2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160349We prove an extension of a theorem of Barta then we make few geometric applications. We extend Cheng's lower eigenvalue estimates of normal geodesic balls. We generalize Cheng-Li-Yau eigenvalue estimates of minimal submanifolds of the space forms. We prove an stability theorem for minimal hypersurfaces of the Euclidean space, giving a converse statement of a result of Schoen. Finally we prove a generalization of a result of Kazdan-Kramer about existence of solutions of certain quasi-linear elliptic equations.23 pages. This paper is an improved version of our paper of the same Titled posted hereDifferential Geometry58C40, 58J50, 58J32text