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A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations

Jérôme Vétois

Abstract

We establish pointwise a priori estimates for solutions in $D^{1,p}(\mathbb{R}^n)$ of equations of type $-Δ_pu=f(x,u)$, where $p\in(1,n)$, $Δ_p:=\mbox{div}\big(\left|\nabla u\right|^{p-2}\nabla u\big)$ is the $p$-Laplace operator, and $f$ is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli-Ramaswamy, we are able to extend a recent result of Damascelli-Merchán-Montoro-Sciunzi on the symmetry of positive solutions in $D^{1,p}(\mathbb{R}^n)$ of the equation $-Δ_pu=u^{p^*-1}$, where $p^*:=np/(n-p)$.

A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations

Abstract

We establish pointwise a priori estimates for solutions in of equations of type , where , is the -Laplace operator, and is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli-Ramaswamy, we are able to extend a recent result of Damascelli-Merchán-Montoro-Sciunzi on the symmetry of positive solutions in of the equation , where .

Paper Structure

This paper contains 1 section, 1 theorem, 4 equations.

Table of Contents

  1. Introduction and main results

Key Result

Theorem 1.1

Let $p\in$1,n$$, $f:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}$ be a Caratheodory function such that Eq2 holds true and $u$ be a solution of Eq1. Then there exists a constant $C_0=C_0$n,p,Λ,u$$ such that for all $x\in\mathbb{R}^n$. If moreover $u\ge0$ in $\mathbb{R}^n$ and $\int_{\mathbb{R}^n}f$x,u$dx>0$, then we have for all $x\in\mathbb{R}^n$, for some constant $C_1=C_1$n,p,λ,Λ,u$>0$, where $\l

Theorems & Definitions (1)

  • Theorem 1.1