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In this paper we consider positive C 1  solutions of the equation - div(|Du| p-2 Du)= f(u) in ℝ N , vanishing at infinity, in the case 1 &lt; 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, s 0 ), s 0  &gt; 0, and improve a previous result of the authors and Pacella [7] where the symmetry was proved under the hypothesis u ∈ C 1 (ℝ N ) ⋂ W 1,p (ℝ N ). In the second case, previously studied when p = 2, we assume that f(u) = O(u α+1 ) (u → O), and prove the radial symmetry of the solution u, providedat infinity with m(α+2-p) &gt; p. These results extend to pLaplace equations, 1 &lt; p &lt; 2, analogous symmetry results previously known in the case of strictly elliptic equations. The proofs exploit some Poincaré and Hardy-Sobolev type inequalities

Symmetry of C 1 Solutions of p-Laplace Equations in ℝ N