I have a white Darth Vader PSP 2000 and the battery was absolutely a spicy pillow when I checked it. I would suggest to judt replace it if you still have the original battery as its at minimum 12 years old now.

I swear I bought a replacement battery for mine but have no idea where its at.

The other thing to look for is a damaged screen. Somehow over the years of sitting on a shelf, mine looks like shitty car window tint thats bubbled up and peeling off.