Existence and asymptotical behavior of positive solutions for the Schrödinger-Poisson system with double quasi-linear terms

In this paper, we consider the following Schrödinger-Poisson system with double quasi-linear terms\begin{document}$ \begin{equation*} \label{1.1} \begin{cases} -\Delta u+V(x)u+\phi u-\frac{1}{2}u\Delta u^2 = \lambda f(x,u),\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ -\triangle\phi-\varepsilon^4\Delta_4\phi = u^{2},\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ \end{cases} \end{equation*} $\end{document}where \begin{document}$ \lambda,\varepsilon $\end{document} are positive parameters. Under suitable assumptions on \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document} , we prove that the above system admits at least one pair of positive solutions for \begin{document}$ \lambda $\end{document} large by using perturbation method and truncation technique. Furthermore, we research the asymptotical behavior of solutions with respect to the parameters \begin{document}$ \lambda $\end{document} and \begin{document}$ \varepsilon $\end{document} respectively. These results extend and improve some existing results in the literature.