Imagine a large chunk of land about half the size of the size of the United States cracked away from the main landmass and floated up into the sky for as high as a mountain (say, 2,800 meters, not as high as mount Everest or some of the highest ones so the condition would still be inhabitable). The continent left a massive crater in the ground, meaning massive cliff sides face in all directions. The landmass is about a thousand metres thick.
This continent has been around forever, has its own ecosystem and resources, and humans have been born and built civilizations both on it and throughout the rest of the world.
I'll try to keep the pretext simple: let’s say that one day the people of the floating continent decided to unite themselves into one single country, and started to act all xenophobic, and eventually a war break out between them and one of the world superpowers below but nearby. The protagonist is a high ranking officer of this superpower’s military force who will have to figure out the logistics and plan of attack. Technology is equivalent to today's NATO countries/forces here on Earth, but nukes are not in play from either side.
The “Ascended” people
Since the country is landlocked, and seafaring vessels cannot reach them, the people of the floating continent are renowned for building some of the largest cargo planes and even helicopters with powerful engines, that can easily transport large amounts of supplies along with some of the most advanced airports in strategic locations all over the continent. This acts as their lifeline. They are still trading with the other countries. So, they don’t have a two front war to contend with and can't be starved into submission.
They can’t overpower the superpower economically or militarily, but they sure can put up a fight and can easily wage a protracted war with their control of the skies. Their soldiers are fanatical due to a belief in racial and geographical superiority. They are not afraid to enact scorched earth policies on some of their own infrastructures.
They have fortified the cliffside facing the superpower's border with a wall of guns in an extended cave system (think of the Maginot line, but they have both dimensions to work with).
They do depend on trade to remain viable economically and with resources.
For simplicity, imagine a clone of America's army today with all of its equipment and ordnance. They have a coast line open to trade, are a juggernaut both militarily and economically, and possess some of the best air force and airborne units in the world, second only to the Ascended. They have no allies.
The superpower’s goal is not to completely occupy the continent, but to control a sizable portion of territory on it (cities or provinces) and shatter the image of superiority of the Ascended people, weakening their position and hopefully bring more allies to their side in the case of any future conflict.
I've planned to have the superpower neutralize the cliff side defenses after an epic battle, but what troubling me the most is the logistics following the battle once they’ve establish footholds on the landmass. Since the superpower can’t bring its massive navy and cargo ships to deliver supplies like the US did at Normandy, everything must be transported by air.
Question: Is there a more efficient solution than aircraft to supply a beach head operation in the sky given the parameters outlined above? Efficiency is the key to the best answer. Efficiency in terms of speed, cost, and defensibility.