I think there may be a cold or broken solder joint in the power barrel.
Try plugging it in and very slowly move the power jack around until the LED lights up. If it lights up at a certain spot, hold it there for a few minutes and see if you can turn it on. Then enable USB charging.
EDIT: It looks like the psp 3000 has a power barrel that is hooked up with wire instead of soldered, so I withdraw my suggestion. I read online that some people solder bridge the power fuse to fix a psp3000 not turning on. Maybe try that.