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

Abstract

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_ν u=-κ&\mbox{on $\partial \Omega$, } \end{cases} \end{equation*} where $1<p<+\infty$, $f\in C^1(\mathbb{R})$, $\Omega \subset {\mathbb{R}}^n$ ($n\ge 2$) is a $C^1$ domain (bounded or unbounded), $\nu$ is the exterior unit normal of $\partial \Omega$ and $\kappa \ge 0$ 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 $F(0)\ge -(p-1){\kappa }^p/p$ (for $p>2$ we also assume that if $F(u_0)=0$, $F(u)=O(|u-u_0{|}^p)$ as $u\to u_0$), then either the mean curvature of $\partial \Omega$ is strictly negative or $\Omega$ is a half-space.