Normalized solutions for a fractional Schrödinger-Poisson system with critical growth

In this paper, we study the fractional critical Schrödinger-Poisson system $$\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u+\lambda \phi u=\alpha u+\mu \mathrm{\mid }u{\mathrm{\mid }}^{q-2}u+\mathrm{\mid }u{\mathrm{\mid }}^{2_s^{\ast }-2}u,& in{\mathbb{R}}^3,\\ (-\mathrm{\Delta }{)}^t\phi =u^2,& in{\mathbb{R}}^3,\end{array}$$ having prescribed mass $${\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=a^2,$$ where $s,t\in (0,1)$ satisfies $2s+2t>3,q\in (2,2_s^{\ast }),a>0$ and $\lambda ,\mu >0$ parameters and $\alpha \in \mathbb{R}$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2,2+\frac{4s}{3})$, we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. For the $L^2$-supercritical perturbation $q\in (2+\frac{4s}{3},2_s^{\ast })$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.