Multiple positive solutions for a bi-nonlocal Kirchhoff-Schrödinger-Poisson system with critical growth

In this article, we study the following bi-nonlocal Kirchhoff-Schr$ \ddot{\mathrm{o}} $dinger-Poisson system with critical growth: $ \begin{equation*} \begin{cases} -\left( \int_{\Omega}|\nabla u|^2dx\right)^r\Delta u+\phi u = u^5+\lambda\left( \int_{\Omega}F(x, u)dx\right)^sf(x, u), & \mathrm{in}\ \ \Omega, \\ -\Delta\phi = u^2, u>0, & \mathrm{in}\ \ \Omega, \\ u = \phi = 0, & \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} $ where $ \Omega\subset \mathbb{R}^3 $ is a smooth bounded domain, $ \lambda > 0 $, $ 0\leq r < 1 $, $ 0 < s < \frac{1-r}{3(r+1)} $ and $ f(x, u) $ satisfies some suitable assumptions. By using the concentration compactness principle, the multiplicity of positive solutions for the above system is established.