Normalized ground states for the fractional Schrödinger–Poisson system with critical nonlinearities

Abstract

In this paper we study the existence and properties of ground states for the fractional Schrödinger–Poisson system with combined power nonlinearities

$$\begin{aligned}{\left\{ \begin{array}{ll}\displaystyle (-\Delta )^su-\phi |u|^{2^*_s-3}u=\lambda u+\mu |u|^{q-2}u+|u|^{2^*_s-2}u, &{}x \in {\mathbb {R}}^{3},\\ (-\Delta )^{s}\phi =|u|^{2^*_s-1}, &{}x \in {\mathbb {R}}^{3},\end{array}\right. } \end{aligned}$$

having prescribed mass

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

and doubly critical growth, where $s\in (0,1)$, $\mu >0$ is a parameter, $2<q<2^*_s$, $2^*_s:=\frac{6}{3-2s}$ is the fractional critical Sobolev exponent and $\lambda \in {\mathbb {R}}$ appears as a Lagrange multiplier. For a $L^2$-subcritical, $L^2$-critical and $L^2$-supercritical perturbation $\mu |u|^{q-2}u$, respectively, we prove several existence, and non-existence results. Furthermore, the qualitative behavior of the ground states as $\mu \rightarrow 0^+$ is also studied. Our results complement and improve the existing ones in several directions, and this study seems to be the first contribution regarding existence of normalized ground states for the fractional Sobolev critical Schrödinger–Poisson system with a critical nonlocal term.