SRB measures for mostly expanding partially hyperbolic diffeomorphisms via the variational approach

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.