you didn't list what you did do, so you might have already tried this, but I think it's always a good idea to try re-seating the RAM and then try different RAM in the computer if you can. The symptoms seem to be identical to those caused by bad RAM sometimes. It might also be a heating or fan problem which, when detected by the motherboard, activates an auto shut down.
Also, make sure the power supply is producing the proper amperage and consistent voltage. some power adapters provide a signal to the computer, along with power, that is critical as well, so there is a slight possibility that the adapter may be causing the problem. Actually I'd follow this further. If the battery is almost dead and the adapter not working, those are the exact symptoms you'd get. Do you know if the battery is charged and the adapter works?
I'm assuming you've removed all external devices and cards or other ad-ons.
Did you try taking the battery out, unplugging it and pushing the power button 5-10 times? This might help reset some settings.
And, if you already have done any of these, please list them so people, like me, don't waste our time giving you incorrect answers, because you didn't take your time to list what you've done.
AND AND AND
Google it first. because
look what popped up.