Multiple Regression where DV is a proportion


I am using multiple regression with 3 continuous predictor Variables, and the Dependent Variables is a proportion (0-1). I'm worrying quite a bit, because im expecting that i will find many cases in my sample where participants record a proportion of 0 (floor effect). There will be a heavy skew in my DV towards 0, with a substantial number of cases specifically on 0, and also on the other end of the spectrum at 1 (ceiling effect).

I'm thinking that if the distribution was even close to normal i would be ok, however with a heavy floor and ceiling effect, im guessing this completely screws my design.

Any advice anyone? or is this a case of flawed design?