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 $D^{1,p}({\mathbb{R}}^n)$ of equations of type $-{\Delta }_{p}u=f(x,u)$, 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 $-{\Delta }_{p}u=u^{p^{\ast }-1}$, where $p^{\ast }:=np/(n-p)$.

Cited by in corpus (3)

• Asymptotic Behaviors of Solutions to quasilinear elliptic Equations with critical Sobolev growth and Hardy potential
• Decay estimates and a vanishing phenomenon for the solutions of critical anisotropic equations
• Monotonicity in half-spaces of positive solutions to $-{\Delta }_{p}u=f(u)$ in the case $p>2$