A nonlinear elliptic problem involving the gradient on a half space

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.