Radial solutions of hydrodynamic model of semiconductors with sonic boundary

The purpose of this paper is to study the multi-dimensional steady hydrodynamic model semiconductors represented by Euler-Poisson equations with sonic boundary. We prove that, boundary possess a unique subsonic solution and at least one supersonic in radial form. adopted approach for proof energy method combining compactness analysis. For n-D solutions, since it more challenging get crucial estimates due effect multiple dimensions restriction boundary, we propose brand new two-steps iteration scheme build up key estimates. This first attempt steady-states results obtained essentially improve develop previous studies one-dimensional case.