Symmetry of C1 Solutions of p-Laplace Equations in ℝN

Abstract

Abstract In this paper we consider positive C1 solutions of the equation - div(|Du|p-2Du)= f(u) in ℝN, vanishing at infinity, in the case 1 < p ≤ 2, f locally Lipschitz continuous in (0, ∞). We prove that the solutions are radially symmetric under two different sets of assumptions on the behaviour of f near zero. In the first case we assume that f is nonincreasing in (0, s0), s0 > 0, and improve a previous result of the authors and Pacella [7] where the symmetry was proved under the hypothesis u ∈ C1(ℝN) ⋂ W1,p(ℝN). In the second case, previously studied when p = 2, we assume that f(u) = O(uα+1) (u → O), and prove the radial symmetry of the solution u, provided at infinity with m(α+2-p) > p. These results extend to pLaplace equations, 1 < p < 2, analogous symmetry results previously known in the case of strictly elliptic equations. The proofs exploit some Poincaré and Hardy-Sobolev type inequalities