Regularity properties for $p-$dead core problems and their asymptotic limit as $p \to \infty$

Abstract

We study regularity issues and the limiting behavior as $p\to\infty$ of nonnegative solutions for elliptic equations of $p-$Laplacian type ($2 \leq p< \infty$) with a strong absorption: $$ -\Delta_p u(x) + \lambda_0(x) u_{+}^q(x) = 0 \quad \text{ in } \quad \Omega \subset \mathbb{R}^N, $$ where $\lambda_0>0$ is a bounded function, $\Omega$ is a bounded domain and $0\leq q<p-1$. When $p$ is fixed, such a model is mathematically interesting since it permits the formation of dead core zones, i.e, a priori unknown regions where non-negative solutions vanish identically. First, we turn our attention to establishing sharp quantitative regularity properties for $p-$dead core solutions. Afterwards, assuming that $\ell \:=\lim_{p \to \infty} q(p)/p \in [0, 1)$ exists, we establish existence for limit solutions as $p\to \infty$, as well as we characterize the corresponding limit operator governing the limit problem. We also establish sharp $C^{\gamma}$ regularity estimates for limit solutions along free boundary points, that is, points on $ \partial \{u>0\} \cap \Omega$ where the sharp regularity exponent is given explicitly by $\gamma = \frac{1}{1-\ell}$. Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved.