Sandy,
How about offering a referral incentive to the current residents (the good ones), and new ones moving in? They'll be happy to get a nice referral credit on their rent, and they get to pick their new neighbors! Maybe give them a $100 credit when the applicant they referred to you signs a lease?
I've used this method before in lease up situations, which is similar to your situation. It usually generates a lot of activity. You can include the qualification criteria when you publicize the referral program, and that should reduce the number of unqualified applicants.
Another strategy is to increase the application fee. If people think the don't have much to lose, they'll file the app hoping they qualify, but knowing they probably won't.
If you're afraid a higher app fee will prevent some of the good applicants from applying, still charge the higher fee but then credit some of it back on their first month's rent.
Matt