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

By Zhaoli Liu, Zhi-Qiang Wang, Jianjun Zhang. Annali di Matematica Pura ed Applicata, 2014

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In this paper, we consider the following Schrödinger–Poisson system -Δu+V(x)u+ϕu=f(u)inR3,-Δϕ=u2inR3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=f(u)&{}\quad \text{ in }\ \mathbb {R}^3,\\ -\Delta \phi =u^2&{}\quad \text{ in }\ \mathbb {R}^3. \end{array} \right. \end{aligned}$$\end{document}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-2u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(u)=|u|^{p-2}u$$\end{document} for the well-studied case p∈(4,6)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (4,6)$$\end{document}, and the less studied case p∈(3,4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (3,4)$$\end{document}, and for the latter case, few existence results are available in the literature.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3b6724d0efc47ad73a57951ceb63751e533a428a

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In Applied Mathematics

  1. Boštjan Gabrovšek
  2. Jijiang Sun
  3. Lin Li
  4. Matija Cencelj

In Mathematical Physics

  1. Boštjan Gabrovšek
  2. Jijiang Sun
  3. Lin Li
  4. Matija Cencelj

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