In this manuscript we study geometric regularity estimates for problems driven by fully nonlinear elliptic operators (which can be either degenerate or singular when “the gradient is small”) under strong absorption conditions of the general form: where the mapping fails to decrease fast enough at the origin, so allowing that nonnegative solutions may create plateau regions, that is, a priori unknown subsets where a given solution vanishes identically. We establish improved geometric regularity along the set (the free boundary of the model), for a sharp value of (obtained explicitly) depending only on structural parameters. Non‐degeneracy among others measure theoretical properties are also obtained. A sharp Liouville result for entire solutions with controlled growth at infinity is proved. We also present a number of applications consequential of our findings.