Abstract: We study a nonlinear equation in the half-space with a Hardy potential, specifically, \[ \displaystyle - \Delta_p u= \lambda \frac{u ^{p-1}}{x_1^p}-x_1^\theta f(u)\ \ {\rm in}\ T,\] where $\Delta_p$ stands for the $p$-Laplacian operator defined by $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$, $p>1$, $\theta > -p$, and $T$ is a half-space $\{x_1>0\}$. When $\lambda > \Theta$ (where $\Theta$ is the Hardy constant), we show that under suitable conditions on $f$ and $\theta$, the equation has a unique positive solution. Moreover, the exact behavior of the unique positive solution as $x_1\to 0^+$, and the symmetric property of the positive solution are obtained.

Key words: p-Lapacian, Hardy potential, symmetry, uniqueness, asymptotic behavior