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.