  In this paper, we prove the existence and stability of subsonic flows for
steady full Euler-Poisson system in a two dimensional nozzle of finite length
when imposing the electric potential difference on non-insulated boundary from
a fixed point at the entrance, and prescribing the pressure at the exit of the
nozzle. The Euler-Poisson system for subsonic flow is a hyperbolic-elliptic
coupled nonlinear system. One of the crucial ingredient of this work is the
combination of Helmholtz decomposition for the velocity field and stream
function formulation together. In terms of the Helmholtz decomposition, the
Euler-Poisson system is rewritten as a second order nonlinear elliptic system
of three equations and transport equations for entropy and pseudo-Bernoulli's
invariant. The associated elliptic system in a Lipschitz domain with nonlinear
boundary conditions is solved with the help of the estimates developed in [2]
based on its nice structure. The transport equations are resolved via the flow
map induced by the stream function formulation. Furthermore, the delicate
estimates for the flow map give the uniqueness of the solutions.
