Radial symmetry and applications for a problem involving the $-{\mathrm{\Delta }}_p(\cdot )$ operator and critical nonlinearity in~${\mathbb{R}}^N$

We consider weak non-negative solutions to the critical $p$-Laplace equation in ${\mathbb{R}}^N$, $-{\mathrm{\Delta }}_{p}u=u^{p^{\ast }-1}$ in the singular case $1<p<2$. We prove that if the nonlinearity is locally Lipschitz continuous, namely $p^{\ast }⩾2$ then all the solutions in ${\mathcal{D}}^{1,p}({\mathbb{R}}^N)$ are radial (and radially decreasing) about some point.