iPhone turns off with half of battery

My phone has been acting weird for some time. When the phone's battery reaches half it will turn off and if I turn it on again it wont stay on. But what I noticed is that if I turn on airplane mode this random turning off wont happen. In airplane mode I even turn on wi-fi and I dont get any problems. It's just when the "phone" (or network or whatever) is turned on that it turns off with half of the battery. I already restored it and it didn't help.