  We study the incompressible stationary Navier-Stokes equations in the
upper-half plane with homogeneous Dirichlet boundary condition and non-zero
external forcing terms. Existence of weak solutions is proved under a suitable
condition on the external forces. Weak-strong uniqueness criteria based on
various growth conditions at the infinity of weak solutions are also given.
This is done by employing an energy estimate and a Hardy's inequality. Several
estimates of stream functions are carried out and two density lemmas with
suitable weights for the homogeneous Sobolev space on 2 dimensional space are
proved.
