Existence, nonexistence, and symmetry results for positive solutions of semilinear elliptic equations whose diffusion is a sum of one-dimensional symmetric 2s-stable fractional operators. Because this anisotropic nonlocal operator is not rotation-invariant and admits no Kelvin transform, solutions can be symmetric without being radial; the results give critical-exponent thresholds for nonexistence of positive supersolutions in the whole space, half space, and ball, proved via maximum principles and a moving-plane method adapted to the operator.
This paper studies positive solutions of the semilinear equation in which the diffusion is a sum of N one-dimensional symmetric 2s-stable fractional operators, an anisotropic, non-rotationally-invari…