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

Abstract

In this paper, we study the fractional critical Schrödinger–Poisson system $$\begin{aligned}{\left\{ \begin{array}{ll} (-\Delta )^su +\lambda \phi u= \alpha u+\mu |u|^{q-2}u+|u|^{2^*_s-2}u,&{}~~ \hbox {in}~{\mathbb {R}}^3,\\ (-\Delta )^t\phi =u^2,&{}~~ \hbox {in}~{\mathbb {R}}^3,\end{array}\right. } \end{aligned}$$ ( - Δ ) s u + λ ϕ u = α u + μ | u | q - 2 u + | u | 2 s ∗ - 2 u , in R 3 , ( - Δ ) t ϕ = u 2 , in R 3 , having prescribed mass $$\begin{aligned} \int _{{\mathbb {R}}^3} |u|^2dx=a^2,\end{aligned}$$ ∫ R 3 | u | 2 d x = a 2 , where $$ s, t \in (0, 1)$$ s , t ∈ ( 0 , 1 ) satisfy $$2\,s+2t> 3, q\in (2,2^*_s), a>0$$ 2 s + 2 t > 3 , q ∈ ( 2 , 2 s ∗ ) , a > 0 and $$\lambda ,\mu >0$$ λ , μ > 0 parameters and $$\alpha \in {\mathbb {R}}$$ α ∈ R is an undetermined parameter. For this problem, under the $$L^2$$ L 2 -subcritical perturbation $$\mu |u|^{q-2}u, q\in (2,2+\frac{4\,s}{3})$$ μ | u | q - 2 u , q ∈ ( 2 , 2 + 4 s 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 $$\mu |u|^{q-2}u,q\in (2+\frac{4\,s}{3}, 2^*_s)$$ μ | u | q - 2 u , q ∈ ( 2 + 4 s 3 , 2 s ∗ ) , we prove two different results of normalized solutions when parameters $$\lambda ,\mu $$ λ , μ 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.