Moving spheres for semilinear spectral fractional Laplacian equations in the half space

In this paper, we introduce a direct method of moving spheres for the spectral fractional Laplacian $ (-\Delta_D)^{\alpha/2} with 0<\alpha<2 on half Euclidean space. As one expected, key ingredient is narrow region maximum principle, which can be obtained via hide monotonicity kernel used in definition Laplacian. Using spheres, establish or symmetry results nonlinear equations