A Global Compact Result for a Fractional Elliptic Problem with Hardy term and critical non-linearity on the whole space

In this paper, we deal with a fractional elliptic equation with critical Sobolev nonlinearity and Hardy term $$(-\mathrm{\Delta }{)}^{\alpha }u-\mu \frac{u}{\mathrm{\mid }x{\mathrm{\mid }}^{2\alpha }}+a(x)u=\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u+k(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u$$ $$u\in H^{\alpha }({\mathbb{R}}^N),$$ where $2<q<2^{\ast }$, $0<\alpha <1$, $N>4\alpha$, $2^{\ast }=2N\mathrm{/}(N-2\alpha )$ is the critical Sobolev exponent, $a(x),k(x)\in C({\mathbb R}^N)$\mathrm{.}Throughacompactnessanalysisofthefunctionalassociatedto$(*)$, we obtain the existence of positive solutions for $(\ast )$ under certain assumptions on $a(x),k(x)$.