A direct method of moving planes for the fractional Laplacian

Abstract

In this paper, we develop a direct method of moving planes for the fractional Laplacian. Instead of conventional extension method introduced by Caffarelli and Silvestre, we work directly on the non-local operator. Using the integral defining the fractional Laplacian, by an elementary approach, we first obtain the key ingredients needed in the method of moving planes either in a bounded domain or in the whole space, such as strong maximum principles for anti-symmetric functions, narrow region principles, and decay at infinity. Then, using a simple example, a semi-linear equation involving the fractional Laplacian, we illustrate how this new method of moving planes can be employed to obtain symmetry and non-existence of positive solutions. We firmly believe that the ideas and methods introduced here can be conveniently applied to study a variety of nonlocal problems with more general operators and more general nonlinearities.

36 pages

References in corpus (1)

• A Liouville Theorem for the Fractional Laplacian

Cited by in corpus (5)

• On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results
• Symmetry of Solutions to Semilinear Equations Involving the Fractional Laplacian on ${\mathbb{R}}^n$ and ${\mathbb{R}}_{+}^n$
• Fractional nonlinear Schrödinger equations with singular potential in ${\mathbf{R}}^n$
• A Liouville Theorem for a Class of Fractional Systems in ${\mathbb{R}}_{+}^n$
• Radial symmetry results for fractional Laplacian systems