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: $$ -Δ_p u(x) + λ_0(x) u_{+}^q(x) = 0 \quad \text{ in } \quad Ω\subset \mathbb{R}^N, $$ where $λ_0>0$ is a bounded function, $Ω$ is a bounded domain and $0\leq q 0\} \cap Ω$ where the sharp regularity exponent is given explicitly by $γ= \frac{1}{1-\ell}$. Finally, some weak geometric and measure theoretical properties as non-degeneracy, uniform positive density, porosity and convergence of the free boundaries are proved.