Some monotonicity results for the fractional Laplacian in unbounded domain

Abstract

In this paper, we develop a direct method of moving planes in ${\mathbb{R}}^n$ without any decay conditions at infinity for solutions for fractional Laplacian. We first prove a monotonicity result for semi-linear equations involving the fractional Laplacian equation in ${\mathbb{R}}^n$, and we also derive a one-dimensional symmetry result, which indicates that fractional De Giorgi conjecture is valid under some conditions. During these processes, we introduce some new ideas: (i) estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum; (ii) analyzing the fractional equations along a sequence of approximate maximum, and then by making translation and taking the limit to derive a limit equation.

Keywords:

• The fractional Laplacian
• narrow region principle
• monotonicity
• De Giorgi conjecture

AMS SUBJECT CLASSIFICATIONS:

• 35S15
• 35B06
• 35J60

Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. 11801446, 11971385), Natural Science Basic Research Plan in Shaanxi Province of China (Program No. 2018JQ1037, Grant No. 2019JM-275).