Multiplicity of solutions for Schrödinger-Poisson system with critical exponent in $\mathbb{R}^{3}$

In this paper, we study the following Schrodinger-Poisson system with critical exponent $ \begin{equation*} \begin{cases} -\Delta u-k(x)\phi u=\lambda h(x)|u|^{p-2}u+s(x)|u|^{4}u, \ ~~x\in\mathbb{R}^{3},\\ -\triangle\phi=k(x)u^{2}, \ \ \ \ \ \ \ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ x\in\mathbb{R}^{3}, \\ \end{cases} \end{equation*} $ where $1 0.$ Under suitable conditions on $k$, $h$ and $s$, we show that there exists $\lambda^{\ast}>0$ such that the above problem possesses infinitely many solutions with negative energy for each $\lambda\in(0, \lambda^{\ast})$. Moreover, we prove the existence of infinitely many solutions with positive energy. The main tools are the concentration compactness principle, $Z_{2}$ index theory and Fountain Theorem. These results extend some existing results in the literature.