Extremals For Fractional Order Hardy-Sobolev-Maz'ya Inequality

TL;DR

This paper investigates positive solutions to a non-local elliptic PDE involving fractional Laplacian, establishing existence, symmetry, and asymptotic behavior based on a fractional Hardy-Sobolev-Maz'ya inequality.

Contribution

It introduces a new fractional Hardy-Sobolev-Maz'ya inequality and analyzes the existence, symmetry, and asymptotics of solutions to related PDEs.

Findings

Existence of positive solutions established.

Symmetry properties of solutions derived.

Asymptotic behavior characterized.

Abstract

In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.