Here's just an idea, if you're looking for stuff to try (not sure if that's your aim). Categorize the values into quantiles (eg quartiles). Since you have so many 0's, you could make the lowest (quartile) all the 0's. Then set the rest of the quartiles up to be evenly distributed.

eg n=100, with 50 zeroes.

Q1 will have 50 subjects, Q2 will have 16, Q3 will have 17, Q4 will have 17.

With the variable now "categorized", you could do polytomous logistic regression instead of OLS.