Monotonicity and symmetry of fractional Lane-Emden-type equation in the parabolic domain

摘要

In this paper, the fractional Lane-Emden-type equation is considered. Instead of using the conventional extension method of Caffarelli and Silvestre or the method of moving planes in integral forms, a new method is employed to consider the monotonicity of positive solutions for the fractional Lane-Emden-type equation in an unbounded parabolic domain. As its application, we obtain the Liouville-type results for the fractional Lane-Emden-type equation under proper conditions. Moreover, we show that the positive solution is radially symmetric about the origin in Rn-1 both in the critical case and in the subcritical case.