Description

• Title:
  On finding the ground state solution to the linearly coupled Brezis-Nirenberg system in high dimensions: the cooperative case
• Creator:
  Wu, Yuanze
• Description:
  \begin{cases} -\Delta u_i+\mu_i u_i=|u_i|^{2^*-2}u_i+\lambda \sum\limits_{j=1,j\not=i}^ku_j &\text{in }\Omega,\\ u_i=0,\quad i=1,\dots,k,&\text{on }\partial\Omega, \end{cases} \end{equation*} where $k\geq2$, $\Omega\subset\bbr^N$ ($N\geq4$) is a bounded domain with smooth boundary $\partial\Omega$, $2^*={2N}/({N-2})$ is the Sobolev critical exponent, $\mu_i\in\bbr$ for all $i=1,\dots,k$ are constants and $\lambda\in\bbr$ is a parameter. By the variational method, we mainly prove that the above system has a ground state for all $\lambda> 0$. Our results reveal some new properties of the above system that imply that the parameter $\lambda$ plays the same role as in the following well- known Brez\'is-Nirenberg equation \begin{equation*} \begin{cases} -\Delta u =\lambda u+ |u|^{2^*-2}u &\text{in }\Omega,\\ u=0 &\text{on }\partial\Omega, \end{cases} \end{equation*} and this system has a very similar structure of solutions as the above Brez\'is-Nirenberg equation for $\lambda$.
• 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); 697 - 729;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.018/17519 ;https://apcz.umk.pl/TMNA/article/view/TMNA.2019.018/17520
• Subject:
  Elliptic system;
  linearly coupled;
  ground state;
  variational method;
  Sobolev critical exponent;
  35B09;
  35B33;
  35J50
• Identifier: https://apcz.umk.pl/TMNA/article/view/TMNA.2019.018
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other