Stable and Minimizing Cones in the Alt-Phillips Problem

We study homogeneous solutions to the Alt-Phillips problem when the exponent γ is close to 1. In dimension d≥3, we show that the radial cone is minimizing when γ is close to 1. In dimension d ≥ 4, we construct an axially symmetric cone whose contact set has with positive density. We show that it is a global minimizer. It is analogous to the De Silva-Jerison {DJ} cone for the Alt-Caffarelli functional which corresponds to exponent γ=0. The cone we construct bifurcates from another minimizing cone whose contact set has zero density, obtained as the trivial extension of the radial solution. This second cone is analogous to a quadratic polynomial solution in the classical obstacle problem which corresponds to exponent γ=1. In particular our results show that, when γ<1 is sufficiently close to 1, there are axis symmetric cones that exhibit the properties of both end point cases γ=0 and γ=1.