On the existence of solutions for nonhomogeneous Schrödinger-Poisson system

In this paper, we study the existence of solutions for the following nonhomogeneous Schrödinger-Poisson systems: (∗){−Δu+V(x)u+K(x)ϕ(x)u=f(x,u)+g(x),x∈R3,−Δϕ=K(x)u2,lim|x|→+∞ϕ(x)=0,x∈R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ (*) \quad \textstyle\begin{cases} -\Delta u +V(x)u+K(x)\phi(x)u =f(x,u)+g(x), &x\in\mathbb{R}^{3}, \\ -\Delta\phi=K(x)u^{2}, \qquad \lim_{|x|\rightarrow+\infty}\phi(x)=0, & x\in\mathbb{R}^{3}, \end{cases} $$\end{document} where f(x,u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x,u)$\end{document} is either sublinear in u as |u|→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$|u|\rightarrow\infty$\end{document} or a combination of concave and convex terms. Under some suitable assumptions, the existence of solutions is proved by using critical point theory.