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

Abstract

In this paper, we consider the following logarithmic Schrödinger–Poisson system

$$\begin{aligned} \left\{ \begin{aligned}&- \Delta u + V(x) u + \lambda K(x)\phi u = f(u) + u \log u^2,&x \in {\mathbb {R}}^{3},\\&- \Delta \phi - \varepsilon ^4 \Delta _4 \phi = \lambda K(x) u^2,&x \in {\mathbb {R}}^{3},\\ \end{aligned} \right. \end{aligned}$$

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 $\varepsilon \rightarrow 0^+$ and $\lambda \rightarrow 0^+$, respectively.