Ground State Solutions of Nehari-Pohozaev Type for Schrödinger–Poisson–Slater Equation with Zero Mass and Critical Growth

In this article, we study the Schrödinger–Poisson–Slater type equation with the critical growth and zero mass: \(\begin{aligned} {\left\{ \begin{array}{ll} {-\Delta } u+\phi u=\mu |u|^{p-2}u+u^5, \ \ \ \ x\in {\mathbb {R}}^{3},\\ {-\Delta } \phi =u^2, \ \ \ \ x\in {\mathbb {R}}^{3}, \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }u+\varphi u={\mu |u|}^{p-2}u+u^5,x\in {\mathbb{R}}^3,\hfill \\ -\mathrm{\Delta }\varphi =u^2,x\in {\mathbb{R}}^3,\hfill \end{array}\right)\end{array}$$ where \(3<p<6\) $3<p<6$ and \(\mu >0\) $\mu >0$ . By combining a new perturbation method and the mountain pass theorem, Liu et al. [J. Diff. Eq., 266 (2019), 5912–5941] prove that the above equation has at least one positive ground state solution for \(p \in (4, 6)\) $p\in (4,6)$ and \(\mu >0\) $\mu >0$ or \(p \in (3, 4]\) $p\in (3,4]$ if \(\mu \) $\mu$ is sufficiently large. By using a much simpler method than the ones used in the above mentioned paper, together with subtle estimates and analyses, we obtain better results on the existence for a ground state solution of Nehari-Pohozaev type.