You can take it to most gaming shops. Gamestop has a farily acceptable service, but some others, like Hollywood Video, do a surprisingly good job. Whatever you do, just don't try to open it yourself, as you'll most likely void the warranty. If all fails, try taking it to a Sony Store near you.
See if it's still under warranty and contact the manufacturer about getting it repaired or refurbished. If you don't want to go that route, perhaps try to find someone on a website like Craigslist who might be able to help you.
A PSP can be fixed at GameStop which is a local game store. GameStop recently took over EB Games so that any place that used to be EB Games is now GameStop.
There are no official places that do the repairs. You can check with any local PC repair place they may have that service available. Otherwise check out your local craigslist and look for someone with good credentials who does it as a personal business. But really stress the credentials checking on that one.