In this paper, we investigate the well-posedness/ill-posedness of the stationary solutions to the isothermal bipolar hydrodynamic model of semiconductors driven by Euler–Poisson equations. Here, the density of electrons is proposed with sonic boundary and considered in interiorly subsonic case or interiorly supersonic case, while the density of holes is considered in fully subsonic case or fully supersonic case. With the developed technique based on the topological degree method, the following four kinds of stationary solutions under some conditions are proved to exist: the interiorly-subsonic-vs-fully-subsonic solution, the interiorly-supersonic-vs-fully-subsonic solution, the interiorly-subsonic-vs-fully-supersonic solution, and the interiorly-supersonic-vs-fully-supersonic solution. The non-existence of the above four kinds of solutions under some conditions is also technically proved. For the existence of these physical solutions, different from the previous studies, where traditional fixed-point argument via energy estimates is used, we recognize that such an approach fails for our cases, due to that the effect of boundary degeneracy for the electrons causes difficulty in estimating the upper and lower bounds for the holes. Instead of it, we use the topological degree method to prove the existence of physical solutions.

中文翻译：

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具有声边界的半导体双极流体动力学模型的亚音速和超音速稳态

在本文中，我们研究了由Euler-Poisson方程驱动的半导体等温双极流体力学模型的固定解的适定性/不适定性。这里，电子的密度被提出具有声边界，并且在内部亚声速情况或内部超音速情况下被考虑，而空穴的密度在完全亚声速情况或完全超音速情况下被考虑。利用基于拓扑度法的先进技术，证明了在一定条件下存在以下四种固定解：内部亚音速与全亚音速解，内部超音速与全亚音速解，内部超音速与全超音速解决方案，以及内部超音速VS -完全超音速溶液。还从技术上证明了上述四种解在一定条件下不存在。对于这些物理解决方案的存在，与以前的研究不同，以前的研究中使用了通过能量估计的传统定点参数，我们认识到这种方法在我们的情况下是失败的，因为电子的边界简并性的影响导致了困难。估计孔的上下限。取而代之的是，我们使用拓扑度方法来证明物理解的存在。