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