In this paper, we study the following double critical Schrödinger-Poisson system involving the fractional p-Laplacian in \(\mathbb {R}^3\) of the form: \(\begin{aligned} {\left\{ \begin{array}{ll}(-\Delta )^{s}_{p} u-\phi |u|^{p_s^\sharp -2}u=\lambda |u|^{p-2}u+\mu |u|^{q-2}u+|u|^{p_s^*-2}u & \text{ in } \mathbb {R}^3, \\ (-\Delta )^s\phi =|u|^{p_s^\sharp }& \text{ in } \mathbb {R}^3, \end{array}\right. } \end{aligned}\) and prescribed mass \(\int _{\mathbb {R}^3} |u|^{p}dx=a^p,\) where \((-\Delta )^{s}_{p}\) is the fractional p-Laplace operator, \(s\in (0,1)\) , \(sp<3\) , \(\mu , a>0,\) \(\lambda \in \mathbb {R}\) , \(q\in (p, p_s^*)\) and \( p_s^*:= \frac{{3p}}{{3-sp}}\) , \(p_s^\sharp :=\frac{p(3+2s)}{2(3-sp)}\) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. For the \(L^p\) -subcritical case, we show the existence of multiple normalized solutions by using the truncation technique and the genus theory. For the \(L^p\) -supercritical case, we obtain a couple of normalized solutions by using the auxiliary functional. For both cases, in order to overcome the loss of compactness of the energy functional due to the double critical growth, the concentration-compactness principle is needed to overcome this difficulty. In a sense, we generalize some of the previous results [2, 24, 33, 54]. As far as we know, this study seems to be the first contribution regarding existence of normalized solutions for double critical Schrödinger-Poisson system involving the fractional p-Laplacian.
