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|^2_s^ * - 3u = λ u + μ |u|^q - 2u + |u|^2_s^ * - 2u, x ∈ℝ^3,( - Δ )^sϕ = |u|^2_s^ * - 1, x ∈ℝ^3,. with prescribed mass ∫_ℝ^3|u|^2dx = a^2 , where a > 0 is a prescribed number, μ > 0 is a paremeter, s ∈ (0, 1), 2 < q < 2* s , and 2_s^ * = 6 3 - 2s is the fractional critical Sobolev exponent. In the L 2 -subcritical case, we show the existence of multiple normalized solutions by using the genus theory and the truncation technique; in the L 2 -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.