Existence results for nonlocal problems governed by the regional fractional Laplacian

TL;DR

This paper investigates the existence of minimizers for the critical fractional Sobolev constant on bounded domains governed by the regional fractional Laplacian, establishing conditions for achievement and positivity of solutions.

Contribution

It provides new existence results for minimizers of the fractional Sobolev constant, including radial solutions in balls and conditions in low dimensions.

Findings

Best constant is achieved under certain fractional parameters.

Positive radial minimizers exist in higher dimensions for balls.

A positive mass condition is required in low dimensions.

Abstract

The aim of the present paper is to study existence results of minimizers of the critical fractional Sobolev constant on bounded domains. Under some values of the fractional parameter we show that the best constant is achieved. If moreover the underlying domain is a ball, we obtain positive radial minimizers for all possible values of the fractional parameter in higher dimension, while we impose a positive mass condition in low dimension.