Existence and nonlinear stability of steady state solutions of the 3-D Navier–Stokes–Poisson equations with non-flat doping profile of large variations in exterior domains

This paper is concerned with an initial boundary value problem of compressible Navier-Stokes-Poisson equations with the non-flat doping profile in a 3-D exterior domain with the general Navier-slip boundary condition for the velocity and the Neumann boundary condition for the electrostatic potential. First, we prove the existence of the steady state solutions (with zero velocity) by constructing the suitable sub and super solutions for the Neumann problem of the associated second order elliptic equation in the 3-D exterior domain. Second, we establish the global existence of strong solutions near a steady state solution by some new techniques to obtain a priori estimates, especially some refined elliptic estimates and the estimates on the boundary. One feature of the problems considered in this paper is that the doping profile is allowed to have arbitrarily large variation and so does the steady state equilibrium.