2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222591In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.some typos corrected, the introduction has been rewritten and several references has been added, to appear in Journal of Functional AnalysisDifferential Geometrytext