Radial symmetry and monotonicity for fractional Hénon equation in Rⁿ

Abstract

In this paper, we consider the following positive solutions of the fractional Hénon equation in  :(Section.Display)

where , and . First, we prove that (1) is equivalent to the following integral equation(Section.Display)

where is the Green’s function of  in  . Then by combining the method of moving planes in integral forms with a certain type of Kelvin transform, in the subcritical case, we obtain that positive solutions for integral Equation (2) are radially symmetric and monotone decreasing about the origin under some integrability condition.

Keywords:

• the fractional Laplacian
• Green’s function
• method of moving planes in integral forms
• radial symmetry
• monotonicity
• Kelvin transform

AMS Subject Classifications:

• 35J40
• 35J61
• 35B53
• 35J91
• 35B06

Acknowledgements

The authors would like to express their sincere thanks to Professor Wenxiong Chen for his helpful discussions. Besides, the authors want to thank the referee for his/her helpful comments on the paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by NSFC [grant number U1304101], [grant number 11171091]; HASTIT [grant number 15HASTIT012].