Local porosity of the free boundary in a minimum problem

Given certain set $\mathcal{K}$ and functions $q$ and $h$, we study geometric properties of the set $\mathrm{\partial }\{x\in \mathrm{\Omega }:u(x)>0\}$ for non-negative minimizers of the functional $\mathcal{J}(u)={\int }_{\mathrm{\Omega }}^{}\left(\frac{1}{p}|\mathrm{\nabla }u{|}^p+q(u^{+}{)}^{\gamma }+hu\right)\text{d}x$ over $\mathcal{K}$, where $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 2)$ is an open bounded domain, $p\in (1,+\mathrm{\infty })$ and $\gamma \in (0,1]$ are constants, $u^{+}$ is the positive part of $u$ and $\mathrm{\partial }\{x\in \mathrm{\Omega }:u(x)>0\}$ is the so-called free boundary. Such a minimum problem arises in physics and chemistry for $\gamma =1$ and $\gamma \in (0,1)$, respectively. Using the comparison principle of $p$-Laplacian equations, we establish first the non-degeneracy of non-negative minimizers near the free boundary, then prove the local porosity of the free boundary.