Near resonance for a Kirchhoff–Schrödinger–Newton system

In this paper, we are interested in the existence and multiplicity of positive solutions for the following Kirchhoff–Schr $$\ddot{\text {o}}$$ o ¨ dinger–Newton system $$\begin{aligned} {\left\{ \begin{array}{ll} -\left( 1+b\displaystyle \int _\Omega |\nabla u|^2dx\right) \Delta u=(\lambda _1+\delta )u+\phi |u|u, &{} \text {in}\ \ \Omega , \\ -\Delta \phi =|u|^3,&{} \text {in}\ \ \Omega , \\ u=\phi =0, &{} \text {on}\ \ \partial \Omega , \end{array}\right. } \end{aligned}$$ - 1 + b ∫ Ω | ∇ u | 2 d x Δ u = ( λ 1 + δ ) u + ϕ | u | u , in Ω , - Δ ϕ = | u | 3 , in Ω , u = ϕ = 0 , on ∂ Ω , where $$\Omega \subset {\mathbb {R}}^{4}$$ Ω ⊂ R 4 is a smooth bounded domain, $$b>0$$ b > 0 , $$\delta >0$$ δ > 0 , $$\lambda _1$$ λ 1 is the first eigenvalue of $$-\Delta$$ - Δ on $$\Omega$$ Ω , and two positive solutions are established via variational method.