why does my sprint S3 say my battery is fully charged unplug charger but it turns off ?

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AC Question

My phone will only get through the entire "boot up" if I have it connected to the charger. If I unplug it as it suggests and shows a full battery it shuts off immediately. When I try to turn it on(not connected to charger)it starts to boot up but starts over right after the the "fancy" Samsung Galaxy 3 font, before the blue swirl and starts again repeating that over and over, which makes me to believe the battery is indeed charged. I cant have my phone plugged in always. Please help.
 
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