In this paper, we consider the following problem: $$ (-\Delta)^{ s} u -\frac{ \zeta u}{ |x|^{ 2s}} = \sum_{ i=1}^{ k} \frac{ |u|^{ 2^{ *}_{ s,\theta_{ i}}-2}u} { |x|^{ \theta_{ i}}} , \mathrm{ ~in~} \mathbb{ R}^{ N}, $$ where $N\geqslant3$, $s\in(0,1)$, $\zeta\in \left[ 0,4^{ s}\frac{ \Gamma(\frac{ N+2s}{ 4})}{ \Gamma(\frac{ N-2s}{ 4})} \right)$, $2^{ *}_{ s,\theta_{ i}}=\frac{ 2(N-\theta_{ i})}{ N-2s}$ are the critical Hardy--Sobolev exponents, the parameters $\theta_{ i}$ satisfy a suitable assumption. By using Morrey space, refinement of Hardy--Sobolev inequality and variational method, we establish the existence of nonnegative solution. Our result generalizes the result obtained by Chen [Electronic J. Differ. Eq. (2018) 1--12].
Categories: math.AP