Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the classification of positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$

Jérôme Vétois

Abstract

In this note, we obtain a classification result for positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$ with $n\ge4$ and $p>p_n$ for some number $p_n\in\left(\frac{n}{3},\frac{n+1}{3}\right)$ such that $p_n\sim\frac{n}{3}+\frac{1}{n}$, which slightly improves upon a similar result recently obtained by Ou under the condition $p\ge\frac{n+1}{3}$.

A note on the classification of positive solutions to the critical p-Laplace equation in $\mathbb{R}^n$

Abstract

In this note, we obtain a classification result for positive solutions to the critical p-Laplace equation in with and for some number such that , which slightly improves upon a similar result recently obtained by Ou under the condition .
Paper Structure (2 sections, 2 theorems, 11 equations)

This paper contains 2 sections, 2 theorems, 11 equations.

Table of Contents

  1. Introduction and main result
  2. Proof of Theorem \ref{['Th']}

Key Result

Theorem 1.1

Assume that $n\ge4$ and $p_n<p<n$, where Then every positive, weak solution $u\in W^{1,p}_{\mathop{\mathrm{loc}}\nolimits}$R^n$\cap L^\infty_{\mathop{\mathrm{loc}}\nolimits}$R^n$$ to IntroEq1 is of the form IntroEq2, i.e. $u\equiv u_{\mu,x_0}$ for some $\mu>0$ and $x_0\in\mathbb{R}^n$.

Theorems & Definitions (2)

  • Theorem 1.1
  • Lemma 2.1