  In this paper, we prove the existence of two-dimensional solutions to the
steady Euler-Poisson system with continuous transonic transitions across sonic
interfaces of codimension 1. First, we establish the well-posedness of a
boundary value problem for a linear second order system that consists of an
elliptic-hyperbolic mixed type equation with a degeneracy occurring on an
interface of codimension 1, and an elliptic equation weakly coupled together.
Then we apply the Schauder fixed point theorem to prove the existence of
two-dimensional solutions to the potential flow model of the steady
Euler-Poisson system with continuous transonic transitions across sonic
interfaces. With the aid of Helmholtz decomposition, established in [6], we
extend the existence result to the full Euler-Poisson system for the case of
nonzero vorticity. Most importantly, the solutions constructed in this paper
are classical solutions to Euler-Poisson system, thus their sonic interfaces
are not weak discontinuities in the sense that all the flow variables are $C^1$
across the interfaces.
