  We consider the finite W-superalgebras for a basic classical Lie superalgebra
g associated with an even nilpotent element in g both over the field of complex
numbers field and and over a filed of positive characteristic. We present the
PBW theorem for these finite W-superalgebrfas. Then we formulate a conjecture
about the minimal dimensional representations of of complex finite
W-superalgebras, and demonstrate it with some examples. Under the assumption
that the conjecture holds, we finally show that the lower bound of dimensions
predicted in the super version of Kac-Weisfeiler conjecture formulated and
proved by Wang-Zhao in [40] for the modular representations of the basic
classical Lie superalgebra with any p-characters can be reached.
