In the first part of this thesis, we are interested in representing homotopy groups by p-harmonic maps with applications to minimal submanifolds of ellipsoids. In the second part, we discuss Liouville-type theorems for p-harmonic or p-stable maps into either a closed upper-half ellipsoid or a p-SSU ellipsoid. In the third part, we are interested in existence and non-existence of stable rectifiable currents on an ellipsoid. In the forth part, we study ellipsoids as geometric applications of Yang-Mills instabilities of convex hypersurfaces. In the fifth part, we verify that all of conclusions in the above topological, analytic and geometric theorems on ellipsoids are still valid on compact convex hypersurfaces. In the last part, we make sharp global integral estimates by a unified method, and find a dichotomy between constancy and infinity of weak sub- and supersolutions of a large class of degenerate and singular nonlinear partial differential equations on complete noncompact Riemannian manifolds.