Description

• Title:
  Positive least energy solutions for coupled nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponent
• Creator:
  You, Song;
  Wang, Qingxun;
  Zhao, Peihao
• Description:
  \begin{cases} \displaystyle -\Delta u+\nu_{1}u=\mu_{1}\bigg(\frac{1}{|x|^{4}}\ast u^{2}\bigg)u +\beta \bigg(\frac{1}{|x|^{4}}\ast v^{2}\bigg)u, & x \in \Omega,\\[10pt] \displaystyle -\Delta v+\nu_{2}v=\mu_{2}\bigg(\frac{1}{|x|^{4}}\ast v^{2}\bigg)v +\beta\bigg (\frac{1}{|x|^{4}}\ast u^{2}\bigg)v, & x \in \Omega,\\[10pt] u,v \geq 0 \quad\text{in }\Omega, \qquad u=v=0 \quad \text{on } \partial\Omega. \end{cases} \end{equation*} Here $\Omega\subset\mathbb{R}^{N}$ is a smooth bounded domain, $-\lambda_{1}(\Omega)< \nu_{1},\nu_{2}< 0, \lambda_{1}(\Omega)$ is the first eigenvalue of $ (-\Delta, H_{0}^{1}(\Omega))$, $\mu_{1},\mu_{2}> 0$ and $\beta\neq 0$ is a coupling constant. We show that the critical nonlocal elliptic system has a positive least energy solution under appropriate conditions on parameters via variational methods. For the case in which $\nu_{1}=\nu_{2}$, we obtain the classification of the positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as $\beta\rightarrow 0$ are studied.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2019 Topological Methods in Nonlinear Analysis
• Date:
  2019.05.10
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 53, No 2 (June 2019); 623 - 657;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.014/17515 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.014/17516
• Subject:
  Coupled Choquard equations;
  least energy solutions;
  Hardy--Littlewood--Sobolev critical exponent;
  35J57;
  35J50
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.014
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other