  We obtain existence, multiplicity, and bifurcation results for the
Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator,
involving critical Hardy-Sobolev exponents. Our results are mainly extend
results in the literature for $\alpha=0$. In the absence of an explicit formula
for a minimizer in the fractional Hardy-Sobolev inequality $\alpha\not= 0$, we
get around this difficulty by working with certain asymptotic estimates for
minimizers recently obtained in \cite{MM}.
