Existence and uniqueness of solutions to a class of quantum hydrodynamic model for semiconductors

In recent years, people begin to concern all kinds of mathematical models for the ultra-small semiconductor systems in physics and quantum theory. Those models include the macroscope and microscope quantum models. The microscope quantum model, which deals with the quantum technology, involves Wigner Equation, Shrodinger-Poinsson Equation etc. The so-called quantum hydrodynamic model (QHD) includs macroscope physical quantities like electron density and current density and it is used in the quantum semiconductor device such as the resonance diode. Thus QHD model has played an important role in the mathematical fields. This paper is concerned with a steady-state quantum hydrodynamic model with the Dirichlet boundary conditions for the electron density and the potential in the isentropic case. The model includes a quantum Bohm potential and a forcing term in the developing system. The electric current density and the particle density are coupled to the Poisson equation. The existence and uniqueness are obtained for a classical positive solution if the electron density is small and no weak solution can exist for a large data. The proofs are based on a reformation of the equations, Leray-Schauder´s fixed point theorem and a truncation technique.