Solutions for a nonlocal elliptic equation involving critical growth and Hardy potential

Abstract: In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left{% \begin{array}{ll} (-\Delta)^{s} u-\ds\frac{\mu u}{|x|^{2s}}=|u|^{2^*_s-2}u+au, & \hbox{$\text{in}~ \Omega$},\vspace{0.1cm} u=0,\,\, &\hbox{$\text{on}~\partial \Omega$}, \ \end{array}% \right. \end{equation*} provided $N>6s$, $\mu\geq0$, $0< s<1$, $2^*_s=\frac{2N}{N-2s}$, $a>0$ is a constant and $\Omega$ is an open bounded domain in $\R^N$ which contains the origin.