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

Abstract

In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system −Δu+V(x)u+K(x)ϕu=f(u),x∈R3,−Δϕ=K(x)u2,x∈R3,\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.

Keywords

• schrödinger-poisson system
• sign-changing solutions
• variational methods
• 35a15
• 35b38
• 35j50