A self-consistent-field model for the adsorption of flexible weak polyelectrolytes onto surfaces with a variable charge is developed. The chain statistics and the short and long range interactions are described using a lattice model. The degree of dissociation of the chargeable polyelectrolyte segments is allowed to vary with the distance from the surface. Electrostatic potential profiles as well as volume density profiles are evaluated numerically as a function of pH, ionic strength, and segment-solvent/segment-surface interaction parameters. For the case of pure electrosorption of a polyelectrolyte from an athermal aqueous electrolyte solution the adsorbed amount of polyelectrolyte is mainly determined by the compensation of the surface charge. The surface charge is affected slightly by the adsorbed polyelectrolyte, which is caused by the higher effective valence of the polyelectrolyte chain as compared to that of the background electrolyte. In absence of specific interactions, no point of zero charge shift is observed. Specific interactions between polyelectrolyte chains and the surface lead to an increase in the adsorbed amount. Further, depending on the conditions, the changes associated with the adsorbed polyelectrolytes compensate or even overcompensate the surface charge. This affects the local electrostatic potential, and thus both components adjust their initial charge significantly. The component that, at a given pH and salt concentration, has the highest initial charge dominates the local electrostatic potential and dictates to a large extent the degree of charging of the other component. A decrease in solvent quality enhances the effects of specific adsorption. Due to increasing overcompensation of the surface charge by the adsorbed polyelectrolyte charge, the fraction of train segments increases with increasing pH, decreasing salt concentration, and increasing adsorption energy.