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By using the variational approach, we prove the existence ofSinai-Ruelle-Bowen measures for partially hyperbolic $C^1$ diffeomorphisms withthe mostly expanding properties. The same conclusion is obtained if one has thedominated splitting $E^{cs}\oplus E^{cu}$, where $\dim E^{cs}=1$ and $E^{cu}$has the mostly expanding properties. When the diffeomorphisms are$C^{1+\alpha}$, we prove the basin covering property for both cases.

On the existence of Sinai-Ruelle-Bowen measures for diffeomorphisms with the mostly expanding properties via the variational approach