Normalized solutions for the critical Schrödinger-Poisson systems with potentials

In this paper, we look for normalized solutions to the following nonlinear Schrödinger-Poisson system with doubly critical growth and non-constant potential \[ \begin{cases} \displaystyle -\Delta u+V(x)u+\lambda u-\phi |u|^3u =\mu|u|^{q-2}u+|u|^4u, & x \in \mathbb R ^{3},\\ -\Delta \phi=|u|^5, & x \in \mathbb R ^{3},\end{cases} \] having prescribed mass \[ \int_{\mathbb R ^3}|u|^2dx=a^2, \] where $u\in H^1(\mathbb R ^3)$, $\mu, a > 0, 2 < q < 6 $, and $\lambda \in \mathbb R $ appears as a Lagrange multiplier. $V$ is an external potential vanishing at infinity. In order to get the existence of normalized solutions, we propose some mild assumptions on $V$, combining the Pohozaev manifold, constrained minimization arguments and some analytical skills. The subcritical perturbation $\mu|u|^{q-2}u$ may be $L^2$-subcritical growth: $q\in(2,\frac{10}{3})$; $L^2$-critical growth: $q=\frac{10}{3}$ and may be $L^2$-supercritical growth: $q\in(\frac{10}{3},6)$, respectively. Simultaneously, we show the exponential decay property of the ground state solution through the Moser iteration.