Geometric regularity estimates for quasi-linear elliptic models in non-divergence form with strong absorption

Abstract

In this manuscript, we investigate geometric regularity estimates for problems governed by quasi-linear elliptic models in non-divergence form, which may exhibit either degenerate or singular behavior when the gradient vanishes, under strong absorption conditions of the form: \[ |\nabla u(x)|^γ Δ_p^{\mathrm{N}} u(x) = f(x, u) \quad \text{in} \quad B_1, \] where ${\hat{I}}^3>-1$, $p\in (1,\infty )$, and the mapping $u\mapsto f(x,u)\lesssim \mathfrak{a}(x)u_{+}^m$ (with $m\in [0,{\hat{I}}^3+1)$) does not decay sufficiently fast at the origin. This condition allows for the emergence of plateau regions, i.e., a priori unknown subsets where the non-negative solution vanishes identically. We establish improved geometric ${\mathrm{C}}^{\hat{I}}{º}_{\text{loc}}$ regularity along the set ${\mathcal{F}}_0=\partial \{u>0\}\cap B_1$ (the free boundary of the model) for a sharp value of $\hat{I}º\gg 1$, which is explicitly determined in terms of the structural parameters. Additionally, we derive non-degeneracy results and other measure-theoretic properties. Furthermore, we prove a sharp Liouville theorem for entire solutions exhibiting controlled growth at infinity.