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

Abstract

We classify non-trivial, non-negative, positively homogeneous solutions of the equation $$ \Delta u=\gamma u^{\gamma-1} $$ 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.