Battery problem or something else?

Hello to all, my iphone turns off when I open an application, or when i get a phone call. I can turn it on when the phone still have 70-100% of battery but when the battery is lower the phone just show the apple logo and turns off again. When this happens i have to connect it to my computer and it stars normally, or sometimes after 10 min later and the phone has like 30 or 40 percent. Also when I use my phone with a power source connected the phone never turns off.

I hope you can help me. THANKS .