Maximum principles for the fractional p-Laplacian and symmetry of solutions

Abstract

In this paper, we consider nonlinear equations involving the fractional p-LaplacianWe prove a maximum principle for anti-symmetric functions and obtain other key ingredients for carrying on the method of moving planes, such as a key boundary estimate lemma. Then we establish radial symmetry and monotonicity for positive solutions to semilinear equations involving the fractional p-Laplacian in a unit ball and in the whole space. We believe that the methods developed here can be applied to a variety of problems involving nonlinear nonlocal operators.Key words The fractional p-Laplacian, maximum principles for anti-symmetric functions, a key boundary estimate, method of moving planes, radial symmetry, monotonicity.

A Hopf lemma and regularity for fractional $ p $-Laplacians

“…The fractional p-Laplacian is a special case in which G(t) = |t| p−2 t and α = sp. This direct method has been successfully applied to obtain symmetry, monotonicity, nonexistence and other qualitative properties of solutions for various nonlocal problems, see e.g., [8,10,11,12,13].…”

Exact behavior around isolated singularity for semilinear elliptic equations with a log-type nonlinearity

“…Meanwhile, C. Li [7] extended the result on the asymptotic radial symmetry of singular solutions of −∆u = g(u) for a more general g(u) considered in [1]. Recently, the asymptotic radial symmetry has been achieved for other operators, such as conformally invariant fully nonlinear equations [5,8], fractional equations [2], and fractional p-laplacian equations [3].…”

Properties for Nonlinear Fractional SubLaplace Equations on the Heisenberg Group

“…But the method has tha limitation that 1 2 ≤ α < 1. Chen, Li and Li [4] developed a direct method of moving planes to handle the problem containing (−∆) α for 0 < α < 1 and this direct method has been used successfully to deduce symmetry, monotonicity and nonexistence for many fractional Laplace equations, see [3,4] and references therein. Recently, Wang and Niu [5] considered (1.2) in H n , and showed properties of solutions.…”

“…The fractional power subLaplace equations can also be studied by generalizing the extension method in [1] to H n , for example, see [13,16,17]. Extending the method of moving planes in [3,4,18] to H n , we establish the Liouville type result of the solution to (1.2) on H n + = {ξ ∈ H n |t > 0}, and the symmetric and monotone of the solution to (1.3) on H n .…”