TY - JOUR
T1 - On Nodal Solutions of the Schrödinger-Poisson System with a Cubic Term
AU - Tang , Ronghua
AU - Guo , Hui
AU - Wang , Tao
JO - Journal of Nonlinear Modeling and Analysis
VL - 3
SP - 623
EP - 642
PY - 2024
DA - 2024/08
SN - 6
DO - http://doi.org/10.12150/jnma.2024.623
UR - https://global-sci.org/intro/article_detail/jnma/23353.html
KW - Schrödinger-Poisson system, nodal solutions, Gersgorin disc theorem, Miranda theorem, blow-up analysis.
AB - <p style="text-align: justify;">In this paper, we consider the following Schrödinger-Poisson system with a cubic term $$\begin{align*}\tag{0.1}\label{0.1} \begin{cases} -\Delta u+V(|x|)u+\lambda\phi u=|u|^2u \ \&nbsp; {\rm in} \ \&nbsp; \mathbb{R}^3,\\ -\Delta\phi=u^2 \ \ {\rm in} \&nbsp; \ \mathbb{R}^3,&nbsp;\end{cases} \end{align*}$$&nbsp;where $λ &gt; 0$ 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.</p>