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

Massimo Grossi, Luigi Montoro, Berardino Sciunzi, Zexi Wang

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.

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

Abstract

In this paper, we study the critical points of stable solutions for the following -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 , satisfies for , and is a smooth bounded domain with non-negative curvature of the boundary. Via a suitable approximation argument, we prove that, a stable solution admits, as its only critical point, the internal absolute maxima and possibly saddle points with zero index. Moreover, is a point or segment.

Paper Structure

This paper contains 4 sections, 15 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction and statement of the main result
  2. The approximation argument
  3. Proof of Theorem \ref{['th1']}
  4. Proof of Theorem \ref{["th2'"]}

Key Result

Theorem 1.1

Assume $f\in C^1([0,+\infty))$ satisfies $f(t)>0$ for $t>0$, and $\Omega\subset\mathbb{R}^2$ is a smooth bounded domain whose boundary has positive curvature. Suppose that $u$ is a stable solution of problem 1. Then the critical points of $u$ has only the internal absolute maxima and possibly saddle

Figures (1)

  • Figure 1: A picture of $N_\theta$ in Proposition \ref{['prop3']}

Theorems & Definitions (27)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • ...and 17 more