An extension of Cabr\'{e}-Chanillo theorem to the $p$-laplacian

TL;DR

This paper extends the Cabré-Chanillo theorem to the p-Laplacian, analyzing stable solutions' critical points in bounded domains, showing maxima are unique or form segments, with saddle points possibly present.

Contribution

It generalizes the Cabré-Chanillo theorem to the p-Laplacian, providing new insights into the structure of stable solutions in two-dimensional domains.

Findings

Stable solutions have only internal maxima and saddle points with zero index.

The maximum set is either a point or a segment.

The results apply to smooth bounded domains with non-negative boundary curvature.

Abstract

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 $p>2$, $f\in C^1([0,+\infty))$ satisfies $f(t)>0$ for $t>0$, and $\Om\subset\R^2$ 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.