Existence and multiplicity of solutions for Hardy nonlocal fractional elliptic equations involving critical nonlinearities

In this paper, systems of fractional Laplacian equations are investigated, which involve critical homogeneous nonlinearities and Hardy-type \textcolor[rgb]{1.00,0.00,0.00}{terms} as follows \begin{eqnarray*} \begin{cases} &(- \Delta)^s u - \gamma \frac{u}{|x|^{2 s}} = \frac{\lambda \mu}{\mu + \eta } \frac{|u|^{\mu - 2} |v|^{\eta} u}{|x|^{\alpha}}+ \frac{Q(x)}{2^*_s (\alpha)}\frac{ H_u(u,v)}{|x|^{\alpha}}, ~ in~ \mathbb{R}^N,\\ \\ & (- \Delta)^s v - \gamma \frac{v}{|x|^{2 s}} = \frac{\lambda \eta}{\mu + \eta } \frac{|u|^{\mu} |v|^{\eta - 2} v}{|x|^{\alpha}} + \frac{Q(x)}{2^*_s (\alpha)}\frac{ H_v(u,v)}{|x|^{\alpha}}, ~ in ~\mathbb{R}^N, \end{cases} \end{eqnarray*} where $0 < s < 1$, $0< \alpha < 2s <N$, $0 < \gamma < \gamma_H$ with \[ \gamma_H =4^s \frac{\Gamma^2(\frac{N+2s}{4})}{\Gamma^2(\frac{N-2s}{4})}, \;\;\; \textrm{for} \; i = 1,\ldots,l \] being the fractional best Hardy constant on $\mathbb{R}^N$, $2^*_s (\alpha) = \frac{2 (N -\alpha)}{N - 2 s}$ is critical Hardy-Sobolev exponent, $\mu, \eta > 1$ with $\mu + \eta = 2^*_s(\alpha)$, $Q$ is $G$-symmetric functions ($\textcolor[rgb]{1.00,0.00,0.00}{G}$ is a closed subgroup of $O(N)$, see Section 2 for details) satisfying some appropriate conditions which will be specified later, $\lambda$ is real parameter, $H_u$, $H_v$ are the partial derivative of the 2-variable $C^1$-functions $H (u,v)$ and $(-\Delta)^s$ is the fractional Laplacian operator which (up to normalization factors) may be defined as \begin{equation*} (- \Delta)^s u (x) = - \frac12\int_ {\mathbb{R}^{N} } \frac{u (x + y) + u (x - y) - 2 u (x) }{|x-y|^{N + 2 s}} dy. \end{equation*} By variational methods and local compactness of Palais-Smale sequences, the extremals of the corresponding best Hardy-Sobolev constant are found and the existence of solutions to the system is established.\