In no particular orde, and these could be related.
The battery could be faulty, the socket where you plug the adapter in may be intermitant/damaged/soldering cracked etc
What if you try battery on it's own... does it attempt to power up.
The AC adapter may be faulty, unable to supply sufficient current etc.
With any issue like this it's worth removing all power AC and battery and holding the power button in for 20 seconds or so to discharge any capacitors and reset the logic.