Radial symmetry for a quasilinear elliptic equation with a critical Sobolev growth and Hardy potential

Abstract

We consider weak positive solutions to the critical $p$-Laplace equation with Hardy potential in ${\mathbb{R}}^N$ $$-{\Delta }_{p}u-\frac{\gamma }{|x{|}^p}{u}^{p-1}=u^{p^{\ast }-1}$$ where $1<p<N$, $0\le \gamma <{\left(\frac{N-p}{p}\right)}^p$ and $p^{\ast }=\frac{Np}{N-p}$. The main result is to show that all the solutions in ${\mathcal{D}}^{1,p}({\mathbb{R}}^N)$ are radial and radially decreasing about the origin.