Do drones fly because of low pressure above them or high pressure below them? I was working on some drone battery life solutions and was stuck with the question whether drones fly because of low pressure created above them by the rotor or due to high pressure created below them.
 A: The same principle applies to an aircraft whether it is kept aloft by a fixed wing or a rotary wing, usually known as a rotor. A combination of high pressure below the wing or rotor, and low pressure above, is what keeps the aircraft airborne.
A: I noticed the other answers phrase the answer in terms of pressure difference.
However, in the case of an aircraft (flying high above the ground) any pressure difference is a side-effect rather than a cause.
The one case where pressure difference is central is with the kind of hovering that a hovercraft does. The air does continuously escape from under the skirt, but if you replenish that air fast enough a sufficiently high pressure is maintained underneath the hovercraft. That air pressure is truly carrrying the hovercraft. 
As a straightforward example of a high flying aircraft with a rotor lets take the case of a helicopter.
When a helicopter is high up, multiple times its own height, then I rather expect that the actual air pressure below it and above it aren't that much different. Sure, air above it is moved down, but surrounding air can easily flow in. Air is moved fast to the space below the helicopter, but again, it can easily move out of the way. 
The rotor is creating lift by accelerating air mass downwards.
To imagine what is taking place imagine the following setup: you are the pilot of a small, one person hovercraft, and you throw something fairly heavy away from you, horizontally. You throw the object, and as consequence there is a recoil, so your own velocity changes a little too, in the opposive direction. By accelerating something else you have accelerated yourself too (in the opposite direction).
That is how a rotor creates lift: by accelerating a lot of air mass downwards you generate a tendency to move upward that counteracts gravity.
Again, pressure difference is a factor only when a body of air is sufficienly trapped, such that you can replenish the air that escapes. In the case of a helicopter it is all about accelerating air mass towards the direction that you want to move away from.

The comment by Ján Lalinský indicates to me that some further discussion is warranted.
Some general remarks about the way a rotor has effect:
For a rotor to have an effect the blades (the airfoils) must be accelerating air mass. Now, of course in the overall process of accelerating air mass there will be some degree of compression of air mass.
Let me compare the following case:
A golfbal that is accelerated by being hit with a golfclub. Slow motion footage of a golfbal that is hit very hard (with a heavy club) shows that the golfbal is deformed a lot. Obviously the elastic deformation of the golfbal plays an important role in the transfer of kinetic energy from the club to the ball. Still, we will say that the ball is accelerated due to being hit by the club. The deformation of the ball is an intermediate stage of the overall transfer process. So we don't say: the ball is accelerated due to the elastic deformation of the ball. We say: the ball is accelerated due to being hit by the club.
It seems to me that any compression of air mass in the process of an airfoil accelerating air mass should likewise be seen as an intermediate stage in an overall transfer process. 
And here is another way of looking at this issue:
Airfoils are optimized for generating lift. For a given power input you want a maximum of lift generated. Question: what would happen if you design to generate maximum pressure difference, at the expense of all other properties of the airfoil. If you change the shape of an airfoil in such a way that the shape will tend to trap air mass then for sure you will get a lot more pressure difference. But obviously such an air trapping airfoil would have terrible performance because the shape would obstruct the air flow. 
Conclusion: If you design to generate maximum pressure difference at the expense of all other properties of the airfoil you lose more than you gain. In fact, you will lose everything. This conclusion corroborates the view that compression of air mass that occurs in the process of generating lift is an intermediate stage in an overall transfer process.
A: Answer: It is both--> it is the pressure difference, top-bottom-difference providing a net upward force.  Pressure over an area is force. More below...
Peter has it correct.  It is ALWAYS a pressure difference pushing up because there is more pushing up on the lower surface than there is pushing down on the upper surface.
Ján also has it correct in that these pressures slowly go back to atmospheric pressure as you move away from the wing. The changes from atmospheric are "local".
However, Cleoins is close, but hasn't quite got it. At a high altitude a helicopter does indeed have more trouble hovering as you reasoned, but there is a maximum hover altitude (actually density).  It is a hover maximum because as you say, the rotor is sitting in the column of air already moving down.  It can't accelerate it any faster and, therefore it can't put any more downward force on it. Go too high and the column is accelerating down as much as the rotor can speed it up (accelerate) it.   The rotor can add only so much velocity and as density decreases, so does the mass and delta V (A) times mass decreases.
Now,
when a heli is in forward flight, you are constantly moving over into new column of stationary air.  Therefore, the rotor can accelerate new air downward.  A fixed wing does this by default, constantly moving into new, still air.
Now for the meat of the lift story.
You must get the desire out of your head to look for some complex story. It is simple and so many people miss the obvious in the search for Bernoulli, Newton's Third, circulation, Kutta, etc.


*

*First, remember Newton's 1st Law is supposed to teach us that a force is required to accelerate a mass (ANY mass).

*Second, Air has mass.

