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

Abstract

In this paper, we study the fractional critical Schrödinger–Poisson system

$$\begin{aligned}{\left\{ \begin{array}{ll} (-\Delta )^su +\lambda \phi u= \alpha u+\mu |u|^{q-2}u+|u|^{2^*_s-2}u,&{}~~ \hbox {in}~{\mathbb {R}}^3,\\ (-\Delta )^t\phi =u^2,&{}~~ \hbox {in}~{\mathbb {R}}^3,\end{array}\right. } \end{aligned}$$

having prescribed mass

$$\begin{aligned} \int _{{\mathbb {R}}^3} |u|^2dx=a^2,\end{aligned}$$

where \( s, t \in (0, 1)\) satisfy \(2\,s+2t> 3, q\in (2,2^*_s), a>0\) and \(\lambda ,\mu >0\) parameters and \(\alpha \in {\mathbb {R}}\) is an undetermined parameter. For this problem, under the \(L^2\)-subcritical perturbation \(\mu |u|^{q-2}u, q\in (2,2+\frac{4\,s}{3})\), we derive the existence of multiple normalized solutions by means of the truncation technique, concentration-compactness principle and the genus theory. In the \(L^2\)-supercritical perturbation \(\mu |u|^{q-2}u,q\in (2+\frac{4\,s}{3}, 2^*_s)\), we prove two different results of normalized solutions when parameters \(\lambda ,\mu \) 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.