I picked up a used one off e-bay since that's the only place I can find the blasted things and I've had nothing but issues with it. No matter what game I play, off the flash card or disc, the PSP will shut off after a few minutes. The battery is full and at first I thought I was hitting the power button but I wasn't.
The buyer won't answer any questions and since the issues didn't turn up until a few weeks after I bought it, I can't return it due to how the policies work.
I have no idea what's wrong with it
Edit: put a new battery in and now its working. Some what. I haven't tried many games but some of the ones I have tried caused the PSP to turn off with in five minutes of playing the game while others, play normally