Hi, i know this is an old question but it seems to be a very persistent problem for the PS Vita community, i had to deal with this for about a year and a half and nothing but replacing the joystick worked for me.
I Love my Vita, but the joystick problems on the slim model are very frustrating, this are the things that i tried for the past year and a half:
- "Recalibrate" the joysticks by moving both clockwise during boot (Useless).
- Blow on the side of the affected joystick (There's a posibility to make the problem worse, moisture from your breath can get on the contacts and cause corrotion).
- Take the vita apart and use a compressed air can or WD40 electronic contact cleaner on the side of the joystick (This seems to work for some people, but for me the problem came back after a few days).
- Replace the joystick (This is the absolute fix for the problem, though it requieres some knowledge or practice on taking apart electronics without damaging them, the slim model is very thightly packed and easy to break).
The replacements can be found on eBay, just look for "ps vita slim joystick replacements", i got mine for just 1 USD from a chinese supplier, they aren't original but that's really not an issue other than aesthetics since they do look very different from the originals.
I hope this can help other people looking for a solution.