  We consider perturbations of the diffusive Hamilton-Jacobi equation
\begin{equation*} %\label{non_pert}
  \left\{ \begin{array}{lcl} \hfill -\Delta u &=& (1+g(x))| \nabla u|^p\qquad
\mbox{ in } \IR^N_+, \\ \hfill u &=& 0 \hfill \mbox{ on } \partial \IR^N_+,
\end{array}\right.
  \end{equation*}
  for $ p>1$. We prove the existence of a classical solution provided $ p \in
(\frac{4}{3},2)$ and $g$ is bounded with uniform radial decay to zero.
