3D Subsonic Solution to Steady Euler–Poisson Equations for Semiconductors in Bounded Domain

Abstract. This paper is concerned with the existence of genuine 3D large amplitude subsonic steady-state for the isentropic Euler–Poisson system when the flow sufficiently approaches the sonic speed, namely, the Mach number [Formula: see text] but sufficiently close to 1. We first choose a subsonic radial solution as a background solution, and then we investigate the existence of a genuine 3D solution as a perturbation of the radial background solution. By utilizing estimates for a quasi-linear elliptic system, we prove that when [Formula: see text] with [Formula: see text] is a positive constant, where [Formula: see text] is the relaxation time and [Formula: see text] is the largest Mach number of the background solution, the perturbation of the background solution exists for the potential flow. Thus, we obtain the unique genuine 3D subsonic steady-state solution [Formula: see text] around the background state with large amplitude in [Formula: see text] with [Formula: see text]. Although the targeted system under consideration is still subsonic—even for the flow with the Mach number sufficiently close to 1—the genuine 3D subsonic solution still exists when the relaxation time is big enough at [Formula: see text]. This result significantly improves and develops the existing studies.