A direct method of moving planes for the fractional Laplacian.(English)Zbl 1362.35320
Summary: In this paper, we develop a directmethod of moving planes for the fractional Laplacian. Instead of using the 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 themethod of moving planes either in a bounded domain or in the whole space, such asstrong maximum principles for anti-symmetric functions,narrow region principles, anddecay at infinity. Then, using simple examples, semilinear equations involving the fractional Laplacian, we illustrate how this newmethod 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.
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.
MSC:
| 35R11 | Fractional partial differential equations |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35B50 | Maximum principles in context of PDEs |
| 35B07 | Axially symmetric solutions to PDEs |
Keywords:
fractional Laplacian;maximum principles for anti-symmetric functions;narrow region principle;decay at infinity;method of moving planes;radial symmetry;monotonicity;non-existence of positive solutionsCite
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