When plugged in, does it run? If so, the battery might be nearing the end of its life cycle. Go grab a new one if it's too old.
Other than that, it might be bricked. If you're unable or unwilling to send it to Sony, maybe try a Pandora's Battery, to put it into service mode.
E: Uh...I don't know too much about servicing PSPs, but my general knowledge is telling me it's bricked, because there's no user serviceable parts left to it, except maybe the memory stick, which you might want to check for corruption on a computer. As I said, maybe try a Pandora's battery. It's meant to get it into service mode to put custom firmware on it for homebrew and all that, but it may save your ROM where the OS of the thing is.
and Google the subject some more.
I must tell you this is risky business, mucking around with a Pandora's battery, but the PSP can't really be worse off than it is. If you're not comfortable with it, give it to Sony. If you're not technically minded, leave it be.
But hey, if you get a Panda battery anyway, and can't save the PSP, you can get another one (maybe) and now you have a working PSP and a Panda battery! Neat-o.
One last think, if you suspect there's a hardware problem, like if it got dropped hard (the new PSPs, that is, 2000 and 3000, have a metal frame that's been reduced in weight and heftiness as compared to the 1000), you might want to take it apart as per iFixit's guide depending on your version. You probably can't fix it, but it might satisfy your curiosity as to what went wrong, if you're that kind of guy, and just want to screw it and get a new one, maybe use this one for parts.
Anyway, I've rambled on long enough.
Good fixin's to you.
When it is plugged in it will not stay on either