 Solution

As $v\to \infty$, $[v(v-1){]}^{-4/\kappa }\sim v^{-8/\kappa }$. ThereforeThus every positive solution is unbounded when κ ≥ 8, including the logarithmically divergent critical case. For $4<\kappa <8$, by contrast,For example the substitution $r=1-1/v$ identifies this normalizing constant as $B(1-4/\kappa ,8/\kappa -1)$, with both beta function parameters positive.