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This is a unique book that provides a comprehensive understanding
of nonlinear equations involving the fractional Laplacian as well
as other nonlocal operators. Beginning from the definition of
fractional Laplacian, it gradually leads the readers to the
frontier of current research in this area. The explanations and
illustrations are elementary enough so that first year graduate
students can follow easily, while it is advanced enough to include
many new ideas, methods, and results that appeared recently in
research literature, which researchers would find helpful. It
focuses on introducing direct methods on the nonlocal problems
without going through extensions, such as the direct methods of
moving planes, direct method of moving spheres, direct blowing up
and rescaling arguments, and so on. Different from most other
books, it emphasizes on illuminating the ideas behind the formal
concepts and proofs, so that readers can quickly grasp the essence.
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