You can downgrade from firmware 3.03 and below.
From 2.81 to 3.03, you need a GTA Liberty City Disc from the first batch that was made. It has the firmware 2.0 on it. If you go and buy the game new today on some store shelves, it wont work. You absolutly need a first edition copy. With some luck you can rent it or find it in some flea market or you can try ebay, theres an handful of them but they mostly go at 50$.
I recently downgraded my 2.81 unit. I paid 65$ dollars for the PSP, 1 gig stick, 1 game and 1 UMD movie but i had to pay 60$ for a boxless copy of GTA.
So what i did is:
-Upgrade my 2.81 with Sony's official 3.03
-Use the 3.03 downgrader (wich needs GTA LC) to downgrade to 1.5
-Upgrade to 3.40 OE after that from the 1.5
So now it's just like the official 3.40 except i can drop iso and cso in the ISO folder and run them from the psp just like a UMD game. Sweet lovin, it was worth all the effort for sure.