Description

• Title:
  On critical pseudo-relativistic Hartree equation with potential well
• Creator:
  Zheng, Yu;
  Yang, Minbo;
  Shen, Zifei
• Description:
  The aim of this paper is to investigate the existence and asymptotic behavior of the solutions for the critical pseudo-relativistic Hartree equation $$ \sqrt{-\Delta+m^{2}}u+(\beta V(x)-\lambda)u =\bigg(\int_{\mathbb{R}^{N}}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg)|u| ^{2_{\mu}^{\ast}-2}u $$% for $\mathbb{R}^{N}$, where $m, \lambda, \beta\in\mathbb{R}^+$, $0< \mu< N$, $N\geq3$, $2_{\mu}^{\ast}=({2N-\mu})/({N-1})$ plays the role of critical exponent due to the Hardy-Littlewood-Sobolev inequality. By transforming the nonlocal problem into a local one via the Dirichlet-to-Neumann map, we are able to obtain the existence of the solutions by variational methods. Suppose that $0< \lambda< \lambda_{1}(\Omega)$ with $\lambda_{1}(\Omega)$ the first eigenvalue and the parameter $\beta$ is large enough, we can prove the existence of ground state solutions. Furthermore, for any sequences $\beta_{n}\rightarrow\infty$, we can show that the ground state solutions $\{u_{n}\}$ converges to a solution of $$ \sqrt{-\Delta+m^{2}}u-\lambda u= \bigg(\int_{\Omega}\frac{|u(z)|^{2_{\mu}^{\ast}}}{|x-z|^{\mu}}dz\bigg) |u|^{2_{\mu}^{\ast}-2}u \quad \mbox{in } \Omega, $$ where $\Omega :=\mbox{int} V^{-1}(0)$ is a nonempty bounded set with smooth boundary. By the way we also establish the existence and nonexistence results for the ground state solutions of the problems set on bounded domain.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2020 Topological Methods in Nonlinear Analysis
• Date:
  2020.03.04
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 55, No 1 (March 2020); 185 - 226;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.094/25004 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.094/25005
• Subject:
  Pseudo-relativistic Hartree equation;
  Brezis-Nirenberg problem;
  Hardy-Littlewood-Sobolev inequality;
  critical exponent;
  potential well;
  Primary;
  35J25;
  35J60;
  Secondary;
  35A15
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.094
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other