In the film The Lobster (2015), the pain in the back is such a defining characteristic of David. Does it have any particular significance? Why did the director decide to give him that burden?
I think the main significance of his back pain was that it required ointment for a part of his body that he couldn't reach. That highlighted just how alone he was, that he didn't have someone to do something so simple as rub the ointment to alleviate his pain.
There was of course the (what I consider) secondary effect, that the character being in semi-constant pain added to the overall oppression of the film.
Aryxus is partly right; the back pain was intended to highlight how advantageous it is to have a partner in life. Taken from Larsen On Film:
So The Lobster backs off a bit from its bitterly comic, anti-romantic first half. Early signs the movie might head this way are the occasions in which we watch David, alone in his room, struggling to apply pain-relief ointment on a hard-to-reach part of his back. At some level, the movie recognizes both our desire for relationship and our need for it. What the movie objects to, it seems, is the way relationship has been propagandized, commoditized and narrowly defined in contemporary society.
Keep in mind, too, such trivialities as a dining room full of "tables for one", and how certain activities (such as masturbation, which lends to the idea that you don't need a partner) are either discouraged or not allowed entirely in the hotel.