Macbook turns off after booting
This sounds like something that happened to my MacBook Pro 15" Early 2011, stayed on white screen for couple minutes then turned off again.
Took it to Apple and they came to the conclusion that my motherboard wasn't working so I had to get it fixed for $500 (£250 ish) and after that ever since works perfectly fine.
The cause of my problem was that it could of been my old charger (was falling apart, and so Apple replaced it for free) could of sent an electric charge that fried my motherboard.

Hope this relates or helps you to fix your issue.