Critical Schrödinger–Bopp–Podolsky System with Prescribed Mass

In this paper, we consider the critical Schrödinger–Bopp–Podolsky system with prescribed mass as follows: -Δu+ϕu=λu+μ|u|p-2u+u5inR3,-Δϕ+Δ2ϕ=4πu2inR3,∫R3u2dx=m2,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{l} -\Delta u+ \phi u=\lambda u+\mu |u|^{p-2}u+u^{5}\ \ \text{ in }\ {\mathbb R}^3,\\ -\Delta \phi +\Delta ^2\phi =4\pi u^2\ \ \text{ in }\ {\mathbb R}^3,\\ \int _{{\mathbb R}^3}u^2\textrm{d}x=m^2, \end{array}\right. \end{aligned}$$\end{document}where λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in {\mathbb R}$$\end{document}, m>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>0$$\end{document}, μ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu >0$$\end{document} is a parameter, 2<p<6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2<p<6$$\end{document}. For p∈(10/3,6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (10/3, 6)$$\end{document}, applying Lagrange multipliers argument and mountain pass theorem, we obtain the existence of positive normalized ground state solutions for above system, and then asymptotic behavior of the solution is also detected. For p∈(2,10/3]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (2, 10/3]$$\end{document}, we obtain the existence of a normalized ground state solution for above system by combining mountain pass theorem with Lebesgue dominated convergence theorem. Finally we prove the existence of infinitely many normalized solutions for above system by the symmetric mountain pass theorem.