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 Schrödinger–Poisson system with exponential and logarithmic nonlinearities \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\phi u =|u|^{p-2}u\log |u|^2 +\lambda f(u) +h(x),&{} \textrm{in} \hspace{5.0pt}\Omega ,\\ -\Delta \phi -\varepsilon ^4 \Delta _4 \phi =u^2,&{} \textrm{in}\hspace{5.0pt}\Omega ,\\ u=\phi =0,&{} \textrm{on}\hspace{5.0pt}\partial \Omega ,\\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+\varphi u={|u|}^{p-2}ulog{|u|}^2+\lambda f(u)+h(x),\hfill & \text{in}\mathrm{\Omega },\hfill \\ -\mathrm{\Delta }\varphi -{\epsilon }^4{\mathrm{\Delta }}_4\varphi =u^2,\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=\varphi =0,\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(4<p<+\infty ,\,\varepsilon ,\,\lambda >0\) $4<p<+\infty ,\epsilon ,\lambda >0$ are parameters, \(\mathrm{\Delta _4 \phi = div(|\nabla \phi |^2 \nabla \phi )}\) ${\mathrm{\Delta }}_4\varphi ={\mathrm{div}(|\mathrm{\nabla }\varphi |}^2\mathrm{\nabla }\varphi )$ , \(\Omega \subset {\mathbb {R}}^2\) $\mathrm{\Omega }\subset {\mathbb{R}}^2$ 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 \(\lambda \) $\lambda$ large enough and \(\varepsilon \) $\epsilon$ fixed. Finally, we research the asymptotic behavior of solutions with respect to the parameters \(\varepsilon \) $\epsilon$ .