|Capitalize if part of the proper name:
Do not capitalize if not part of the proper name.
Sahara, Gobi, Negev should be written without desert because these names themselves mean desert. Write:
the Sahara not the Sahara Desert; Sahara means "desert" in Arabic.
the Gobi not Gobi Desert; Gobi means "desert" in Mongolian.
the Negev not Negev Desert; Negev means "desert" in Hebrew.
although if necessary for clarity desert may be used lowercased: