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 (R) of equations of type−∆pu = f (x, u), where p ∈ (1, n), ∆p := div ( |∇u|∇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 D (R) of the equation −∆pu = u −1, where p∗ := np/ (n− p).