We will apply Theorem 4.1 in section 6 to construct singular Kähler-Einstein metrics on manifolds of general type. This will follow from the resolution of $(\omega+dd^{c}\varphi)^{n}=e^{t\varphi}\mu$ for large enough values of $t>0$. The Monge-Ampère equations

$$(\omega+dd^{c}\varphi)^{n}=e^{-t\varphi}\mu,\;\;t>0,$$

can also be solved with a similar method, but only for small values of $t<t_{X}$. The critical exponent $t_{X}$ depends on the manifold $X$, and may be too small to produce Kähler-Einstein metrics when $c_{1}(X)>0$: even smooth manifolds of positive scalar curvature do not necessarily admit Kähler-Einstein metrics (see [T]). Since technical details are much more involved in this case, we postpone this study to a forthcoming article.