Blow up limits of the fractional Laplacian and their applications to the fractional Nirenberg problem

TL;DR

This paper studies the blow-up behavior of the fractional Laplacian and applies these findings to construct solutions to the fractional Nirenberg problem that exhibit blow-up in regions with negative prescribed functions, revealing new phenomena.

Contribution

It establishes a convergence result for the fractional Laplacian on unbounded sequences and demonstrates a novel blow-up phenomenon in the fractional Nirenberg problem.

Findings

Convergence of fractional Laplacian for nonnegative functions without uniform boundedness.

Construction of solutions blowing up where prescribed functions are negative.

Identification of a new blow-up phenomenon distinct from the classical case.

Abstract

We show a convergence result of the fractional Laplacian for sequences of nonnegative functions without uniform boundedness near infinity. As an application, we construct a sequence of solutions to the fractional Nirenberg problem that blows up in the region where the prescribed functions are negative. This is a different phenomenon from the classical Nirenberg problem.