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

We establish pointwise a priori estimates for solutions in D1,p(Rn)D1,p(Rn) of equations of type −Δpu=f(x,u)−Δpu=f(x,u), where p∈(1,n)p∈(1,n), Δp:=div(|∇u|p−2∇u)Δp:=div(|∇u|p−2∇u) is the p-Laplace operator, 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 [6], we are able to extend a recent result of Damascelli–Merchán–Montoro–Sciunzi [7] on the symmetry of positive solutions in D1,p(Rn)D1,p(Rn) of the equation −Δpu=up⁎−1−Δpu=up⁎−1, where p⁎:=np/(n−p)p⁎:=np/(n−p).