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.

中文翻译：

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某些对数薛定谔-泊松系统正解的存在性和渐近行为

在本文中，我们考虑以下对数薛定谔-泊松系统

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

由于泊松方程中能量泛函和四阶项的不确定性，它越来越受到人们的关注。利用变分法证明了正解的存在性和多重性。最后，我们得到正解的渐近行为分别为\(\varepsilon \rightarrow 0^+\)和\(\lambda \rightarrow 0^+\)。