Classification of global solutions of a free boundary problem in the plane

Abstract

We classify nontrivial, nonnegative, positively homogeneous solutions of the equation \begin{equation*} Δu=γu^{γ-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.