Non-trivial solutions for Schrödinger-Poisson systems involving critical nonlocal term and potential vanishing at infinity

The present study is concerned with the following Schrödinger-Poisson system involving critical nonlocal term $$\begin{array}{} \displaystyle \begin{cases} -\triangle u+V(x)u-l(x)\phi|u|^{3}u=\eta K(x)f(u),~~\mbox{in}~~\mathbb{R}^{3}, \notag\\ -\triangle\phi=l(x)|u|^{5},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\mbox{in}~~\mathbb{R}^{3}, \end{cases} \end{array}$$(1.1) where the potential V(x) and K(x) are positive continuous functions that vanish at infinity, and l(x) is bounded, nonnegative continuous function. Under some simple assumptions on V, K, l and f, we prove that the problem (1.1) has a non-trivial solution.