Direct Methods for Problems Involving the Nonlocal Elliptic Operators
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In this thesis, we consider problems involving the fractional Laplacians and the uniformly elliptic nonlocal operators. The thesis is mostly devoted to presenting our original work on the progress obtained in the development of direct methods that can effectively deal with the nonlocal problems. It includes the direct method of moving planes, the direct method of moving spheres and the direct blowing up and re-scaling argument, the Pohozaev identity, a direct analysis via the potential theory. We illustrate how these methods work by applying them to various types of problems, such as the Dirichelt problems on bounded and unbounded domains, the Navier problems in the upper-half space and problems in the whole Euclidean space Rn. We conclude the thesis with the applications of the method of moving planes in integral forms to derive symmetry and nonexistence results for fractional equations.