Abstract. In this talk we establish sharp C1,β regularity estimates for bounded solutions loc of a class of fully nonlinear elliptic PDEs with non-homogeneous degeneracy, whose model equation is given by
[|Du|p +a(x)|Du|q]M+λ,Λ(D2u)=f(x,u) in Ω,for a bounded domain Ω ⊂ RN, and suitable data p,q ∈ (0,∞), a and f. Our approach is based on geometric tangential methods and makes use of a refined oscillation mechanism combined with compactness and scaling techniques. Finally, we present some connections of our findings with a variety of nonlinear free boundary problems in the theory of elliptic PDEs, which may have their own interest.