I am not sure if PSN has removed everything for the psp online network, but on the playstation website they do list little big planet as one of the games that no longer have online servers. So you might have luck on other games? But little big planet is gone sadly.
It sucks but to my knowledge all servers for PSP are long and gone. You can still play some games over virtual system link online but most have ping limits so you are stuck playing gems like Shrek Smash and Crash Racing.
It's on a game-by-game basis, PSN still works in general since Sony doesn't seem to know what it means to discontinue online services, but when a company hosted the servers rather than have Sony do it for them, it's up to them to shut them down once the game gets too old. For example, Test Drive Unlimited's servers were shut down 5 years after release or LBP's servers I think was two years ago and so on, while other games like Monster Hunter (literally ALL of them) have their servers up and working still just fine!