Asymptotic symmetry and local behaviors of solutions to a class of anisotropic elliptic equations

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Let 0 ≤ a 0. Equation (0.1) is arising from the celebrated Caffarelli-Kohn-Nirenberg inequality. For a = b < 0 and p = 2N N−2 , we show there is no positive solution of the equation (0.2)  −div(|x|−2a∇u) = |x|−bpup−1 in Ω, u = 0 on ∂Ω. in D a (Ω), where Ω ⊆ R+ be a cone domain satisfying Ω = R+ × ω with ω ⊂ SN−1 being star-shaped with respect to the North pole on SN−1.