Least-energy sign-changing solutions for Kirchhoff–Schrödinger–Poisson systems in R3 $\mathbb{R}^{3}$

Abstract

Abstract In this paper, we study the following Kirchhoff–Schrödinger–Poisson systems: {−(a+b∫R3|∇u|2dx)Δu+V(x)u+ϕu=f(u),x∈R3,−Δϕ=u2,x∈R3, $$\textstyle\begin{cases} -(a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx)\Delta u+V(x)u+\phi u=f(u), &x \in \mathbb{R}^{3}, \\ -\Delta \phi =u^{2}, &x\in \mathbb{R}^{3}, \end{cases} $$ where a, b are positive constants, V∈C(R3,R+) $V\in \mathcal{C}(\mathbb{R} ^{3},\mathbb{R}^{+})$. By using constraint variational method and the quantitative deformation lemma, we obtain a least-energy sign-changing (or nodal) solution ub $u_{b}$ to this problem, and study the energy property of ub $u_{b}$. Moreover, we investigate the asymptotic behavior of ub $u_{b}$ as the parameter b↘0 ${b\searrow 0}$.

Keywords

• Sign-changing solution
• Nonlocal term
• Variation methods