Multiple positive solutions for a bi-nonlocal Kirchhoff-Schr $\ddot{\mathrm{o}}$dinger-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:

$$\{\begin{array}{ll}-{\left({\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^{2}dx\right)}^r\mathrm{\Delta }u+\varphi u=u^5+\lambda {\left({\int }_{\mathrm{\Omega }}F(x,u)dx\right)}^{s}f(x,u),& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ -\mathrm{\Delta }\varphi =u^2,u>0,& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=\varphi =0,& \mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega },\end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^3$ is a smooth bounded domain, $\lambda >0$, $0\le r<1$, $0<s<\frac{1-r}{3(r+1)}$ 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.