How old is your battery? always look for the easy stuff first :) and than make sure that the battery makes good contact. If I remember right the PSP has fuses on the MB. It has 2 fuses (LD) One marked TB. The other is located beside the power cable disconnect.So by using your car charger you may just have set off one of these. Of course to fix it you will have to open it up and get to them. If you are under warranty with your PSP, don't do it. Go to Sony and replace it. Otherwise, you don't have anything to lose, try to fix it and let us know how it is going. Good luck
http://www.yall.net/Forum/Repair-Center/Video-Game-Repair-Center/PSP-/-PSP-2000-/-PSP-3000/The-PSP-and-Fuse-Repair.html and http://www.youtube.com/watch?v=h-TEeWRvlwg might help you.