The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II

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 establish a global branch of positive log-periodic solutions along which blow-up 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 scale-invariant upper bound can hold throughout the half-space. This behavior contrasts sharply with the classical interior singularity theory of Caffarelli–Gidas–Spruck (1989).