So when my iPhone 5S shuts down over night because of low battery, I have following behavior the next day, when I connect it to my laptop or my car's usb:
It turns on, tries to boot, but then dies because of a the low battery. This happens again and again for about 30 minutes. Then it will stay on with some percent of battery, charge as it should, and then last about one day.
I think this is because the battery is too low to boot. But Why does it turn on automatically? It's really annoying.
Even if the battery is broken, it just should not turn on.