Topological Effects on Bubbling in the Critical Dirichlet Problem on Hyperbolic Domains: $3\le N\le 5$

Let $\mathrm{\Omega }⋐{\mathbb{H}}^N$, $N\in \{3,4,5\}$, be a bounded connected $C^2$ domain. We prove that the pure critical Dirichlet problem $$-{\mathrm{\Delta }}_{\mathbb{H}}u=u^{\frac{N+2}{N-2}}\text{in }\mathrm{\Omega },u=0\text{on }\mathrm{\partial }\mathrm{\Omega }$$ admits a positive solution whenever $H_d(\mathrm{\Omega };{\mathbb{F}}_2)\mathrm{\ne }0$ for some $1\le d\le N-1$. This gives a hyperbolic Bahri-Coron theorem for $3\le N\le 5$ under $C^2$ boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential.