Least energy sign-changing solutions for Kirchhoff–Poisson systems

The paper deals with the following Kirchhoff–Poisson systems: 0.1{−(1+b∫R3|∇u|2dx)Δu+u+k(x)ϕu+λ|u|p−2u=h(x)|u|q−2u,x∈R3,−Δϕ=k(x)u2,x∈R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \textstyle\begin{cases} - ( {1+b\int _{{\mathbb{R}}^{3}} { \vert \nabla u \vert ^{2}\,dx} } ) \Delta u+u+k(x)\phi u+\lambda \vert u \vert ^{p-2}u=h(x) \vert u \vert ^{q-2}u, & x\in {\mathbb{R}}^{3}, \\ -\Delta \phi =k(x)u^{2}, & x\in {\mathbb{R}}^{3}, \end{cases} $$\end{document} where the functions k and h are nonnegative, 0≤λ,b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0\le \lambda , b$\end{document}; 2≤p≤4<q<6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$2\le p\le 4< q<6$\end{document}. Via a constraint variational method combined with a quantitative lemma, some existence results on one least energy sign-changing solution with two nodal domains to the above systems are obtained. Moreover, the convergence property of ub\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u_{b}$\end{document} as b↘0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b \searrow 0$\end{document} is established.