2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/100899Given a compact four dimensional manifold, we prove existence of conformal metrics with constant $Q$-curvature under generic assumptions. The problem amounts to solving a fourth-order nonlinear elliptic equation with variational structure. Since the corresponding Euler functional is in general unbounded from above and from below, we employ topological methods and minimax schemes, jointly with a compactness result by the second author.36 pages, revised version. To appear in Annals of MathematicsAnalysis of PDEs35B33; 35J35; 53A30; 53C21text