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

Abstract

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

arXiv admin note: text overlap with arXiv:1703.00076