Optimality conjecture for the Hausdorff-dimension bound on the singular set

Let $f$ be a nonlinearity in the smooth asymptotic class of the dimensional-bound theorem, let $T_f$ denote its blow-up level, and define

$$q_f:=1+2\liminf_{t\uparrow T_f}\frac{\log f(t)+\int^t\sqrt{\frac{f”(s)}{f(s)}}\,ds}{\log f'(t)}.$$

For a corresponding stable solution $u^\star$, write $\Sigma(u^\star)$ for its singular set. Optimality conjecture. Within this smooth asymptotic class, the bound

$$\dim_{\mathcal H}\Sigma(u^\star)\leq n-2q_f$$

cannot in general be improved.

The quantity $q_f$ is obtained from the higher-integrability estimates for $f'(u^\star)$ and yields the current dimensional bound on the singular set. The conjecture asserts that, among nonlinearities in the stated class, this bound is sharp in general; the source provides no resolution.