*Third, it is a pressure difference between two regions that accelerates air.  This 'difference' is called "pressure gradient" because it changes gradually from point A to B. The higher pressure accelerates air toward the lower pressure region. That is WHY, in the commonly cited case, it is moving faster when it gets to the lower pressure region.  A higher pressure (the net difference) accelerated it there.
THEREFORE, TA-DA!
The pressures which provide the upward lift force on the wing, are the VERY SAME PRESSURES that accelerate the air downward behind the wing/rotor.
Under-wing it is a bit more obvious. Looking at a wing with a moderate angle of Attack, it is easy to see that as the wing and air approach each other (Either in a wind tunnel or flight) the pressure must increase.  It is like the wind hitting me, or me running into the wind. This is really easy to feel in water. You really must push against the water to run! 
YOU are pushing on the air/water and, therefore, this relative motion is increasing the pressure under...  This pressure on the bottom surface pushes BOTH up on the wing and down on the air below it.  This causes that air to follow a curved path downward.
Here, we see that there is a higher pressure on the outside of the curved flow which is the centripetal force as the say "turning" the air. This accelerates that air into the downwash.  
Above wing, The air must follow the curve (I won't go into detail, but in "normal" flight we know it does).  Think of being on a merry-go-round.  Your inertia tries to keep you going straight and you "pull" away from the center.  Above the wing - SAME THING.  This relative motion of the curved AIRFLOW lowers the pressure at the surface. [mote it's not speed causing it, but curved flow] ... No Bernoulli, or anything else.  
---- EDIT: Forgot to mention.
The low pressure above (that is caused by the curved flow) does three things:


*

*It is the lesser pressure 'half' of the difference for lift.

*It allows the higher pressure ahead of the wing to push, accelerate air toward the trailing edge. This is the acceleration that has the air moving 'faster' in the wind tunnel view.  ***

*It allows the atmospheric pressure farther above the wing to push air down from well above the wing adding to the downwash. Downwash air comes from above and below the wing.
*** Data from a wing flying through stationary air shows that the upper air actually reverses direction as it 'crests' the upper camber. It starts being pushed forward (and up) in the bow wave ahead of the wing, then because of the pressure gradient (lower over the wing) it is yanked rearward, totally reversing direction to travel toward the trailing edge. This is shown in a diagram in my blog, linked in this posting.
NOTE ALSO: Acceleration is the "change in velocity" talked about in "Bernoulli's Principle", NOT SPEED!
People like to call pressure a scalar; that is, it has no direction.  However, in a sense, it pushes in all directions.  ANY higher pressure region pushes toward all nearby lower pressure regions. This includes nearby air and surfaces. 
This easily explains the bow-wave effect (up-wash) ahead of the wing.  With more pressure under than above, air is pushed in front of the wing so it is "trying" to get up to the top - same way it is around the tips - up and over, so to speak. 
--- END EDIT ---
IN ADDITION; PLEASE NOTE that we are dealing with ACCELERATIONS OF AIR.  Acceleration is the same whether we observe this from a moving air, fixed-wing reference frame in a wind tunnel, or in still air, moving-wing frame.  Accelerations are not dependent on speed of observer, whereas speeds are. 
THEREFORE, it is the same science in the wind tunnel as flight.  The talk about Faster air above the wing is frame of reference dependent. 
PLEASE NOTE that I said "AIR FLOW". This is important because it isn't simply wing shape, but flow shape!  A completely flat wing has the very, very similar kind of curved flow around it.!.!.!.  SAME John Anderson's book shows this.
For my full explanation (with pictures) with all the misconceptions explained, go to my Understanding Lift Blog: https://www.quora.com/profile/Steve-Noskowicz/Understanding-Lift 
How do I know this?  I took responsibility for a sophisticated full cockpit flight simulator some years ago and decided to brush-up.  I saw so much misconception on the web, I figured-out just about all of this myself, then talked with two experts actually in the field (not just people repeating others), Boeing's Doug McLean and Embry-Riddle's Charles Eastlake, to verify it and clear some things up. (because many years ago in pilot training I heard the equal transit time story and didn't look into it though it didn't satisfy me)
Best Regards, Steve
P.S.
A Cessna 172 has a wing loading of only 0.1 PSI. That is the difference in lower-upper pressure. With a wing area of 25,000 square inches, this gives PxA = 0.1 x 25,000 = 2,500 pounds loaded aircraft weight.
It turns out that the Decrease above is greater than the Increase below, this is something like 0.04 psi more below the wing and 0.06 psi less above.  So the changes in pressure is barely One-Half Percent at low altitudes. --> 0.06/14 psi = 0.429% change from atmospheric pressure.  
Airliners are about 1 psi loading.  So flight is really a very small effect, so to speak.  Why we need large wings!
This difference MUST be the same at any altitude, so at higher altitudes it is a larger percentage change due to the less dense / lower absolute-pressure air and, therefore higher air-speeds are required to get the same lift.
Also please note that the density change (compressability) is negligible and not a factor in normal flight.
Cheers
