Airlines in the developed world almost never own planes, especially in the dimensions of any typical commercial airliner. The other problems that plane ownership brings (besides routine maintenance, which the airlines do own) are outside the desired core competency of airlines. The leasing of aircraft to commercial airlines is quite an industry in itself, and the transactions often involve complex, multilateral bond financing. There are almost always complex financing arrangements involved. Few buy a 737 with a bag of cash anymore than one buys a skyscraper outright. It is done by some of the world's less creditworthy airlines, and American has a penchant for owning, but it is increasingly not the normal industrial practice. Investors go in on planes just as they do on any other massively expensive capital goods.
Thus, the key for a drug cartel would be to get the various intermediaries involved in the aircraft leasing and financing process (some of which are surely quite corruptible) to lease one to ShellCorp, or buy on the secondary market, as some have suggested.
The bigger mystery for me is how they manage to fly in the context of overall international airspace control and air traffic control. Like someone else said, a 747 can't take off and land on a desolate patch of grass. Law enforcement and intelligence aren't stupid; they know how much money is in the "air charter" business, and it definitely won't pay for a 747. Charter companies use much smaller aircraft.
Edit: Unlike airlines, drug cartels actually do have the cash to buy a $138m plane outright, yes. But that would set off enormous alarm bells, because nobody otherwise obscure does that.
