Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic model of semiconductors

We consider a one-dimensional bipolar hydrodynamic model of semiconductors. Although some results exist for the bipolar case, almost their conditions (the boundary condition, the doping profile, etc.) are far from practical application. In the present paper, under a condition appropriate for engineering, we shall prove the existence and the uniqueness of classical solutions for the stationary problem. The most difficult point is to obtain the bounded estimate and the energy estimate.