Classification of global solutions of a free boundary problem in the plane (Q6082523)

Summary: 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.