Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities

In this paper, we investigate the following fractional Sobolev critical nonlinear Schrödinger (NLS) coupled systems: \begin{equation*} \left{\begin{array}{lll} (-Δ)^{s} u=μ_{1} u+|u|^{2^{}{s}-2}u+η{1}|u|^{p-2}u+γα|u|^{α-2}u|v|^β ~ $\text{in}$~ $\mathbb{R}$^{N},\ (-Δ)^{s} v=μ_{2} v+|v|^{2^{}{s}-2}v+η{2}|v|^{q-2}v+γβ|u|^α|v|^{β-2}v ~~$\text{in}$~ $\mathbb{R}$^{N},\ |u|^{2}{L^{2}}=m{1}^{2} ~$\text{and}$~ |v|^{2}{L^{2}}=m{2}^{2}, \end{array}\right. \end{equation*} where $(-\mathrm{\Delta }{)}^s$ is the fractional Laplacian, $N=3,4$, $s\in (0,1)$, ${\mu }_1,{\mu }_2\in \mathbb{R}$ are unknown constants, which will appear as Lagrange multipliers, $2_s^{\ast }$ is the fractional Sobolev critical index, ${\eta }_1,{\eta }_2,\gamma ,m_1,m_2>0$, $\alpha >1,\beta >1$, $p,q,\alpha +\beta \in (2+4s\mathrm{/}N,2_s^{\ast }]$. Firstly, if $p,q,\alpha +\beta <2_s^{\ast }$, we obtain the existence of positive normalized solution when $\gamma$ is big enough. Secondly, if $p=q=\alpha +\beta =2_s^{\ast }$, we show that nonexistence of positive normalized solution. The main ideas and methods of this paper are scaling transformation, classification discussion and concentration-compactness principle.