Near resonance for a Kirchhoff–Schrödinger–Newton system

Abstract

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.