In this paper, we summarize some of the recent developments in the area of fractional equations with focus on the ideas and direct methods on fractional non-local operators. These results have more or less appeared in a series of previous literature, in which the ideas were usually submerged in detailed calculations. What we are trying to do here is to single out these ideas and illustrate the inner connections among them, so that the readers can see the whole picture and quickly grasp the essence of these useful methods and apply them to a variety of problems in this area.

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In this paper, we study qualitative properties of the fractional p-Laplacian. Specifically, we establish a Hopf type lemma for positive weak super-solutions of the fractional p−Laplacian equation with Dirichlet condition. Moreover, an optimal condition is obtained to ensure (−△) s p u ∈ C 1 (R n ) for smooth functions u.

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