Multiplicity of normalized solutions for the fractional Schrödinger-Poisson system with doubly critical growth

In this paper, we are concerned with solutions to the fractional Schrödinger-Poisson system {(−Δ)su−ϕ|u|2s∗−3u=λu+μ|u|q−2u+|u|2s∗−2u,x∈ℝ3,(−Δ)sϕ=|u|2s∗−1,x∈ℝ3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{{{( - \Delta )}^s}u - \phi |u{|^{2_s^ * - 3}}u = \lambda u + \mu |u{|^{q - 2}}u + |u{|^{2_s^ * - 2}}u,} \hfill & {x \in {\mathbb{R}^3},} \hfill \cr {{{( - \Delta )}^s}\phi = |u{|^{2_s^ * - 1}},} \hfill & {x \in {\mathbb{R}^3},} \hfill \cr } } \right.$$\end{document} with prescribed mass ∫ℝ3|u|2dx=a2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int_{{\mathbb{R}^3}} {|u{|^2}{\rm{d}}x = {a^2}} $$\end{document}, where a > 0 is a prescribed number, μ > 0 is a paremeter, s ∈ (0, 1), 2 < q < 2*s, and 2s∗=63−2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2_s^ * = {6 \over {3 - 2s}}$$\end{document} is the fractional critical Sobolev exponent. In the L2-subcritical case, we show the existence of multiple normalized solutions by using the genus theory and the truncation technique; in the L2-supercritical case, we obtain a couple of normalized solutions by developing a fiber map. Under both cases, to recover the loss of compactness of the energy functional caused by the doubly critical growth, we need to adopt the concentration-compactness principle. Our results complement and improve upon some existing studies on the fractional Schrödinger-Poisson system with a nonlocal critical term.