Fully nonlinear singularly perturbed models with non-homogeneous degeneracy

Abstract

This work is devoted to studying non-variational, nonlinear singularly perturbed elliptic models enjoying a double degeneracy character with prescribed boundary value in a domain. In its simplest form, for each $\epsilon >0$ fixed, we seek a non-negative function $u^{\epsilon }$ satisfying

in the viscosity sense for suitable data $p,q\in (0,\infty )$, $\mathfrak{a}$ and $g$, where ${\zeta }_{\epsilon }$ behaves singularly as $\text{O}({\epsilon }^{-1})$ near $\epsilon$-level surfaces. In such a context, we establish the existence of certain 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. In particular, for a restricted class of nonlinearities, we prove the finiteness of the $(N-1)$-dimensional Hausdorff measure of level sets. We also address a complete and in-deep analysis concerning the asymptotic limit as $\epsilon \to 0^{+}$, which is related to one-phase solutions of inhomogeneous nonlinear free boundary problems in flame propagation and combustion theory. Finally, we present some fundamental regularity tools in the theory of doubly degenerate fully nonlinear elliptic PDEs, which may have their own mathematical interest.