Description

• Title:
  Semiclassical states for singularly perturbed Schrödinger-Poisson systems with a general Berestycki-Lions or critical nonlinearity
• Creator:
  Chen, Sitong;
  Tang, Xianhua;
  Zhang, Ning
• Description:
  This paper is concerned with two classes of singularly perturbed Schrödinger-Poisson systems of the form \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+u+ \phi u=f(u), & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2, & x\in \mathbb{R}^3, \end{cases} \end{equation*} and \begin{equation*} \begin{cases} -\varepsilon^2\triangle u+V(x)u+ \phi u=g(x,u)+K(x)u^5, & x\in {\mathbb{R}}^{3},\\ -\triangle \phi=u^2,& x\in \mathbb{R}^3, \end{cases} \end{equation*} where $\ep> 0$ is a small parameter. We prove that: (1) the first system admits a concentrating bounded state for small $\ep$, where $f\in \mathcal{C}(\mathbb{R},\mathbb{R})$ satisfies Berestycki-Lions assumptions which are almost necessary; (2) there exists a constant $\ep_0> 0$ determined by $V,K$ and $g$ such that for any $\ep\in (0,\ep_0]$ the second system has a nontrivial solution, where $V,K\in \mathcal{C}(\mathbb{R}^3,\mathbb{R})$, $V(x)\ge 0$, $K(x)> 0$, $g\in \mathcal{C}(\mathbb{R}^3\times \mathbb{R},\mathbb{R})$ is an indefinite function. Our results improve and complement the previous ones in the literature.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2019 Topological Methods in Nonlinear Analysis
• Date:
  2019.11.09
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 54, No 2 (December 2019); 665 - 683;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.060/23626 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.060/23627
• Subject:
  35J60;
  35J20;
  35Q40;
  Semiclassical states;
  Schrödinger--Poisson system;
  Berestycki--Lions type assumptions;
  critical Sobolev exponent;
  variational method
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.060
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other