In this short article, we state a Hopf type lemma for fractional equations and the outline of its proof. We believe that it will become a powerful tool in applying the method of moving planes on fractional equations to obtain qualitative properties of solutions.

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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