Existence and asymptotic behavior of solutions for nonhomogeneous Schrödinger-Poisson system with exponential and logarithmic nonlinearities

In this paper, we consider the following nonhomogeneous quasilinear Schr & ouml;dinger-Poisson system with exponential and logarithmic nonlinearities -Delta u+phi u=|u|p-2ulog|u|2+lambda f(u)+h(x),in Omega,-Delta phi-epsilon 4 Delta 4 phi=u2,in Omega,u=phi=0,on partial derivative Omega,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\phi u =|u|<^>{p-2}u\log |u|<^>2 +\lambda f(u) +h(x),&{} \textrm{in} \hspace{5.0pt}\Omega ,\\ -\Delta \phi -\varepsilon <^>4 \Delta _4 \phi =u<^>2,&{} \textrm{in}\hspace{5.0pt}\Omega ,\\ u=\phi =0,&{} \textrm{on}\hspace{5.0pt}\partial \Omega ,\\ \end{array} \right. \end{aligned}$$\end{document}where 4<p<+infinity,epsilon,lambda>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$4<p<+\infty ,\,\varepsilon ,\,\lambda >0$$\end{document} are parameters, Delta 4 phi=div(|del phi|2 del phi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{\Delta _4 \phi = div(|\nabla \phi |<^>2 \nabla \phi )}$$\end{document}, Omega subset of R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}<^>2$$\end{document} is a bounded domain, and f has exponential critical growth. First, using reduction argument, truncation technique, Ekeland's variational principle, and the Mountain Pass theorem, we obtain that the above system admits at least two solutions with different energy for lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} large enough and 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} fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters 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}.