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 $\mathbb{R}^N$, $-\Delta_p u =u^{p^*-1}$ 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. 
Lucio Damascelli, Susana Merchan, Luigi Montoro and Berardino Sciunzi
math.AP