Least energy sign-changing solutions for Schrödinger-Poisson systems with potential well

Abstract

Abstract In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system − Δ u + V ( x ) u + K ( x ) ϕ u = f ( u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u+K\left(x)\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ \hspace{1.0em}\end{array}\right. where the functions V ( x ) , K ( x ) V\left(x),K\left(x) have finite limits as ∣ x ∣ → ∞ | x| \to \infty satisfying some mild assumptions. By combining variational methods with the global compactness lemma, we obtain a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.