Solutions for Schrödinger-Poisson system involving nonlocal term and critical exponent

In this paper, we consider the following Kirchhoff-Schrödinger-Poisson system:

$$\left\{ {\matrix{{ - (a + b\int_{{\mathbb{R}^3}} {|\nabla u{|^2})\Delta u + u + \phi u = \mu Q(x)|u{|^{q - 2}}u + K(x)|u{|^4}u,} } \hfill & {{\rm{in}}\,\,\,{\mathbb{R}^3},} \hfill \cr { - \Delta \phi = {u^2},} \hfill & {{\rm{in}}\,\,\,{\mathbb{R}^3},} \hfill \cr } } \right.$$

the nonlinear growth of ∣u∣⁴ u reaches the Sobolev critical exponent. By combining the variational method with the concentration-compactness principle of Lions, we establish the existence of a positive solution and a positive radial solution to this problem under some suitable conditions. The nonlinear term includes the nonlinearity f(u) ∼∣u∣^q−2u for the well-studied case q ∈ [4, 6), and the less-studied case q ∈ (2, 3), we adopt two different strategies to handle these cases. Our result improves and extends some related works in the literature.