The fundamental solution of the fractional p-laplacian

TL;DR

This paper derives the fundamental solution for the fractional p-Laplacian operator and applies it to establish Liouville-type theorems, including non-existence results for certain fractional p-harmonic functions and equations.

Contribution

It provides the first explicit fundamental solution for the fractional p-Laplacian and uses it to prove new Liouville-type theorems for fractional p-harmonic functions and equations.

Findings

Established the fundamental solution of the fractional p-Laplacian.

Proved a non-existence Liouville theorem for p-superharmonic functions.

Derived Liouville-type results for fractional Emden-Fowler equations.

Abstract

In this article, we find the fundamental solution of the fractional p-laplacian and use them to prove two different Liouville-type theorems. A non-existence classical Liouville-type theorem for p-superharmonic and a Louville type results for an Emden-Folder type equation with the fractional p-laplacian.