In this paper we study the following p-Laplacian Schrödinger-Poisson system with Sobolev critical exponent and mixed nonlinearities \( {\left\{ \begin{array}{ll} -\Delta _{p} u + \lambda u^{p-1} + \kappa \bigl (|x|^{-1}*|u|^{p}\bigr )u^{p-1} = |u|^{p^{*}-2}u + \mu |u|^{q-2}u, & x\in \mathbb {R}^{3},\\ \displaystyle \int _{\mathbb {R}^{3}}|u|^{p}\,dx = a^{p}, \end{array}\right. } \) where \(a>0\) is prescribed, \(\kappa >0\) , \(\mu \in \mathbb {R}\) , \(p<q<p^{*}=\frac{3p}{3-p}\) is the Sobolev critical exponent, and \(\lambda \in \mathbb {R}\) appears as a Lagrange multiplier. We prove the existence of two normalized solutions, including a ground state, under the conditions \(\mu > 0\) and \(q \in \bigl (p, \frac{4p}{3}\bigr )\) . It is worth emphasizing that unlike the case \(\kappa =0\) , the Schwarz spherical rearrangement method fails when \(\kappa >0\) , and the derivation of a strict subadditivity inequality for the minimal energy becomes delicate in order to exclude dichotomy of minimizing sequences. As far as we know, the existence of a second solution for the above problem has not been studied before. We also prove that no positive normalized solution exists when \(\mu \le 0\) and \(q \in (p, p^{*})\) , which extends earlier related results in the literature [35–37].
