Normalized Solutions for Schrödinger–Poisson Type Systems with Critical Nonlocal Term

This paper focuses on the existence of normalized ground state solution for Schrödinger–Poisson system with doubly critical growth \(\begin{aligned}{\left\{ \begin{array}{ll}\displaystyle -\Delta u-\phi |u|^3u=\lambda u+f\left( u\right) +|u|^4u, & x \in {\mathbb {R}}^{3},\\ -\Delta \phi =|u|^5, & x \in {\mathbb {R}}^{3},~~\end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle -\mathrm{\Delta }u-{\varphi |u|}^{3}u=\lambda u+f\left(u\right)+{|u|}^{4}u,}\hfill & x\in {\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }\varphi ={|u|}^5,\hfill & x\in {\mathbb{R}}^3,\hfill \end{array}\right)\end{array}$$ having prescribed mass \(\begin{aligned} \int _{{\mathbb {R}}^3}|u|^2dx=m^2,\end{aligned}\) $$\begin{array}{c}\hfill {\int }_{{\mathbb{R}}^3}{|u|}^{2}dx=m^2,\end{array}$$ where \(m>0\) $m>0$ is a constant, \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ is unknown and appears as a Lagrange multiplier, f is a nonlinear term of Sobolev subcritical and is mass supercritical. We show that there exists a normalized ground state solution for the above system when the mass m is large enough. In addition, we also discuss the continuity and monotonicity of the ground state energy \(E_m\) $E_m$ , and explain the asymptotic property of \(E_m\) $E_m$ when m is large enough. Our studies complement and extend the researches of Meng and He (Methods Nonlinear Anal 62:509–534, 2023), in the sense we deal with the effect of mass-changing on the existence of normalized solutions.