We investigate the distribution of the electron density and the potential in a quantum wire coupled to reservoirs, treating this structure as a unified quantum system and taking into account the Coulomb interaction of electrons. The chemical potential difference that exists between a decoupled, isolated quantum wire and the reservoirs gives rise to charge transfer in the coupled system. We show that the quantum wire can be charged positively or negatively or remain neutral as a whole, depending on such factors as the wire radius and the background charge density in the wire. The magnitude of the charge and its sign are to a large extent determined by the exchange interaction of the electrons in the wire. Using a Hartree-Fock approach, we develop a model of a quantum wire which includes the reservoirs. This model allows us to find the self-consistent distribution of the electron density and the potential in the wire both at equilibrium and in the presence of transport. The linear conductance is investigated as a function of the chemical potential. The nonadiabatic transition from the reservoirs to the wire leads to conductance oscillations caused by multiple scattering of electron waves. The period of the oscillations depends on the charge acquired by the wire and the exchange energy. We find that the exchange interaction strongly enhances the Friedel oscillations near the contacts. However, they do not noticeably suppress the conductance because the wire has a finite length and is charged. Under far from equilibrium conditions, which appear when the applied voltage exceeds the Fermi energy in the wire, the system becomes unstable with respect to fluctuations of the electric potential and the electron density. The instability results in the appearance of multistable electron states.