Chosen Solution

I recently acquired an iPhone 5s. After taking it home and getting a new SIM card for it, I noticed that the display in the top right corner was popping out. After looking around on Google, I believe that this is an effect of the battery swelling. Also, looking at these forums, I noticed many people were able to get a free replacement even if their phone was out of warranty. Will Apple still give a free replacement or will I have to replace the battery myself?

Your best bet is to take it to the store you got it from and see. If you bought the phone from an apple store or a service provider who deals with Apple then you have a chance but if you bought it from someone unauthorized by Apple you don’t.