I had the same problem.....
Someone on another website mentioned that this notebook must not be powered up on a bed, or something soft because it blocks the airflow coming in from the bottom, which will cause it to overheat and quickly start then shut down.
I replaced a bad fan in mine and pulled out all the lint blocking the airflow the 1st time I had this problem.
But the 2nd time it happened,(brief flash of lights-then black screen, nothing) I changed the CMOS battery and Voila! it worked. You have to reset the Date/Time but that's easy, a lot easier than spending $275.00+ for a motherboard. The process is on the Dell Website but DO NOT attempt it unless your idea of fun is to completely disassemble/re-assemble dozens of parts,ribbon cables,etc. If you are patient, have neuro-surgeon hands and have well learned the lesson NOT TO FORCE something that doesn't easily come apart-well, then give it a try. 5 bucks tops for the battery, and 1 1/2 hours time(for me) and my 'boat anchor' was purring like a kitten.