In your first post you said "It shuts down after starting up...",
which I took to mean that the computer was shutting off completely...no power, no activity at all. This would be a hardware shutdown, possibly due to cpu overheating.
If it is still sitting there idling, albeit at a blu screen, then you probably have something else going on. ~Jethro~ knows laptop hardware better than I do.
Do you have access to a bootable disc? Like an operating system disc or maybe a copy of the Ultimate Boot CD? I'd like to see if either could load into memory successfully.