Yes they do - had one drop the other day. These are pretty easy to take on. Just remember that all of these are social and will jump in if one of their OWN kind are being attacked. Attacking a Greater Fire Elemental near A Earth Elemental will not cause the Earth elemental to attack. But if you try it near another Fire they will jump in.
Fire Elementals have Fire Damage Shields Earth Elementls cast Root Air and Water are the easiest to kill in my opinion...
After spending days on this island, I have the following information. The best chance you have at getting any "greater" elemental (swift, beserk, gigantic, Aero, Hydro, Terra, and Inferno, etc) is to simply kill as many elementals as you can, as quickly as possible. Every time an elemental spawns, there is a chance that a "greater" elemental will spawn. Each "greater" elemental drops either a lesser or greater item, always. Once you kill a gigantic elemental or a named elemental of a particular type, you will become KOS for that particular type immediately. Otherwise, your faction goes down slowly each time you kill one, and goes up with the people in the castle on the island. Air Elementals are sometimes invisible, so having see invisible is a plus. It seems to me that killing the elementals near the Large Rocks in the middle of the island seems best, as the "greaters" always seem to spawn near these rocks, but, simply killing them all is best anyway. If you have a full group, the loot is plentiful, and for NON UBER folk, very good. I was there for 5 hours with a full group the other night, and everyone except me looted 2 items (I have all the items that dropped, and they are lore). In all the time I have spent on this island, which is well over 36 hours, I have only seen 1 named Elemental, Aero, once. I also know he spawned on patch day, after a server re-pop. If this helped, please rate it. Thanks.
Iomhair Othar 47 Level Ranger of Tunare Fennin Ro
All that is needed for the triumph of evil is for good men to do nothing.