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}\\Omega,\\\\ u=v=0 & \\text{on }\\partial\\Omega, \\end{cases} \\end{equation*} where $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^{N}$, $N\\geq 3$, $2^{*}:=\\frac{2N}{N-2}$ is the critical Sobolev exponent, $\\mu_{1},\\mu_{2}>0$, $\\alpha, \\beta>1$, $\\alpha+\\beta =2^{*}$ and $\\lambda\\in\\mathbb{R}$.\n  We establish the existence of a 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solutions to a weakly coupled purely critical elliptic system in bounded domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jorge Faya, M\\'onica Clapp","submitted_at":"2018-05-25T18:05:22Z","abstract_excerpt":"We study the weakly coupled critical elliptic system \\begin{equation*} \\begin{cases} -\\Delta u=\\mu_{1}|u|^{2^{*}-2}u+\\lambda\\alpha |u|^{\\alpha-2}|v|^{\\beta}u & \\text{in }\\Omega,\\\\ -\\Delta v=\\mu_{2}|v|^{2^{*}-2}v+\\lambda\\beta |u|^{\\alpha}|v|^{\\beta-2}v & \\text{in }\\Omega,\\\\ u=v=0 & \\text{on }\\partial\\Omega, \\end{cases} \\end{equation*} where $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^{N}$, $N\\geq 3$, $2^{*}:=\\frac{2N}{N-2}$ is the critical Sobolev exponent, $\\mu_{1},\\mu_{2}>0$, $\\alpha, \\beta>1$, $\\alpha+\\beta =2^{*}$ and $\\lambda\\in\\mathbb{R}$.\n  We establish the existence of a 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