A remark on doubly critical elliptic systems
type: Item
instance of: scholarly article
title: A remark on doubly critical elliptic systems
zbMATH Open document ID: 1298.35063
author: Zhijie Chen
author: Wenming Zou
published in: Calculus of Variations and Partial Differential Equations
publication date: 2014-08-01T00:00:00Z
review text: The paper deals with the weakly coupled nonlinear elliptic system \[ \begin{cases} -\Delta u-\frac{\lambda_1}{|x|^2}u-|u|^{2^*-2}u=\nu h(x)\alpha|u|^{\alpha-2}|v|^{\beta}u,\,\,\,x\in\mathbb{R}^N,\\ -\Delta v-\frac{\lambda_2}{|x|^2}v-|v|^{2^*-2}v=\nu h(x)\beta|u|^{\alpha}|v|^{\beta-2}v,\,\,\,x\in\mathbb{R}^N,\\ u(x),\,\,v(x)\in D^{1,2}(\mathbb{R}^N),\end{cases} \leqno(1) \] where \(N\geq 3\), \(\lambda_1,\,\lambda_2\in (0,\Lambda_N)\), \(\Lambda_N:=\frac{(N-2)^2}{4}\), \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(\nu>0\) is a parameter, \(\alpha,\,\beta>1\), \(\alpha+\beta\leq 2^*\) and \(D^{1,2}(\mathbb{R}^N)\) is the completion of \(C_0^{\infty}(\mathbb{R}^N)\) with respect to the norm \(\|u\|=\left(\int_{\mathbb{R}^N}|\nabla u|^2\,dx\right)^{1/2}\). By using some variational methods, the authors investigate the existence and properties of the positive solutions of problem \((1)\).
Mathematics Subject Classification ID: 35J50
Mathematics Subject Classification ID: 35B09
zbMATH DE Number: 6324017
zbMATH Keywords: coupled nonlinear elliptic system
zbMATH Keywords: positive solutions
zbMATH Keywords: ground state solutions
reviewed by: Rodica Luca
MaRDI profile type: Publication
full work available at URL: s00526-013-0662-0
OpenAlex ID: W2022524944
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DOI: 10.1007/S00526-013-0662-0
description: scientific article