Modica type estimates and curvature results for overdetermined $p$-Laplace problems

In this paper we prove Modica type estimates for the following overdetermined $p$-Laplace problem \begin{equation*} \begin{cases} \mathrm{div} \left(|\nabla u|^{p-2}\nabla u\right)+f(u) =0& \mbox{in $\Omega$, } u>0 &\mbox{in $\Omega$, } u=0 &\mbox{on $\partial\Omega$, } \partial_{\nu} u=-\kappa &\mbox{on $\partial\Omega$, } \end{cases} \end{equation*} where $1<p<+\infty$, $f\in C^1(\mathbb{R})$, $\Omega \subset \mathbb{R}^n$ ($n\geq 2$) is a $C^1$ domain (bounded or unbounded), $\nu$ is the exterior unit normal of $\partial \Omega$ and $\kappa\geq 0$ is a constant. Based on Modica type estimates, we obtain rigidity results for bounded solutions. In particular, we prove that if there exists a nonpositive primitive $F$ of $f$ satisfying $F(0)\geq -(p-1)\kappa^p / p$ (for $p>2$ we also assume that if $F(u_0)=0$, $F(u)=O(|u-u_0|^p)$ as $u\rightarrow u_0$), then either the mean curvature of $\partial \Omega$ is strictly negative or $\Omega$ is a half-space.