  We analyse a stationary problem for the two dimensional model of morphogen
transport introduced by Hufnagel et al. The model consists of one linear
elliptic PDE posed on $(-1,1)\times(0,h)$ which is coupled via a nonlinear
boundary condition with a nonlinear elliptic PDE posed on $(-1,1)\times\{0\}$.
The main result is that the system has a unique steady state for all ranges of
parameters present in the system. Moreover we consider the problem of the
dimension reduction. After introducing an appropriate scaling in the model we
prove that, as $h\to0$, the stationary solution converges to the unique steady
state of the one dimensional simplification of the model which was analysed in
the first part of the paper. The main difficulty in obtaining appropriate
estimates stems from the presence of a measure source term in the boundary
condition.
