Abstract As shown in [17] , [18] , for the hydrodynamic model of semiconductors represented by Euler-Poisson equations with sonic boundary and subsonic/supersonic doping profile, the structure of stationary solutions are very complicated. It may possess various solutions like subsonic/supersonic/transonic flows. In this paper, we consider a more challenging case where the doping profile is transonic, which is categorized into two types: subsonic-dominated and supersonic-dominated. In the subsonic-dominated case, we show that the system has a unique interior subsonic solution, at least one interior supersonic solution and infinitely many transonic solutions under the suitable assumptions. However, the difference with the case of subsonic doping profile is that the interior subsonic solution and interior supersonic solution may not exist in special cases when the relaxation time is small. In the supersonic-dominated case, the non-existence and existence of all types of solutions are also obtained. The approach adopted is the technical compactness analysis combining the Green's function method. Here, the results obtained perfectly develop the existing studies.

中文翻译：

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具有声波边界和跨声速掺杂分布的半导体稳态流体动力学模型

摘要 如[17]、[18]所示，对于以欧拉-泊松方程为代表的具有声波边界和亚声/超音速掺杂分布的半导体流体动力学模型，其稳态解的结构非常复杂。它可能拥有各种解决方案，如亚音速/超音速/跨音速流动。在本文中，我们考虑了一个更具挑战性的情况，其中掺杂分布是跨音速的，分为两种类型：亚音速主导和超音速主导。在亚音速主导的情况下，我们证明了系统在适当的假设下具有唯一的内部亚音速解、至少一个内部超音速解和无限多个跨音速解。然而，与亚音速掺杂分布的情况不同的是，在弛豫时间小的特殊情况下，内部亚音速溶液和内部超声速溶液可能不存在。在超音速占优的情况下，也得到了所有类型解的不存在和存在。采用的方法是结合格林函数法的技术紧凑性分析。在这里，获得的结果完美地发展了现有的研究。