What May Be Causing iPhone to Not Stay On?

I got an iPhone 5 from a friend, and it has one small issue. It has a dent on the back, which is right about where the battery connector would be, but not over the battery itself. Though this big dent is over the logic board, the phone still powers on when you hold the power button or plug it in, but it will shut down after about 3 seconds of being powered on. If you plug the phone in, you see the Apple logo for three seconds, it goes black, then the cycle starts over. I'm going to open the phone and make sure the battery connection still looks sound, but I'd appreciate any suggestions on what to fix or what could be going wrong. Thanks!