A system of equations involving the fractional $p$-Laplacian and doubly critical nonlinearities

This paper deals with existence of solutions to the following fractional $p$-Laplacian system of equations \begin{equation*} %\tag{$\mathcal{P}$}\label{MAT1} \begin{cases} (-Δ_p)^s u =|u|^{p^_s-2}u+ \frac{γα}{p_s^}|u|^{α-2}u|v|^β;;$\text{in}$;Ω, (-Δ_p)^s v =|v|^{p^_s-2}v+ \frac{γβ}{p_s^}|v|^{β-2}v|u|^α;;$\text{in}$;Ω, % % u,;v\in\wsp, \end{cases} \end{equation*} where $s\in (0,1)$, $p\in (1,\mathrm{\infty })$ with $N>sp$, $\alpha ,\beta >1$ such that $\alpha +\beta =p_s^{\ast }:=\frac{Np}{N-sp}$ and $\mathrm{\Omega }={\mathbb{R}}^N$ or smooth bounded domains in ${\mathbb{R}}^N$. For $\mathrm{\Omega }={\mathbb{R}}^N$ and $\gamma =1$, we show that any ground state solution of the above system has the form $(\lambda U,\tau \lambda V)$ for certain $\tau >0$ and $U,V$ are two positive ground state solutions of $(-{\mathrm{\Delta }}_p{)}^{s}u=\mathrm{\mid }u{\mathrm{\mid }}^{p_s^{\ast }-2}u$ in ${\mathbb{R}}^N$. For all $\gamma >0$, we establish existence of a positive radial solution to the above system in balls. For $\mathrm{\Omega }={\mathbb{R}}^N$, we also establish existence of positive radial solutions to the above system in various ranges of $\gamma$.