2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/72768We prove compactness of solutions to some fourth order equations with exponential nonlinearities on four manifolds. The proof is based on a refined bubbling analysis, for which the main estimates are given in integral form. Our result is used in a subsequent paper to find critical points (via minimax arguments) of some geometric functional, which give rise to conformal metrics of constant $Q$-curvature. As a byproduct of our method, we also obtain compactness of such metrics.32 pages, fixed some bug in the previous versionAnalysis of PDEs35B33; 35J35; 53A30; 53C21text