I found my old PSP (epic) and I had to buy a new battery and charger to see if it'd work (the links will be here too) but for some when I try to turn it on it doesn't unless I plug into the charger. Once I unplug it switches from orange to green for a millisecond and then shuts off, no response. There has only been one time where it turned on without being plugged in but eventually shut off. My PSP is a 3000 by the way.
Charger: AC Adapter Home Wall Charger Power Supply For Sony PSP 1000 2000 3000 Slim Lite
Battery: Battery Sony PlayStation Portable PSP 2001 3001 PSP-S110 Rechargeable 1200mAh
Charger: AC Adapter Home Wall Charger Power Supply For Sony PSP 1000 2000 3000 Slim Lite
Battery: Battery Sony PlayStation Portable PSP 2001 3001 PSP-S110 Rechargeable 1200mAh