Radial solutions of the hydrodynamic model of semiconductors with sonic boundary

The purpose of this paper is to study radial solutions for steady hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary. The existence and uniqueness of radial subsonic solution, and the existence of radial supersonic solutions are derived by using the energy method and the compactness method, but under a general condition of the doping profile. In particular, for radial supersonic solutions, it is more difficult to get the related estimates by the effect of high dimensional space and the sonic boundary, so we apply a special iteration to complete the proofs. The results obtained essentially improve and develop the previous studies in the one-dimensional case.