  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.
