A Degenerate One-Phase Free Boundary Problem Arising From the Alt-Phillips Equation for Negative Powers

We study viscosity solutions for a class of degenerate one-phase free boundary problems of the form $w\Delta w = h(\nabla w)$. We assume the existence of a star-shaped domain $D$ such that $h < 0$ in $D$, $h = 0$ on $\partial D$, and $h > 0$ in $\bar{D}^{c}$. This class of degenerate one-phase free boundary problems arises when a canonical transformation is performed to a semilinear equation $\Delta u = f(u)$, and $f$ behaves like $-\gamma u^{-(\gamma + 1)}$ for some $\gamma \in (0,2)$. In this case, known as the Alt-Phillips equation for negative power potentials, $h(\rho) = c(|\rho|^2 - 1)$. We show existence of a viscosity solution, Lipschitz regularity, and regularity of the free boundary at flat points. Additionally, we show that as $\gamma$ converges to $2$, the free boundary converges to a minimal surface.