Seminar| Institute of Mathematical Sciences
Time: Wednesday, December 31th, 2025,14:30-15:30
Location: RS408, IMS
Speaker: Zhipeng Yang, Yunnan Normal University
Abstract: We consider the critical Dirichlet problem on bounded $C^2$ domains in the hyperbolic space. Working in the Poincar\'e ball model, we develop a critical point at infinity theory for the associated energy functional. We prove a Palais--Smale decomposition and energy quantization in multiples of the Talenti energy with no boundary bubbles, and we construct a multi-bubble manifold on which a Lyapunov--Schmidt reduction and a pseudo-gradient flow can be carried out. The resulting reduced functional admits sharp single-bubble and multi-bubble expansions in terms of the hyperbolic Green and Robin functions, which relate the topology and geometry of $\Omega$ to the existence and distribution of positive solutions of the problem.