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

Abstract

We establish pointwise a priori estimates for solutions in D 1,p (R n ) of equations of type −∆ p u = f (x, u), where p ∈ (1, n), ∆ p := 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 [5], we are able to extend a recent result of Damascelli-Merchán-Montoro-Sciunzi [6] on the symmetry of positive solutions in D 1,p

Classification of positive D1,p(RN)-solutions to the critical p

“…It has been showed in fact in [25, Theorem 1.1] that, under our assumptions (some more general estimates are also considered in [25]), it holds that…”

“…For the reader's convenience we provide an outline of the proofs: i) In Section 2 (see Theorem 2.2 ) we provide a lower bound on the decay rate of |∇u|, namely we show that |∇u(x)| ≥C|x| − N−1 p−1 for some constantC, in R N \ {B R 0 } (for some R 0 > 0). This is the counterpart to the upper bound obtained in [25]. We use here a new technique based on the study of the limiting profile at infinity and we also take advantage of the a priori estimates proved in [25] .…”

“…This is the counterpart to the upper bound obtained in [25]. We use here a new technique based on the study of the limiting profile at infinity and we also take advantage of the a priori estimates proved in [25] .…”

“…Since the case 1 < p < 2 is already known [4,25], we only consider the case p > 2. We have the following: Theorem 3.1.…”

“…This requires that 2N N +2 ≤ p < 2. The result has been extended to the case 1 < p < 2 in [25] exploiting a fine analysis of the behaviour of the solutions at infinity that allows to exploit the moving plane method as developed in [5,6](see also [19]). …”