If the PSU provides power to boot the notebook, but the notebook suddenly shuts off, it could be either the PSU or the motherboard. You can check that the PSU is functioning normally but simply measuring it's output with a multimeter. PSUs are actually pretty easy to fix for notebooks, considering they're external, and 95% of the time, the only problem they have is a bad connection (which doesn't appear to be occurring here).
The motherboard is a bit harder to diagnose. If you test the PSU and it's fine, you're going to need to dismantle the machine and breadboard it, removing every bit of hardware you don't need to have the machine boot. If it boots fine, then one of the pieces you removed is malfunctioning, but if it still doesn't work, you may indeed have a dead board.