Fully nonlinear singularly perturbed models with non-homogeneous degeneracy

This work is devoted to studying non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in domain. In its simplest form, for each $\varepsilon>0$ fixed, we seek non-negative function $u^{\varepsilon}$ satisfying $$ \begin{cases} \left\[|\nabla u^{\varepsilon}|^p + \mathfrak{a}(x)|\nabla u^{\varepsilon}|^q \right] \Delta u^{\varepsilon} = \zeta\_{\varepsilon}(x, u^{\varepsilon}) & \operatorname{in } \Omega,\ u^{\varepsilon}(x) g(x) \operatorname{on \partial \Omega, \end{cases} the viscosity sense suitable data $p, q \in (0, \infty)$, $\mathfrak{a}$ and $g$, where $\zeta\_{\varepsilon}$ behaves as $\text{O} (\varepsilon^{-1})$ near $\varepsilon$-level surfaces. such context, establish existence of certain solutions. We also prove that solutions are locally (uniformly) Lipschitz continuous, they grow linear fashion. Moreover, their free boundaries possess sort measure-theoretic weak geometric properties. particular, restricted class nonlinearities, finiteness $(N-1)$-dimensional Hausdorff measure level sets. address complete in-deep analysis concerning asymptotic limit $\varepsilon \to 0^{+}$, which related one-phase inhomogeneous problems flame propagation combustion theory. Finally, present some fundamental regularity tools theory doubly degenerate fully PDEs, may have own mathematical interest.