Exact sign-transition exponent for the cubic reduced coefficient

Determine the exact transition value, if any, in the exponent interval $(1,2)$ at which the cubic coefficient $\kappa(p)$ of the exact reduced nonlinearity changes sign.

Background

In the critical dimension $N=7$, the exact reduced nonlinearity has expansion $\mathfrak q(a)=C_p+\kappa(p)a+O(a^2)$. The paper proves that $\kappa(p)>0$ for every $p\ge2$ and that $\kappa(p)<0$ for $p>1$ sufficiently close to $1$, establishing a sign transition somewhere in $(1,2)$.

The sign determines the behavior of the small radial branch of the critical reduced equation: positive $\kappa$ yields a positive entire branch and corresponding non-one-dimensional half-space solutions, whereas negative $\kappa$ rules out a small positive entire radial branch of the exact reduced equation. The precise transition location remains unresolved.