  In this paper, we deal with existence, uniqueness and exact rate of boundary
behavior of blow-up solutions for a class of logistic type quasilinear problem
in a smooth bounded domain involving the $p$-Laplacian operator, where the
nonlinearity can have a singular behavior. In the proof of the existence of
solution, we have used the sub and super solution method in conjunction with
variational techniques and comparison principles. Related to the rate on
boundary and uniqueness, we combine a comparison principle proved in the
present paper together with the our result of existence of solution.
