We investigate a class of PDEs featuring nonlocal degeneracies arising from self-dependent regions determined by the solution's Hölder semi-norm. Notably, this framework unifies two classical settings: free boundary problems, where degeneracy occurs along the nodal set, and critical-point degenerate PDEs, both recast as extrema (local) problems within our formulation. For models where an elliptic PDE is only activated beyond a given positive threshold \( \kappa>0 \), we establish the local Hölder continuity of solutions, which is the optimal regularity possible, and prove a result of Krylov-Safonov type for operators with coefficients, yielding universal continuity estimates for solutions, independent of coefficient regularity. In the globally degenerate case \( \kappa=0 \), we develop a \( C^{1,\beta} \)-regularity theory, which sharply interpolates the known estimates for the extrema local problems. Beyond its intrinsic relevance, the framework developed in this paper provides new perspectives on the classical extrema models, offering insights that are not accessible through previously known approaches.