  We classify nontrivial, nonnegative, positively homogeneous solutions of the
equation \begin{equation*} \Delta u=\gamma u^{\gamma-1} \end{equation*} in the
plane.
  The problem is motivated by the analysis of the classical Alt-Phillips free
boundary problem, but considered here with negative exponents $\gamma$.
  The proof relies on several bespoke results for ordinary differential
equations.
