Normalized solutions for critical Schrödinger–Poisson system involving p-Laplacian in \({\mathbb {R}}^3\)

In this paper, we study the following critical Schrödinger–Poisson system involving p-Laplacian in \({\mathbb {R}}^3\) ${\mathbb{R}}^3$ of the form: \(\begin{aligned} {\left\{ \begin{array}{ll}-\Delta _{p} u-\phi |u|^{p-2}u=\lambda |u|^{p-2}u+\mu |u|^{q-2}u+|u|^{p^*-2}u & \text{ in } {\mathbb {R}}^3, \\ -\Delta \phi =|u|^{p}& \text{ in } {\mathbb {R}}^3, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-{\mathrm{\Delta }}_{p}u-{\varphi |u|}^{p-2}u={\lambda |u|}^{p-2}u+{\mu |u|}^{q-2}u+{|u|}^{p^{\ast }-2}u\hfill & \text{in}{\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }\varphi ={|u|}^p\hfill & \text{in}{\mathbb{R}}^3,\hfill \end{array}\right)\end{array}$$ and prescribed mass \(\begin{aligned} \int \limits _{{\mathbb {R}}^3} |u|^{p}{\textrm{d}}x=a^p, \end{aligned}\) $$\begin{array}{c}\hfill \underset{{\mathbb{R}}^3}{\int }{|u|}^p\text{d}x=a^p,\end{array}$$ where \(\Delta _pu =\text {div} (|\nabla u|^{p-2}\nabla u)\) ${\mathrm{\Delta }}_{p}u={\text{div}(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u)$ is the p-Laplacian operator, \(p\in (\frac{1}{2},3)\) $p\in (\frac{1}{2},3)$ , \(\mu , a>0,\) $\mu ,a>0,$ \(\lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ , \(q\in (p, p^*)\) $q\in (p,p^{\ast })$ and \( p^*:= \frac{{3p}}{{3-p}}\) $p^{\ast }:=\frac{3p}{3-p}$ . For the \(L^p\) $L^p$ -subcritical case, we show the existence of multiple normalized solutions by using the truncation technique and the genus theory. For the \(L^p\) $L^p$ -supercritical case, we obtain a couple of normalized solutions by developing the auxiliary functional. For both cases, in order to overcome the loss of compactness of the energy functional due to the critical growth, the concentration–compactness principle is needed to overcome this difficulty. In a sense, we generalize some of the previous results.