On the mass concentration of \(L^2\)-constrained minimizers for a class of Schrödinger–Poisson equations

In this paper, we study the mass concentration behavior of positive solutions with prescribed \(L^2\)-norm for a class of Schrodinger–Poisson equations in \({\mathbb R}^3\) $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-\mu u+\phi _uu-|u|^{p-2}u=0,\,\,\, &{} x\in {\mathbb R}^3,~\mu \in {\mathbb R},\\ -\Delta \phi _u=|u|^2,\,\,\, &{} \end{array} \right. \end{aligned}$$ where \(p\in (2,6)\). We show that positive solutions with prescribed \(L^2\)-norm as which tends to 0 (in some cases) or to \(+\,\infty \) (in others), behave like the positive solution of Schrodinger equation \(-\Delta u+u=|u|^{p-2}u\) in \({\mathbb R}^3\).