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| On Nodal Solutions of the Schr\"odinger-Poisson System with a Cubic Term |
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| KeyWord:Schr\"odinger-Poisson system, nodal solutions, Gersgorin disc theorem, Miranda theorem, blow-up analysis |
| Author Name | Affiliation | | Ronghua Tang | Information Department, Dongguan Light Industry School, Dongguan, Guangdong 523000, China | | Hui Guo | Department of Mathematics and Finance, Hunan University of Humanities, Science and Technology, Loudi, Hunan 417000, China | | Tao Wang | College of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan, Hunan 411201, China |
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| Abstract: |
| In this paper, we consider the following Schr\"odinger-Poisson system with a cubic term
\begin{equation}\label{000}\left\{\begin{aligned}
&-\Delta u+V(|x|)u+ \lambda\phi u=|u|^2u\quad\mbox{in }\mathbb{R}^3,\&-\Delta \phi=u^{2}\quad\mbox{in }\mathbb{R}^3,
\end{aligned}\right.\end{equation}
where $\lambda>0$ 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 \eqref{000} admits a radial nodal solution $U_{k,4}^\lambda$ that changes sign exactly $k$ times. Furthermore, the energy of $U_{k,4}^{\lambda}$ is strictly increasing in $k$ and the asymptotic behavior of $U_{k,4^{\lambda}$ as $\lambda\to 0_+$ is established. These results extend the existing ones from the super-cubic case in \cite{Kim-Seok} to the cubic case. |
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