Why couldn't my phone turn on without charging it?

My phone is all fine but it can be used ONLY when it is charging and it will automatically switch off when the charger is off, I can't quite figure out if the problem is with the battery or the software? I've reset the phone (using the volume up, power and home key) hoping it'll be okay again without the charger, apparently, that is only formatting the phone and its content. I tried to connect the phone to the laptop, and only the Samsung logo appeared and it'll continue charging. The phone can only be used normally when it is plugged to the charger on a socket, it doesn't seem to be supported by powerbanks or cables to laptop.