  The main concern of this paper is to study large-time behavior of the sheath
to the full Euler-Poisson system. As is well known, the monotone stationary
solution under the Bohm criterion can be referred to as the sheath which is
formed by interactions of plasma with wall. So far, the existence and
asymptotic stability of stationary solutions in one-dimensional half space to
the full Euler-Poisson system have been proved in \cite{DYZ2021}. In the
present paper, we extend the results in \cite{DYZ2021} to $N$-dimensional
($N$=1,2,3) half space. By assuming that the velocity of the positive ion
satisfies the Bohm criterion at the far field, we establish the global unique
existence and the large time asymptotic stability of the sheath in some
weighted Sobolev spaces by weighted energy method. Moreover, the time-decay
rates are also obtained. A key different point from \cite{DYZ2021} is to derive
some boundary estimates on the derivative of the potential in the
$x_1$-direction.
