In this paper, we study the fractional critical Schr\"{ o}dinger-Poisson system \[\begin{ cases} (-\Delta)^su +\lambda\phi u= \alpha u+\mu|u|^{ q-2}u+|u|^{ 2^*_s-2}u,&~~ \mbox{ in}~{ \mathbb R}^3,\\ (-\Delta)^t\phi=u^2,&~~ \mbox{ in}~{ \mathbb R}^3,\end{ cases} \] having prescribed mass \[\int_{ \mathbb R^3} |u|^2dx=a^2,\] where $ s, t \in (0, 1)$ satisfies $2s+2t > 3, q\in(2,2^*_s), a>0$ 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 $q\in (2+\frac{ 4s}{ 3}, 2^*_s)$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.
Categories: math.AP