In this talk, we consider the center Lyapunov exponents of a large class of partially hyperbolic diffeomorphisms with one-dimensional center, which includes the classical examples of robustly transitive perturbations of skew-products, derived from Anosov diffeomorphisms and the time one-map of transitive Anosov flows, as well as the new anomalous examples. We consider the level sets of center Lyapunov exponents, and show that the entropy varies continuously with respect to the center Lyapunov exponent. We also obtain the restricted variational principle, except at the zero-level set. This is a joint work with L. Diaz, K. Gelfert and B. Santiago.