We study homogeneous solutions to the Alt-Phillips problem when the exponent is close to 1. In dimension d3, we show that the radial cone is minimizing when 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 =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 =1. In particular our results show that, when <1 is sufficiently close to 1, there are axis symmetric cones that exhibit the properties of both end point cases =0 and =1.
Savin et al. (Tue,) studied this question.