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H^1_0(\\Omega), \\quad i=1,...,m $$\n  where $\\Omega$ is a bounded domain, $\\beta>0$ and $a_i\\geq 0\\ \\forall i.$ Moreover, for $a_i=0$, we show a relation between critical energies associated with this system and the optimal partition problem\n  $$ \\mathop{\\inf_{\\omega_i\\subset \\Omega \\text{open}}}_{\\omega_i\\cap \\omega_j=\\emptyset\\forall i\\neq j} 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