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

We consider weak non-negative solutions to the critical $p$-Laplace equation in $\mathbb{R}^N$ \begin{equation}\nonumber -\Delta_p u =u^{p^*-1}\,, \end{equation} in the singular case $1<p<2$. We prove that if %the nonlinearity is locally Lipschitz continuous, namely $p^*\geqslant2$ then all the solutions in ${\mathcal D}^{1,p}(\R^N)$ are radial (and radially decreasing) about some point.