[Assess graduate-level mastery of sharp Sobolev inequalities and the existence/uniqueness (up to symmetries) of extremal functions for critical nonlinear elliptic PDEs. Evaluate ability to compute and justify best constants, apply concentration-compactness and profile decomposition, analyze blow-up/compactness scenarios, use variational methods (Palais–Smale, constrained minimization), and deploy PDE tools (moving planes, symmetrization) to prove existence, nonexistence, and uniqueness/classification results for model problems such as critical Sobolev embeddings and Yamabe-type equations.]