  By using the variational approach, we prove the existence of
Sinai-Ruelle-Bowen measures for partially hyperbolic $\mathcal C^1$
diffeomorphisms with mostly expanding properties. The same conclusion holds
true if one considers a dominated splitting $E\oplus F$, where $\dim E=1$ and
$F$ is mostly expanding. When the diffeomorphisms are $\mathcal C^{1+\alpha}$,
we prove the basin covering property for both cases.
