= Solution
As $v\to\infty$, $[v(v-1)]^{-4/\kappa}\sim v^{-8/\kappa}$. Therefore
$$
s_\kappa(z)\sim
\begin{cases}
\dfrac{z^{1-8/\kappa}}{1-8/\kappa},&\kappa>8,\\
\log z,&\kappa=8.
\end{cases}
$$
Thus \b[every positive solution is unbounded when κ ≥ 8], including the logarithmically divergent critical case. For $4<\kappa<8$, by contrast,
$$
S_\kappa:=s_\kappa(\infty)<\infty.
$$
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.
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