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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-Hardyexponent, $2^*=2N/(N-2\alpha)$ is the critical Sobolev exponent, $a(x),k(x)\inC({\mathbb R}^N)$. Through a compactness analysis of the functional associatedto the problem, we obtain the existence of positive solutions under certainassumptions on $a(x),k(x)$.

A Global Compact Result for a Fractional Elliptic Problem with Critical Sobolev-Hardy Nonlinearities on ${\mathbb R}^N$