  We study a nonlinear equation in the half-space $\{x_1>0\}$ with a Hardy
potential, specifically \[-\Delta u
-\frac{\mu}{x_1^2}u+u^p=0\quad\text{in}\quad \mathbb R^n_+,\] where $p>1$ and
$-\infty<\mu<1/4$. The admissible boundary behavior of the positive solutions
is either $O(x_1^{-2/(p-1)})$ as $x_1\to 0$, or is determined by the solutions
of the linear problem $-\Delta h -\frac{\mu}{x_1^2}h=0$. In the first part we
study in full detail the separable solutions of the linear equations for the
whole range of $\mu$. In the second part, by means of sub and supersolutions we
construct separable solutions of the nonlinear problem which behave like
$O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have
exactly the same asymptotic behavior as the separable solutions of the linear
problem. In the last part we construct solutions that behave like
$O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest
of the boundary the solutions decay or blowup at a slower rate determined by
the linear part of the equation.
