In this thesis, we study the continuity method on Calabi-Yau fibrations, which is e-quivalent to studying certain degenerate complex Monge-Ampère equations. Firstly, for a general Calabi-Yau fibration, we prove the weak limit of continuity method is the pullback of generalized Kähler-Einstein current on the base. Moreover, the convergence takes place in C⁰-topology away from singular fibers. Secondly, we show for certain Calabi-Yau fibration that the metric completion of the regular part of generalized Kahler-Einstein current on the base is compact and is the u-nique Gromov-Hausdorff limit of the continuity method, confirming some special cases of conjectures in Analytic Minimal Model Program.