  We study degenerate complex Monge-Amp\`ere equations of the form
$(\omega+dd^c \varphi)^n = e^{t \varphi} \mu$ where $\omega$ is a big
semi-positive form on a compact K\"ahler manifold $X$ of dimension $n$, $t \in
\R^+$, and $\mu=f\omega^n$ is a positive measure with density $f\in
L^p(X,\omega^n)$, $p>1$. We prove the existence and unicity of bounded
$\omega$-plurisubharmonic solutions. We also prove that the solution is
continuous under a further technical condition.
  In case $X$ is projective and $\omega=\psi^*\omega'$, where $\psi:X\to V$ is
a proper birational morphism to a normal projective variety, $[\omega']\in
NS_{\R} (V)$ is an ample class and $\mu$ has only algebraic singularities, we
prove that the solution is smooth in the regular locus of the equation.
  We use these results to construct singular K\"ahler-Einstein metrics of
non-positive curvature on projective klt pairs, in particular on canonical
models of algebraic varieties of general type.
