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

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