first, but rather obvious, make sure you are using the right button to power it on. Just pressing (not holding) the front button (not top) should turn it on. I had a similar issue, mine was still in warranty, and I contacted gopro support and got a new battery sent to me. That resolved my issue.
1st, make sure you're doing everything right. Follow all the instructions in the manual. If none of this works, call GoPro up and tell them your problem. Either they will help you fix the problem over the phone or send you a return label so they can either fix the problem or exchange the camera. Good luck!!
Sometimes mine freezes up so I take out the battery, give it five minutes to cool down and then put the battery back in. Works every time but it kills all your settings - date, time, video quality, auto-off etc). If you don't set the auto-off, it never turns off unless you push the button and that can really kill your battery charge. Don't know if that helps.