Infinitely many sign-changing solutions for the nonlinear Schrödinger–Poisson system

  • Zhaoli Liu
  • Zhi-Qiang WangEmail author
  • Jianjun Zhang


In this paper, we consider the following Schrödinger–Poisson system
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=f(u)&{}\quad \text{ in }\ \mathbb {R}^3,\\ -\Delta \phi =u^2&{}\quad \text{ in }\ \mathbb {R}^3. \end{array} \right. \end{aligned}$$
We investigate the existence of multiple bound state solutions, in particular sign-changing solutions. By using the method of invariant sets of descending flow, we prove that this system has infinitely many sign-changing solutions. In particular, the nonlinear term includes the power-type nonlinearity \(f(u)=|u|^{p-2}u\) for the well-studied case \(p\in (4,6)\), and the less studied case \(p\in (3,4)\), and for the latter case, few existence results are available in the literature.

Mathematics Subject Classification

35J20 35J60 



J. Zhang thanks Dr. Zhengping Wang and Prof. Huansong Zhou for letting him know their work [30]. Z. Liu is supported by NSFC-11271265, NSFC-11331010 and BCMIIS. Z.-Q. Wang is partially supported by NSFC-11271201 and BCMIIS. J. Zhang is supported by CPSF-2013M530868.


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Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesCapital Normal UniversityBeijingPeople‘s Republic of China
  2. 2.Center for Applied MathematicsTianjin UniversityTianjinPeople‘s Republic of China
  3. 3.Department of Mathematics and StatisticsUtah State UniversityLoganUSA
  4. 4.Chern Institute of MathematicsNankai UniversityTianjinPeople‘s Republic of China

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