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

In this paper, we are concerned with the following type of elliptic problems: $$(-\mathrm{\Delta }{)}^{\alpha }u+a(x)u=\frac{\mathrm{\mid }u{\mathrm{\mid }}^{2_s^{\ast }-2}u}{\mathrm{\mid }x{\mathrm{\mid }}^s}+k(x)\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u,u\in H^{\alpha }({\mathbb{R}}^N),$$ where $2<q<2^{\ast }$, $0<\alpha <1$, $0<s<2\alpha$, $2_s^{\ast }=2(N-s)\mathrm{/}(N-2\alpha )$ is the critical Sobolev-Hardy exponent, $2^{\ast }=2N\mathrm{/}(N-2\alpha )$ 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 $a(x),k(x)$.