This is a unique book that provides comprehensive understanding of nonlinear equations involving the fractional Laplacian as well as other nonlocal operators. Beginning from the definition of the fractional Laplacian, it gradually leads the readers to the frontier of the current research in this area. The explanations and illustrations are elementary enough so that the 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 the professionals 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, and 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 the readers can quickly grasp the essence. Key Features: oThe subjects discussed here are currently hot in the research community oDifferent from most other books, it emphasizes on illuminating the ideas behind the formal concepts and proofs, so that the readers can quickly grasp the essence oIt accumulates the authors as well as other researchers’ experience and understanding in their research process and tries to present these in a vivid and intuitive way, so that the readers can follow easily

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