Existence and asymptotic behavior of positive solutions to some logarithmic Schrödinger-Poisson system

In this paper, we consider the following logarithmic Schrodinger-Poisson system -Delta u+V(x)u+lambda K(x)phi u=f(u)+ulogu2,x is an element of R3,-Delta phi-epsilon 4 Delta 4 phi=lambda 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}$$\begin{aligned} \left\{ \begin{aligned}&- \Delta u + V(x) u + \lambda K(x)\phi u = f(u) + u \log u<^>2,&x \in {\mathbb {R}}<^>{3},\\&- \Delta \phi - \varepsilon <^>4 \Delta _4 \phi = \lambda K(x) u<^>2,&x \in {\mathbb {R}}<^>{3},\\ \end{aligned} \right. \end{aligned}$$\end{document}which has increasingly received interest due to the indefiniteness of the energy functional and fourth-order term in Poisson equation. By using variational method, we prove the existence and multiplicity of positive solutions. Finally, we obtain the asymptotic behavior of positive solutions as epsilon -> 0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0<^>+$$\end{document} and lambda -> 0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \rightarrow 0<^>+$$\end{document}, respectively.