The Halacha permits instructing a non-Jew to perform a Melacha for the sake of a Misva only in a case of "Shvut D'shvut"- a double Rabbinic prohibition. Asking the non-Jew in itself is a Rabbinic prohibition, but the Melacha he is being asked to perform must also be only a Rabbinic violation. The double leniency of two Rabbinic prohibitions allows instructing the non-Jew. Therefore, asking a non-Jew to turn on a light, even for a Misva such as learning Torah, is prohibited, since turning on lights, according to most authorities, is a Torah prohibition.
However, the Pri Megadim (R. Yosef Ben Meir Teomim, 1727-1792, Poland-Germany) has a Hidush (novel Halachic approach) and permits instructing a non-Jew to perform a Torah prohibition with a Shinui (in an unusual fashion). He applies this to telling a non-Jew to pick an Etrog off the tree (A Torah violation) with his teeth (a Shinui) to perform the Misva of Lulav and Etrog.
The Shach (R. Shabtai b. Meir HaKohen, 1621–1662, Eastern Europe) in his Nekudot HaKasef (198:18) deals with a case of a woman who needs to immerse in the Mikveh on Friday night forgot to cut her nails before Shabbat. He permits her to instruct a non-Jew to cut her nails with a Shinui.
A practical example would be telling a non-Jew to turn on the lights by flicking the switch with his elbow in order for the Jew to learn Torah.
All these cases are permitted since a Melacha is only a Torah prohibition if performed in the standard method. Even if a Jew would turn on the light switch with his elbow, it would only be a Rabbinic prohibition. Therefore, performing an act classified as a Torah prohibition in an unusual fashion constitutes a "Shvut D'shvut."
The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), concur with this ruling, whereas the Melahayich Omnayich cites Rav Elyashiv as being stringent. Hacham Ovadia (in Hazon Ovadia) and Hazon David rules in accordance with the lenient opinion.
This leniency is a solution for the elderly who live on high floors in a building and are unable to walk down all the stairs to attend Bet Knesset. While it is prohibited to instruct a non-Jew to operate the elevator on their behalf, it would be permitted to tell the non-Jew to press the button with a Shinui, such as with their elbow. The elderly person would also be permitted to ascend in the elevator after Tefila in the same manner. Even though he has already completed the Misva of Tefila, remaining in the lobby for the rest of Shabbat is considered Sa'ar (suffering) which is tantamount to a Misva.
When applying leniencies such as this, Hacham David recommends instructing the non-Jew before Shabbat.
It should be emphasized the leniency only applies to cases of Misva or Sa'ar (suffering), but not to someone who wants a light to read secular books and magazines!
SUMMARY
It is permitted to instruct a non-Jew on Shabbat to turn on a light with his elbow to enable someone to learn Torah or pray.
