Sliding method and one-dimensional symmetry for p-Laplace equations

Let f be a sign-changing locally Lipschitz continuous function which satisfies some mild assumptions. We show that every nonnegative solution to the equation \(-\Delta _p u = f(u)\) $-{\mathrm{\Delta }}_{p}u=f(u)$ in \({\mathbb {R}}^N_+\) ${\mathbb{R}}_{+}^N$ with zero boundary condition and a uniform limit must be monotone increasing in \(x_N\) $x_N$ and depends only on \(x_N\) $x_N$ . Both cases \(f(0)\ge 0\) $f(0)\ge 0$ and \(f(0)<0\) $f(0)<0$ will be treated. We exploit a variant of the sliding method for the p-Laplacian to prove our results.