In this work we study the existence of positive solutions to the
following fractional elliptic systems with Hardy-type singular
potentials, and coupled by critical homogeneous nonlinearities
\begin{equation*} \begin{cases}
(-\Delta)^{s}u-\mu_{1}\frac{u}{|x|^{2s}}=|u|^{2^{\ast}_{s}-2}u+\frac{\eta\alpha}{2^{\ast}_{s}}|u|^{\alpha-2}
|v|^{\beta}u+\frac{1}{2}Q_{u}(u,v)
\ \ in \
\Omega, \\[2mm]
(-\Delta)^{s}v-\mu_{2}\frac{v}{|x|^{2s}}=|v|^{2^{\ast}_{s}-2}v+\frac{\eta\beta}{2^{\ast}_{s}}|u|^{\alpha}
|v|^{\beta-2}v+\frac{1}{2}Q_{v}(u,v)
\ \ in \
\Omega, \\[2mm]
\ \ u, \ v>0
\ \ \ \
\ in \ \
\Omega, \\[2mm]
\ u=v=0 \ \
\ \ in \ \
\mathbb{R}^{N}\backslash\Omega,
\end{cases} \end{equation*} where
$(-\Delta)^{s}$ denotes the fractional Laplace
operator,
$\Omega\subset\mathbb{R}^{N}$
is a smooth bounded domain such that
$0\in\Omega$,
$\mu_{1}, \mu_{2}\in
[0,\Lambda_{N,s})$,
$\Lambda_{N,s}=2^{2s}\frac{\Gamma^{2}(\frac{N+2s}{4})}{\Gamma^{2}(\frac{N-2s}{4})}$
is the best constant of the fractional Hardy inequality and
$2^{*}_{s}=\frac{2N}{N-2s}$ is the
fractional critical Sobolev exponent. In order to prove the main result,
we establish some refined estimates on the extremal functions of the
fractional Hardy-Sobolev type inequalities and we get the existence of
positive solutions to the systems through variational methods.
Non-variational elliptic systems in dimension two: a priori bounds and existence of positive solutions