The analysis of semilinear variational elliptic equations becomes essential because of several significant problems emerging in differential geometry. I aim to focus on such equations in one of the geometric spaces, the simplest model motivated by the concern of prescribing curvature in the non-compact setting called the Hyperbolic Space. The conformal ball model of the hyperbolic space can be described as a Riemannian manifold. The hyperbolic space is very rich in the sense that it retains some properties of the Euclidean space and has some beautiful properties of its own. Furthermore, the Yamabe problem is one of the most historically significant examples that arise in differential geometry and has been a motivation to study critical exponent problems. During my Ph.D. I want to explore the complete profile decomposition of solutions to problems like the Brezis-Nirenberg and the Choquard equations on the hyperbolic space.