In biomolecular and colloidal systems, the dielectric environment often depends on the local ionic concentrations. Recent experiments and molecular dynamics simulations have revealed quantitatively such dependence, and indicated that the electrostatic interaction in a system with such a dependence can be significantly different from that predicted by theories assuming a uniform dielectric constant. In this work, we develop a mean-field model for the electrostatic interactions with ionic concentration-dependent dielectrics. We minimize the electrostatic free energy of ionic concentrations using Poisson's equation with a concentration-dependent dielectric coefficient to determine the electrostatic potential. Our analysis leads to the the corresponding generalized Boltzmann distributions of the equilibrium ionic concentrations that is quite different from the usual ones with a uniform dielectric constant We show by constructing an example that the free-energy functional is in fact nonconvex. This implies the existence of multiple local minimizers, a property that can be of physical significance. By numerical computations using our continuum model, we find many interesting phenomena such as the non-monotone ionic concentration profile near a charged surface, and the unusual shift of concentration peak due to the increase of surface charge density. It is clear that the effect of local dielectrics has a significant impact on the system and our models and results have potential applications to large biological systems. This is joint work with Bo Li and Shenggao Zhou.