilians) fly in on military aircraft which aren’t tracked on here.
The RAF Mount Pleasant airport is kind of a let-down since it is neither on a “mount” nor even remotely “pleasant”. 
Perhaps you should try Googling the purpose of a “Discussion” forum.
Downside of Discussion format is the inability to upvote a comment.
More’s the pity in this case.
If the OP had wanted a discussion and not just an answer to a very easily answered question, there would have been a discussion.
Monty Python Argument sketch.
He enters room 12.
Angry man: WHADDAYOU WANT?
Man: Well, Well, I was told outside that…
Angry man: DON’T GIVE ME THAT, YOU SNOTTY-FACED HEAP OF PARROT DROPPINGS!
Man: What?
A: SHUT YOUR FESTERING GOB, YOU TIT! YOUR TYPE MAKES ME PUKE!
YOU VACUOUS STUFFY-NOSED MALODOROUS PERVERT!!!
M: Yes, but I came here for an argument!!
A: OH! Oh! I’m sorry! This is abuse!
M: Oh! Oh I see!
A: Aha! No, you want room 12A, next door.
M: Oh…Sorry…
A: Not at all!
A: (under his breath) stupid git.
You must be a New Yorker!
Native? Nope. Originally from IN. Upstate now? You bet. Point? Oh that’s right… we have to guess as to the point of your post instead of offering a direct, lucid question…
For the purposes of discussion only, did you try searching for Malvinas?
During the height of the Falklands War in '82 I was getting my stock broker’s license courtesy of Prudential/Bache in NYC along with about 50 others who came from all over the USA and around the world, including students from the UK and Argentina.
One of the Argentinian students was put on the spot by our instructor and asked what the best investment for his clients would be at this time given their country’s state of war.
Unfortunately, my stage whispered answer of “Una casa en las Malvinas”, which broke up the decorum of the class, also carried to the instructor’s ears and I was dismissed from the class for the day.