Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent

In this paper, we study the existence of multiple positive solutions of the following Schrödinger-Poisson system with critical exponent $\begin{equation*}\begin{cases}-Δ u-l(x)φ u=λ h(x)|u|^{q-2}u+|u|^{4}u,\ \text{in}\ \mathbb{R}^{3}, \\-Δφ=l(x)u^{2},\ \text{in}\ \mathbb{R}^{3},\end{cases}\end{equation*}$ where $1 < q < 2 $ and $λ>0 $. Under some appropriate conditions on $ l$ and $h $, we show that there exists $λ^{*}>0 $ such that the above problem has at least two positive solutions for each $λ∈(0,λ^{*}) $ by using the Mountain Pass Theorem and Ekeland's Variational Principle.