The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II

Abstract

Continuing the work of the second author (2017), we study the Sobolev critical semilinear elliptic equation in the half-space with an isolated boundary singularity and zero Dirichlet boundary condition. This paper addresses two open questions in this setting: the existence of Delaunay type log-periodic solutions posed by del Pino--Musso--Pacard (2007), and the asymptotic classification of singular solutions posed by Bidaut-Véron--Ponce--Véron (2007). We construct a global continuum of positive log-periodic solutions containing the local bifurcation branch and prove that blow-up along this continuum occurs at a uniquely determined period. We also construct the corresponding concentrating family and prove its local uniqueness. Consequently, the expected stationary asymptotic classification fails, and no universal critical scaling-invariant upper bound can hold throughout the half-space. This behavior contrasts sharply with the classical interior singularity theory of Caffarelli--Gidas--Spruck (1989).

56 pages