For 3<=k<=n-1, 1<p<n, the deficit in the Hardy-Sobolev-Maz'ya inequality is bounded below by a constant times the gradient distance to the extremal manifold raised to the power max{2,p}, and this exponent is optimal.
Best constant and extremal functions for a class Hardy-Sobolev-Maz'ya inequalities
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abstract
We derive an integral identity for a class $p$-Laplace equation, and then classify all positive finite energy cylindrically symmetric solutions of the equation (\ref{1.2}) for $3\leq k\leq n-1,$ with the help of some a prior estimates. Combining this with the result of Secchi-Smets-Willem{\cite{SSW03}}, as a consequence, we obtain the best constant and extremal functions for the related Hardy-Sobolev-Maz'ya inequalities.
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Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities
For 3<=k<=n-1, 1<p<n, the deficit in the Hardy-Sobolev-Maz'ya inequality is bounded below by a constant times the gradient distance to the extremal manifold raised to the power max{2,p}, and this exponent is optimal.