If it works fine when it's plugged in, then it's a battery issue. Either fault of the battery itself or the connection. If it shows as being fully charged, and shutting off anyway, it is likely an issue with the connection. Worst case scenario the PSP itself is basically fucked, but you can try and wiggle the battery, tape it in place if need be. If the battery refuses to work no matter what you do, then have fun either replacing it or cracking it open. If you have a 3000 I would recommend replacing it with a 1000. The 3000 model might have more vibrant colors, but the think is fragile as all hell, and that fragility might be demonstrated by the issue you're having now.
