I had this problem in August and paid my local Apple store to replace the screen. It was cheaper than a new phone. In December, the problem reappeared. I was told this time that the problem was a swelling battery that was a known problem. Unfortunately, the three-year from original purchase date warranty had expired TWO MONTHS AFTER I PAID APPLE TO REPLACE THE SCREEN. Seems like they might have told me, at the time, the problem was the battery, and replaced it. I would have been happy to pay for that replacement.
I ended up buying a new phone because they offered me 1) a "new" 5S or 2) $47 toward a new phone. My old 5 is a phone on which I'd already had to replace the home button (another common defect). I don't know what's happened to Apple's quality/service but this is not the company I dealt with when first switching from PCs to Apple products. Now, I'm so heavily invested in Apple, it would be difficult and time consuming to make the switch. If my new 6S fails to deliver, I may choose difficult and time consuming.
The Apple device that caused me to switch was the iPod click wheel, for which I had to get a firewire card for my desktop PC to be able to use. I've owned many iPods and my last one, the iPod Classic died a few months ago of hard drive failure. Apple could make a 160GB iPod with a solid state drive but has made it clear that streaming is what I need to embrace and get over the fact that I once had an elegant device that was designed to do only one thing and do it extremely well. Ditto for the removal of CD/DVD drives from everything.
iPhones are where Apple makes its money these days and I guess with the end of carrier subsidies, Apple's willingness to give people a break is pretty much over as well.