Multiple positive solutions for Schrodinger-Poisson systems involving critical nonlocal term

Abstract: The present study is concerned with the following Schr\"{o}dinger-Poisson system involving critical nonlocal term $$ \left{ \begin{array}{ll} -\Delta u+u-K(x)\phi |u|^3u=\lambda f(x)|u|^{q-2}u, & x\in\mathbb{R}^3, -\Delta \phi=K(x)|u|^5, & x\in\mathbb{R}^3,\ \end{array} \right. $$ where $1<q\<2$ and $\lambda\>0$ is a parameter. Under suitable assumptions on $K(x)$ and $f(x)$, there exists $\lambda_0=\lambda_0(q,S,f,K)>0$ such that for any $\lambda\in(0,\lambda_0)$, the above Schr\"{o}dinger-Poisson system possesses at least two positive solutions by standard variational method, where a positive least energy solution will also be obtained.