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establish the existence of a positive fully nontrivial solution $(u,v)$ to the weakly coupled elliptic system% \\[ \\left\\{ \\begin{tabular} [c]{l}% $-\\Delta u=\\mu_{1}|u|^{{2}^{\\ast}-2}u+\\lambda\\alpha|u|^{\\alpha-2}|v|^{\\beta }u,$\\\\ $-\\Delta v=\\mu_{2}|v|^{{2}^{\\ast}-2}v+\\lambda\\beta|u|^{\\alpha}|v|^{\\beta{-2}% }v,$\\\\ $u,v\\in D^{1,2}(\\mathbb{R}^{N}),$% \\end{tabular} \\ \\right. \\] where $N\\geq4,$ $2^{\\ast}:=\\frac{2N}{N-2}$ is the critical Sobolev exponent, $\\alpha,\\beta\\in(1,2],$ $\\alpha+\\beta=2^{\\ast},$ $\\mu_{1},\\mu_{2}>0,$ and $\\lambda<0.$ We show that these solutions exhibit phase separation as ","authors_text":"Angela Pistoia, M\\'onica Clapp","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-08-31T13:31:34Z","title":"Existence and phase separation of entire solutions to a pure critical competitive elliptic 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