Classification of solutions of an elliptic Hamilton-Jacobi equation

We show that any classical solution of the diffusive Hamilton-Jacobi (DHJ) equation $-\mathrm{\Delta }u=\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^p$ in a half-space with zero boundary conditions for $1<p\le 2$ is necessarily one-dimensional. This improves the previously known result, which required an extra assumption of boundedness from above. Combined with the existing analogous result for $p>2$, our result completes the full classification picture of the Dirichlet problem for equation (DHJ) in a half-space.