In this paper, we are interested in the existence and multiplicity of positive solutions for the following Kirchhoff–Schr\(\ddot{\text {o}}\)dinger–Newton system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( 1+b\displaystyle \int _\Omega |\nabla u|^2dx\right) \Delta u=(\lambda _1+\delta )u+\phi |u|u, &{} \text {in}\ \ \Omega , \\ -\Delta \phi =|u|^3,&{} \text {in}\ \ \Omega , \\ u=\phi =0, &{} \text {on}\ \ \partial \Omega , \end{array}\right. } \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^{4}\) is a smooth bounded domain, \(b>0\), \(\delta >0\), \(\lambda _1\) is the first eigenvalue of \(-\Delta\) on \(\Omega\), and two positive solutions are established via variational method.

中文翻译：

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基尔霍夫-薛定谔-牛顿系统的近共振

在本文中，我们对以下 Kirchhoff–Schr \(\ddot{\text {o}}\) dinger–Newton 系统的正解的存在性和多重性感兴趣

$$\begin{aligned} {\left\{ \begin{array}{ll} -\left( 1+b\displaystyle \int _\Omega |\nabla u|^2dx\right) \Delta u=(\ lambda _1+\delta )u+\phi |u|u, &{} \text {in}\ \ \Omega , \\ -\Delta \phi =|u|^3,&{} \text {in}\ \ \Omega , \\ u=\phi =0, &{} \text {on}\ \ \partial \Omega , \end{array}\right. } \end{对齐}$$

其中\(\Omega \subset {\mathbb {R}}^{4}\)是一个光滑的有界域，\(b>0\) , \(\delta >0\) , \(\lambda _1\)是\(-\Delta\)在\(\Omega\)上的第一个特征值，通过变分方法建立了两个正解。