Hi!
This is a common issue with PSPs, their memory chips get corrupted if not used too long. There are thankfully quite a few solutions online. You can try them in any order, but if none work you may need to replace the board.
* Try rebooting without the memory card in
* Try removing battery for ten seconds. Then put it back in, plug in the charger, turn the unit on while holding the two shoulder buttons. If something comes up, choose the Recovery folder.
* Plug it in. When powering it on hold the power button up until something happens.
Also, there could be a problem with the backlight of the display. Can you hear any sounds at all if you press buttons while the screen is black? Or if you shine a bright light on the blank screen, can you see the menu? Then you will have to replace the display.