Best constant and extremal functions for a class Hardy-Sobolev-Maz'ya inequalities

TL;DR

This paper derives the best constants and extremal functions for Hardy-Sobolev-Maz'ya inequalities by classifying positive finite energy solutions of a p-Laplace equation with cylindrical symmetry.

Contribution

It introduces a new integral identity and classification method for solutions, leading to optimal constants and extremal functions for a class of inequalities.

Findings

Derived integral identity for p-Laplace equations.

Classified positive finite energy solutions with cylindrical symmetry.

Established best constants and extremal functions for Hardy-Sobolev-Maz'ya inequalities.

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.