The stationary solution of a one-dimensional bipolar quantum hydrodynamic model

Abstract

In this paper, we consider the existence and uniqueness of stationary solution to the bipolar quantum hydrodynamic model in one dimensional space with general non-constant doping profile. The existence of the stationary solution is proved by Leray-Schauder fixed-point theorem and a crucial truncation technique is used to derive the positive upper and lower bounds of the stationary solution. The uniqueness of the stationary solution is shown by a delicate energy estimate.