Quantitative stratification and sharp regularity estimates for supercritical semilinear elliptic equations

In this paper, we investigate the interior regularity theory for stationary solutions of the supercritical nonlinear elliptic equation $$-\Delta u=|u|^{p-1}u\quad\text{in }\Omega,\quad p>\frac{n+2}{n-2},$$ where $\Omega\subset\mathbb{R}^n$ is a bounded domain with $n\geq 3$. Our primary focus is on the structure of stratification for the singular sets. We define the $k$-th stratification $S^k(u)$ of $u$ based on the tangent functions and measures. We show that the Hausdorff dimension of $S^k(u)$ is at most $k$ and $S^k(u)$ is $k$-rectifiable, and establish estimates for volumes associated with points that have lower bounds on the regular scales. These estimates enable us to derive sharp interior estimates for the solutions. Specifically, if $\alpha_p=\frac{2(p+1)}{p-1}$ is not an integer, then for any $j\in\mathbb{Z}_{\geq 0}$, we have $$D^ju\in L_{\operatorname{loc}}^{q_j,\infty}(\Omega),$$ which implies that for any $\Omega'\subset\subset\Omega$, $$\sup\{\lambda>0:\lambda^{q_j}\mathcal{L}^n(\{x\in\Omega':|D^ju(x)|>\lambda\})\}<+\infty,$$ where $\mathcal{L}^n(\cdot)$ is the $n$-dimensional Lebesgue measure, and $$q_j=\frac{(p-1)(\lfloor\alpha_p\rfloor+1)}{2+j(p-1)},$$ with $\lfloor\alpha_p\rfloor$ being the integer part of $\alpha_p$. The proofs of these results rely on Reifenberg-type theorems developed by A. Naber and D. Valtorta to study the stratification of harmonic maps.