Big cat habitat roundup (in order of descending size)
Tigers do not live in the desert.
Lions live in the Namib deset, as mentioned above, and also on the fringes of the Sahara. They used to live all through Iran, Middle East and North Africa, much of it desert scrub. They also used to live in the deserts of India, though they are now confined mostly to the Gir Forest, a semi-arid forest area. Many of Lion's favorite prey specialize in dry areas like the Etosha Pan, such as Eland, Kudu, and Hartebeest.
Jaguars are from the Americas, and never lived in Egypt. They do, however, live in desert areas like Arizona and the Sonoran desert of Mexico, but mostly in wetter mountainous parts of the desert. Not really a desert creature.
Leopards have the greatest habitat variation of any of the big cats. They live every where Lions can live, and additionally still exist in North Africa and Arabia. They specialize in ambush, so they tend to be in areas where there is at least some scrubby growth to hide in.
Cheetahs could be considered desert specialists. They are designed to hunt antelope, and antelope are designed to take advantage of semi-arid and arid plains. In addition to being present in all the major deserts of Africa, Cheetahs still live in Iran and until recently in the deserts of India. They also were distributed farther north into central asia than either Lions or Leopards because of the plentiful Goitered gazelle.
Snow Leopard lives in many areas that are very dry (Tibet) but aren't really considered deserts the way you are thinking of. But they can very much live in desert mountains.
(Extra Bonus) Cougar is the largest small cat. But we'll throw it in here because they are bigger than Leopards. They also have a very wide range of habitats, and live in the Desert southwest. However, unlike in Africa and Eurasia, there aren't a lot of things to eat in the Desert southwest; pretty much only pronghorn (which are too fast) and the occasionaly feral donkey. There are some desert bighorn sheep, but they are nearly extinct. Cougars would probably make a desert comeback if more big mammals were around, like bison, more bighorn sheep, and more feral donkeys.