A priori estimates and application to the symmetry of solutions for critical $p$-Laplace equations

We establish pointwise a priori estimates for solutions in $D^{1,p}({\mathbb{R}}^n)$ of equations of type $-{\mathrm{\Delta }}_{p}u=f(x,u)$, where $p\in (1,n)$, $\Delta_p:=\mbox{div}\big(\left|\nabla u\right|^{p-2}\nabla u\big)$isthe$p$-Laplaceoperator,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-Merch'an-Montoro-Sciunzi on the symmetry of positive solutions in $D^{1,p}({\mathbb{R}}^n)$ of the equation $-{\mathrm{\Delta }}_{p}u=u^{p^{\ast }-1}$, where $p^{\ast }:=np\mathrm{/}(n-p)$.