Description

• Title:
  Positive solutions for Schrödinger-Poisson-Slater system with coercive potential
• Creator:
  Jiang, Yongsheng;
  Wang, Zhengping;
  Zhou, Huan-Song
• Description:
  We study the following Schrödinger-Poisson-Slater type system: \begin{equation}\label{eq:0.1} \begin{cases} -\Delta u + V(x) u + \lambda\phi (x) u =|u|^{p-1}u,& x\in \mathbb{R}^3, \\ -\Delta\phi = u^2, \quad \lim\limits_{|x|\to +\infty}\phi (x)=0, \end{cases} \end{equation} where $\lambda> 0$ is a parameter, $p\in (1,2)$, $V\in C(\mathbb{R}^N,\mathbb{R}_0^+)$. If $V$ is a constant, or $p \in (2,5)$ with $V\in L^{\infty}(\mathbb{R}^3)$, the above system has been considered in many papers. In this paper, we are interested in the case: $p\in (1,2)$ and $V$ being a coercive potential, i.e., $\lim\limits_{|x|\to +\infty} V(x)=\infty$. We prove that there exists $\lambda_0> 0$ such that system (\ref{eq:0.1}) has at least two positive solutions $u^0_\lambda$ and $u^1_\lambda$ for any $\lambda \in (0, \lambda_0)$. Moreover, $u^0_\lambda$ is a ground state (i.e., the least energy solution) which must blow up as $\lambda \rightarrow 0$. Particularly, when $p\in ({11}/{7},2)$ and $\lambda> 0$ is small enough, we show that the ground state of (\ref{eq:0.1}) must be non-radially symmetric even if $V(x)=V(|x|)$, such as $V(x)=|x|^2$.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2021 Topological Methods in Nonlinear Analysis
• Date:
  2021.02.20
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 57, No 2 (June 2021); 427 - 439;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2020.041/28256 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2020.041/28257
• Subject:
  Schrödinger-Poisson-Slater system;
  coercive potential;
  ground state;
  Primary;
  35J20;
  Secondary;
  35J60
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2020.041
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other