Iphone 5 battery replacement

The battery replacement program is presented in: https://www.apple.com/support/iphone5-battery/

Note the Additional notes at the bottom of the page about limitations on the country of replacement. You can try taking the iPhone to an Apple store and see what they say.

If no joy is found officially over there (a pity, since the replacement program is free), and if strong of heart and firm of hand, you can do the replacement yourself. See here (note they carry a full kit with all needed tools for $30+shipping).