Was it ever wet? If not the Os may be corrupt. You can try replacing it if you can get it to be seen by a Pc via the USB. Try downloading Desktop manager from RIM( free) and try to connect to the BB. Worse case the BB is defective. Another thing, check the connectors on the case. Maybe they are dirty or not fully touching.
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I had those problem... Your battery is too low to start BlackBerry. The device will automatically on by the time we plug-in the battery. If the battery have not enough power, it will turn off / restart & it will automatically on again.
You need to charge the battery on independent charger. Good luck