I've got an Inspire 4G that will boot up, you can maybe open up the app drawer or something for a few seconds before it completely turns itself off (not a reboot). This does not occur when the phone is on the charger though. I have checked the battery levels as well as the connection and they are fine. I just did a factory reset back to stock (not even rooted) Android 2.3 and am still having this issue. Any ideas?