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

Abstract

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 the half Euclidean space. As one expected, the key ingredient is the narrow region maximum principle, which can be obtained via the hide monotonicity of the kernel used in the definition of the spectral fractional Laplacian. Using this direct method of moving spheres, we establish monotonicity or symmetry results for nonlinear spectral Laplacian equations on the half Euclidean space.