AI 中文总结
本文研究半空间中带孤立边界奇点的临界半线性椭圆型方程，解决两个开放问题，建立正对数周期解全局分支，构造集中族并证局部唯一性，其行为与经典内部奇点理论形成鲜明对比。
AI 中文摘要
继第二作者（2017）的工作之后，我们研究半空间中具有孤立边界奇点和零狄利克雷边界条件的索伯列夫临界半线性椭圆型方程。本文解决了该背景下的两个开放问题：del Pino-Musso-Pacard（2007）提出的Delaunay型对数周期解的存在性，以及Bidaut-Véron-Ponce-Véron（2007）提出的奇异解的渐近分类。我们建立了正对数周期解的全局分支，其 blow-up 发生在唯一确定的周期处；还构造了对应的集中族并证明其局部唯一性。因此，预期的定常渐近分类不成立，且在整个半空间中不存在通用的临界尺度不变上界，该行为与Caffarelli-Gidas-Spruck（1989）的经典内部奇点理论形成鲜明对比。
英文摘要
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).
Comments56 pages
