Year: 2024
Author: Ronghua Tang, Hui Guo, Tao Wang
Journal of Nonlinear Modeling and Analysis, Vol. 6 (2024), Iss. 3 : pp. 623–642
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.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.12150/jnma.2024.623
Journal of Nonlinear Modeling and Analysis, Vol. 6 (2024), Iss. 3 : pp. 623–642
Published online: 2024-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Schrödinger-Poisson system nodal solutions Gersgorin disc theorem Miranda theorem blow-up analysis.