We study the Dirichlet problem in half-space for the equation
Δ
u
+
g
(
u
)
|
∇
u
|
2
=
0
,
{\Delta u+g(u)|\nabla u|^2=0,}
where
g
g
is continuous or has a power singularity (in the latter case positive solutions are considered). The results presented give necessary and sufficient conditions for the existence of (pointwise or uniform) limit of the solution as
y
→
∞
,
y\to \infty ,
where
y
y
denotes the spatial variable, orthogonal to the hyperplane of boundary-value data. These conditions are given in terms of integral means of the boundary-value function.