Solutions of a quasilinear Schrödinger-Poisson system with linearly bounded nonlinearities

In this paper, we are concerned with the following quasilinear Schrodinger-Poisson system -Delta u+V(x)u+K(x)phi u=f(x,u),x is an element of R3,-Delta phi-epsilon 4 Delta 4 phi=K(x)u2,x is an element of R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+V(x)u+ K(x)\phi u=f(x,u),\quad &{}x\in {\mathbb {R}}<^>3,\\ -\Delta \phi -\varepsilon <^>4\Delta _4\phi = K(x) u<^>2, &{}x\in {\mathbb {R}}<^>3, \end{array}\right. } \end{aligned}$$\end{document}where epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} is a positive parameter and f is linearly bounded in u at infinity. Under suitable assumptions on V, K and f, we establish the existence and asymptotic behavior of ground state solutions to the system. We prove that they converge to the solutions of the classic Schrodinger-Poisson system associated as epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} tends to zero.