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

In this paper, we consider the following logarithmic Schrödinger–Poisson system \(\begin{aligned} \left\{ \begin{aligned}&- \Delta u + V(x) u + \lambda K(x)\phi u = f(u) + u \log u^2,&x \in {\mathbb {R}}^{3},\\&- \Delta \phi - \varepsilon ^4 \Delta _4 \phi = \lambda K(x) u^2,&x \in {\mathbb {R}}^{3},\\ \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& -\mathrm{\Delta }u+V(x)u+\lambda K(x)\varphi u=f(u)+ulog{u}^2,\hfill & \hfill x\in {\mathbb{R}}^3,\\ \hfill & -\mathrm{\Delta }\varphi -{\epsilon }^4{\mathrm{\Delta }}_4\varphi =\lambda K(x)u^2,\hfill & \hfill x\in {\mathbb{R}}^3,\\ \hfill \end{array}\right)\end{array}$$ 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 \(\varepsilon \rightarrow 0^+\) $\epsilon \to 0^{+}$ and \(\lambda \rightarrow 0^+\) $\lambda \to 0^{+}$ , respectively.