On Nodal Solutions of the Schrödinger-Poisson System with a Cubic Term

Abstract

In this paper, we consider the following Schrödinger-Poisson system with a cubic term $$\begin{align*}\tag{0.1}\label{0.1} \begin{cases} -\Delta u+V(|x|)u+\lambda\phi u=|u|^2u \ \  {\rm in} \ \  \mathbb{R}^3,\\ -\Delta\phi=u^2 \ \ {\rm in} \  \ \mathbb{R}^3, \end{cases} \end{align*}$$ where $λ > 0$ and the radial function $V (x)$ is an external potential. By taking advantage of the Gersgorin disc theorem and Miranda theorem, via the variational method and blow up analysis, we prove that for each positive integer $k,$ problem (0.1) admits a radial nodal solution $U^λ_{k,4}$ that changes sign exactly $k$ times. Furthermore, the energy of $U^λ_{k,4}$ is strictly increasing in $k$ and the asymptotic behavior of $U^λ_{k,4}$ as $λ → 0_+$ is established. These results extend the existing ones from the super-cubic case in [17] to the cubic case.