This paper deals with the existence and nonexistence of normalized solutions for a type of fractional Schrödinger–Choquard system with critical nonlinearity \(\left\{ \begin{aligned} (-\Delta )^s u&= \lambda _1 u + \mu _1|u|^{p-2}u+(I_\alpha *|u|^{2_{\alpha ,s}^{*}})|u|^{2_{\alpha ,s}^{*}-2}u+\beta r_{1}|u|^{r_1-2}u|v|^{r_2}, \\ (-\Delta )^s v&= \lambda _2 v + \mu _2|v|^{q-2}v+(I_\alpha *|v|^{2_{\alpha ,s}^{*}})|v|^{2_{\alpha ,s}^{*}-2}v+\beta r_{2}|u|^{r_1}|v|^{r_2 -2}v \end{aligned} \right. \) with the restrictions \(\int _{\mathbb {R}^N}|u|^2\textrm{d}x=a\) and \(\int _{\mathbb {R}^N}|v|^2\textrm{d}x=b\) , where \(a,b>0\) are prescribed, \(\frac{1}{2}\le s<1\) , \(2\le N\le 4s\) , \(\alpha \in (0,N)\) , \(I_\alpha (x):=\frac{\Gamma (\frac{\alpha }{2})}{\Gamma (\frac{N-\alpha }{2})\pi ^{N/2}2^{N-\alpha }|x|^{\alpha }}\) , \(x\in \mathbb {R}^N\setminus \{0\}\) is the Riesz potential, \(\mu _1\) , \(\mu _2\) , \(\beta >0\) , \(r_1,r_2>1\) and \(2_{\alpha ,s}^{*}:=\frac{2N-\alpha }{N-2s}\) . The frequencies \(\lambda _{1}\) and \(\lambda _{2}\) appear as Lagrange multipliers. \((-\Delta )^s\) is the fractional Laplace operator. In the literature, any (u, v) solving the above system (for some \(\lambda _1\) , \(\lambda _2\) ) is called a normalized solution. In the case where \(2+\frac{4s}{N}<p,q,r_1+r_2<2_s^*:=\frac{2N}{N-2s}\) and sufficiently large \(\beta >0\) , we prove the existence of a positive normalized solution. For the triple Sobolev-critical growth case \(p=q=r_1+r_2=2_s^*\) , we obtain the nonexistence of a positive normalized solution.
