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

In this paper, we study the fractional critical Schrödinger-Poisson system (-Δ)^su +λϕ u= α u+μ|u|^q-2u+|u|^2^*_s-2u,    ℝ^3, (-Δ)^tϕ=u^2,    ℝ^3, having prescribed mass ∫_ℝ^3 |u|^2dx=a^2, where s, t ∈ (0, 1) satisfies 2s+2t > 3, q∈(2,2^*_s), a>0 and λ,μ>0 parameters and α∈ℝ is an undetermined parameter. Under the L^2-subcritical perturbation q∈ (2, 2+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∈ (2+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.