We derive a differential identity for a class p-Laplace equation, and then classify all positive finite energy cylindrically symmetric solutions of the equation (1) for 3≤k≤n−1, with the help of some a priori estimates. The Euler–Lagrange equation associated to the inequality is

- Δ p u = u p * ( 1 ) - 1 ‖ y ‖, in ℝ n (1)

among positive functions u belonging to D1,p(ℝn), where

p * (1) = p ( n - 1 ) n - p, x = (y, z), and ℝ n = ℝ k \times ℝ n - k .

As a consequence, we obtain the best constant and the extremal function for the related Hardy–Mazya–Sobolev inequalities. When p=2, the corresponding results was obtained by Mancini–Fabbri–Sandeep in 2006, and Alvino–Ferone–Trombetti posed a conjecture in 2006 for 1<p<n. This is joint work with Daowen Lin.