  In our work we study non-variational, nonlinear singularly perturbed elliptic
models enjoying a double degeneracy character with prescribed boundary value in
a domain. In such a scenario, we establish the existence of solutions. We also
prove that solutions are locally (uniformly) Lipschitz continuous, and they
grow in a linear fashion. Moreover, solutions and their free boundaries possess
a sort of measure-theoretic and weak geometric properties. Moreover, for a
restricted class of non-linearities, we prove the finiteness of the
(N-1)-dimensional Hausdorff measure of level sets. We also address a complete
analysis concerning the asymptotic limit as the singular parameter, which is
related to one-phase solutions of inhomogeneous nonlinear free boundary
problems in flame propagation and combustion theory.
