In this paper, we investigate the existence and nonexistence of nontrivial solutions for the following Schrödinger-Poisson type problem with a critical nonlocal term: \( \left\{ \begin{array}{ll} -\triangle u+ u+\lambda \phi |u|^3u=|u|^{p-2}u, \ & \hbox {in}\ \mathbb {R}^3,\\ -\triangle \phi =|u|^5, \ & \hbox {in}\ \mathbb {R}^3, \end{array} \right. \) where \(\lambda >0\) and \(p\in (2,6)\) . By constructing a function \(\hat{w} \in H_{\textrm{rad}}^1(\mathbb {R}^3)\) with negative energy, we obtain a Palais-Smale sequence in \(H_{\textrm{rad}}^1(\mathbb {R}^3)\) by applying the Ekeland variational principle and the Mountain-pass theorem, respectively. We employ refined analytical techniques to prove that the above system admits two nontrivial solutions when \(\lambda \in (0, \lambda _1)\) and has no nontrivial solutions when \(\lambda \ge \lambda _2\) , where \(\lambda _1\) and \(\lambda _2\) are explicitly defined in terms of p. Our results improve upon those obtained in [Y. Li, F. Li, J. Shi, Calc. Var. Partial Differential Equations, 56(2017), 134, 17].
