On Schrödinger-Poisson equations with a critical nonlocal term

In this paper, we study the following non-autonomous Schrödinger-Poisson equation with a critical nonlocal term and a critical nonlinearity:

$$\{\begin{array}{rl}& -\mathrm{\Delta }u+V(x)u+\lambda \varphi |u{|}^{3}u=f(u)+(u^{+}{)}^5,\mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ & -\mathrm{\Delta }\varphi =|u{|}^5,\mathrm{i}\mathrm{n}{\mathbb{R}}^3.\end{array}$$

First, we consider the case that the nonlinearity satisfies the Berestycki-Lions type condition with critical growth. Second, we consider the case that $\mathrm{i}\mathrm{n}\mathrm{t}{V}^{-1}(0)$ is contained in a spherical shell. By using variational methods, we obtain the existence and asymptotic behavior of positive solutions.