数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (5): 1947-1970.doi: 10.1007/s10473-022-0513-1

• 论文 • 上一篇    

LOCALIZED NODAL SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS

Xing WANG, Rui HE, Xiangqing LIU   

  1. Department of Mathematics, Yunnan Normal University, Kunming, 650500, China
  • 收稿日期:2020-11-30 修回日期:2022-05-23 发布日期:2022-11-02
  • 通讯作者: Xiangqing Liu,E-mail:lxq8u8@163.com E-mail:lxq8u8@163.com
  • 基金资助:
    Supported by NSFC (12161093).

LOCALIZED NODAL SOLUTIONS FOR SCHRÖDINGER-POISSON SYSTEMS

Xing WANG, Rui HE, Xiangqing LIU   

  1. Department of Mathematics, Yunnan Normal University, Kunming, 650500, China
  • Received:2020-11-30 Revised:2022-05-23 Published:2022-11-02
  • Contact: Xiangqing Liu,E-mail:lxq8u8@163.com E-mail:lxq8u8@163.com
  • Supported by:
    Supported by NSFC (12161093).

摘要: In this paper, we study the existence of localized nodal solutions for Schrödinger-Poisson systems with critical growth \begin{equation*} \left\{ \begin{aligned} &-\varepsilon^2\Delta v+V(x)v+\lambda \psi v=v^{5}+\mu|v|^{q-2}v, \ \ \ \text{in}\,\,\mathbb{R}^3,\\ &-\varepsilon^2\Delta \psi=v^2, \ \ \ \text{in}\,\,\mathbb{R}^3; \,\,v(x)\rightarrow 0,\,\psi(x)\rightarrow 0\quad\text{as}\,\,|x| \rightarrow\infty. \end{aligned} \right. \end{equation*} We establish, for small $\varepsilon$, the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method, and employ some new analytical skills to overcome the obstacles caused by the nonlocal term $\varphi_u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}{\rm d}y$. Our results improve and extend related ones in the literature.

关键词: Schrödinger-Poisson systems, localized nodal solutions, perturbation method

Abstract: In this paper, we study the existence of localized nodal solutions for Schrödinger-Poisson systems with critical growth \begin{equation*} \left\{ \begin{aligned} &-\varepsilon^2\Delta v+V(x)v+\lambda \psi v=v^{5}+\mu|v|^{q-2}v, \ \ \ \text{in}\,\,\mathbb{R}^3,\\ &-\varepsilon^2\Delta \psi=v^2, \ \ \ \text{in}\,\,\mathbb{R}^3; \,\,v(x)\rightarrow 0,\,\psi(x)\rightarrow 0\quad\text{as}\,\,|x| \rightarrow\infty. \end{aligned} \right. \end{equation*} We establish, for small $\varepsilon$, the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function via the perturbation method, and employ some new analytical skills to overcome the obstacles caused by the nonlocal term $\varphi_u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}{\rm d}y$. Our results improve and extend related ones in the literature.

Key words: Schrödinger-Poisson systems, localized nodal solutions, perturbation method

中图分类号: 

  • 35B05