An extension of Cabré-Chanillo theorem to the $p$-laplacian

In this paper, we study the critical points of stable solutions for the following $p$-laplacian equation \begin{equation*} \begin{cases} -div\big(|\nabla u|^{p-2}\nabla u\big)=f(u)&in\ \Om,\ u>0&in\ \Om,\ u=0&on\ \partial\Om, \end{cases} \end{equation*} where $p>2$, $f\in C^1([0,+\mathrm{\infty }))$ satisfies $f(t)>0$ for $t>0$, and $\text{Om}\subset {\mathbb{R}}^2$ is a smooth bounded domain with non-negative curvature of the boundary. Via a suitable approximation argument, we prove that, a stable solution $u$ admits, as its only critical point, the internal absolute maxima and possibly saddle points with zero index. Moreover, $Argmax(u)$ is a point or segment.