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arXiv · 2608.16106

The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II

Hua-Yang Wang·Jingang Xiong
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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-V\'eron--Ponce--V\'eron (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).

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Hua-Yang Wang, Jingang Xiong. 2026-08-17. The Critical Semilinear Elliptic Equation with Isolated Boundary Singularities II. https://arxiv.org/abs/2608.16106

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