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

Abstract

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