Symmetry of bounded solutions to quasilinear elliptic equations in a half-space

Let $u$ be a bounded positive solution to the problem $-{\mathrm{\Delta }}_{p}u=f(u)$ in ${\mathbb{R}}_{+}^N$ with zero Dirichlet boundary condition, where $p>1$ and $f$ is a locally Lipschitz continuous function. Among other things, we show that if $f({\sup }_{{\mathbb{R}}_{+}^N}u)=0$ and $f$ satisfies some other mild conditions, then $u$ depends only on $x_N$ and monotone increasing in the $x_N$-direction. Our result partially extends a classical result of Berestycki, Caffarelli and Nirenberg in 1993 to the $p$-Laplacian.