As far as i know photoshop elements 3 won't run on mountain lion.
Your best option is to upgrade to photoshop elements 11 if you still want a workable version of photoshop elements.
PSE 3 hasn't been able to be installed in OS X for several versions now. 10.4 was the last one, IIRC. PSE 6 is the very earliest version that will work in 10.8.