Steady Hydrodynamic Model of Semiconductors with Sonic Boundary: (I) Subsonic Doping Profile

This series of papers concerns the structure of stationary solutions to the hydrodynamic model of semiconductors with sonic boundary represented by Euler--Poisson equations. The physical solutions are characterized according to different types of doping profiles. In the first part of the series, we consider the case of the subsonic doping profile and prove that the steady-state equations with sonic boundary possess a unique interior subsonic solution, at least one interior supersonic solution, infinitely many shock transonic solutions when the relaxation time is large, and infinitely many C¹-smooth transonic solutions when the relaxation time is small. In particular, the interior subsonic/supersonic solutions are proved to be globally C1/2 Hölder continuous, and the Hölder exponent 1/2 is optimal. The regularity of transonic solutions is dependent on the size of the relaxation time, equivalently, the effect of semiconductors. The proof of the existence of subsonic/supersonic solutions is the technical compactness analysis combining the energy method and the phase-plane analysis, while the approach for the existence of multiple shock/smooth transonic solutions is the artful construction. The results obtained significantly improve and develop existing studies.