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

In this paper, we study the fractional critical Schrödinger–Poisson system \(\begin{aligned}{\left\{ \begin{array}{ll} (-\Delta )^su +\lambda \phi u= \alpha u+\mu |u|^{q-2}u+|u|^{2^*_s-2}u,&{}~~ \hbox {in}~{\mathbb {R}}^3,\\ (-\Delta )^t\phi =u^2,&{}~~ \hbox {in}~{\mathbb {R}}^3,\end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-\mathrm{\Delta })}^{s}u+\lambda \varphi u=\alpha u+{\mu |u|}^{q-2}u+{|u|}^{2_s^{\ast }-2}u,\hfill & \text{in}{\mathbb{R}}^3,\hfill \\ {(-\mathrm{\Delta })}^t\varphi =u^2,\hfill & \text{in}{\mathbb{R}}^3,\hfill \end{array}\right)\end{array}$$ having prescribed mass \(\begin{aligned} \int _{{\mathbb {R}}^3} |u|^2dx=a^2,\end{aligned}\) $$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}{|u|}^{2}dx=a^2,\end{array}$$ where \( s, t \in (0, 1)\) $s,t\in (0,1)$ satisfy \(2\,s+2t> 3, q\in (2,2^*_s), a>0\) $2s+2t>3,q\in (2,2_s^{\ast }),a>0$ and \(\lambda ,\mu >0\) $\lambda ,\mu >0$ parameters and \(\alpha \in {\mathbb {R}}\) $\alpha \in \mathbb{R}$ is an undetermined parameter. For this problem, under the \(L^2\) $L^2$ -subcritical perturbation \(\mu |u|^{q-2}u, q\in (2,2+\frac{4\,s}{3})\) ${\mu |u|}^{q-2}u,q\in (2,2+\frac{4s}{3})$ , we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. In the \(L^2\) $L^2$ -supercritical perturbation \(\mu |u|^{q-2}u,q\in (2+\frac{4\,s}{3}, 2^*_s)\) ${\mu |u|}^{q-2}u,q\in (2+\frac{4s}{3},2_s^{\ast })$ , we prove two different results of normalized solutions when parameters \(\lambda ,\mu \) $\lambda ,\mu$ satisfy different assumptions, by applying the constrained variational methods and the mountain pass theorem. Our results extend and improve some previous ones of Zhang et al. (Adv Nonlinear Stud 16:15–30, 2016); and of Teng (J Differ Equ 261:3061–3106, 2016), since we are concerned with normalized solutions.