Classification and asymptotics for the boundary Yamabe equation with an isolated singularity

Abstract

We classify positive singular solutions of the boundary Yamabe equation on the punctured half-space in every dimension $n\ge3$. Previous symmetry results reduce this problem to a nonlinear PDE in two variables, for which classical phase-plane methods are unavailable. We develop a Hamiltonian method that combines local bifurcation with compactness to prove that every nonstationary cylindrical profile of a singular solution is periodic and uniquely determined, up to translation, by its Hamiltonian energy. At each stationary or periodic profile, the space of solutions of the linearized equation is two-dimensional. Finally, every positive solution in a half-ball with a nonremovable isolated boundary singularity is asymptotic to a unique global singular solution, with an algebraically decaying relative error.