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

Abstract

In this paper, we study the fractional critical Schrödinger-Poisson system \[\begin{cases} (-Δ)^su +λϕu= αu+μ|u|^{q-2}u+|u|^{2^*_s-2}u,&~~ \mbox{in}~{\mathbb R}^3,\\ (-Δ)^tϕ=u^2,&~~ \mbox{in}~{\mathbb R}^3,\end{cases} \] having prescribed mass \[\int_{\mathbb R^3} |u|^2dx=a^2,\] where $s,t\in (0,1)$ satisfies $2s+2t>3,q\in (2,2_s^{\ast }),a>0$ and $\lambda ,\mu >0$ parameters and $\alpha \in \mathbb{R}$ is an undetermined parameter. Under the $L^2$-subcritical perturbation $q\in (2,2+\frac{4s}{3})$, 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 $q\in (2+\frac{4s}{3},2_s^{\ast })$, by applying the constrain variational methods and the mountain pass theorem, we show the existence of positive normalized ground state solutions.

43 pages