  In this paper, we are concerned with the following type of elliptic problems:
$$
  (-\Delta)^{\alpha} u+a(x) u=\frac{|u|^{2^*_{s}-2}u}{|x|^s}+k(x)|u|^{q-2}u,
u\,\in\,H^\alpha({\mathbb R}^N), $$ where $2<q< 2^*$, $0<\alpha<1$,
$0<s<2\alpha$, $2^*_{s}=2(N-s)/(N-2\alpha)$ is the critical Sobolev-Hardy
exponent, $2^*=2N/(N-2\alpha)$ is the critical Sobolev exponent, $a(x),k(x)\in
C({\mathbb R}^N)$. Through a compactness analysis of the functional associated
to the problem, we obtain the existence of positive solutions under certain
assumptions on $a(x),k(x)$.
