A potential flow formulation of the hydrodynamic equations with the quantum Bohm potential for the particle density and the current density is given. The equations are selfconsistently coupled to Poisson's equation for the electric potential. The stationary model consists of nonlinear elliptic equations of degenerate type with a quadratic growth of the gradient. Physically motivated Dirichlet boundary conditions are prescribed. The existence of solutions is proved under the assumption that the electric energy is small compared to the thermal energy. The proof is based on Leray-Schauder's fixed point theorem and a truncation method. The main difficulty is to find a uniform lower bound for the density. For sufficiently large electric energy, there exists a generalized solution (of a simplified system), where the density vanishes at some point. Finally, uniqueness of the solution is shown for a sufficiently large scaled Planck constant.