If it is the button needing to be repushed over and over, then it is likely the button that is the problem. Were it the battery, then it would only occur when on battery and not when the charger is connected. Assuming it happens during both (you didn't mention), then it is likely the button.
Also, were it the battery, it wouldn't turn on at all, not even after a few tries. Now it could be the motherboard, but lets hope not.
If the device is under warranty, contact the manufacturer. If it isn't, then try a local tech.