Direct Method of Moving Planes for Tempered Fractional Laplacian Equations

In this paper, we extend the direct method of moving planes to tempered fractional Laplacian equations. To carry out the method, a narrow region principle and a decay at infinity lemma are established firstly. Moreover, a boundary estimate lemma is also discussed. Then, we obtain the radial symmetry result for positive solutions of the tempered fractional Laplacian equations both in a unit ball and on the whole space.