4 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) of equations of type -Δₚu=f(x,u), where p∈(1,n), 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, we are able to extend a recent result of Damascelli-Merch\'an-Montoro-Sciunzi on the symmetry of positive solutions in D1,p(Rⁿ) of the equation -Δₚu=up^*-1, where p^*:=np/(n-p).