By Felli V., Pistoia A.

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26) and for any j = 1, . . , N ∗ RN 2 −1 j [k(x) − k(ξ0 )] Wδ,ξ ψδ,ξ dx ∗ 2∗ = (N − 2)α2N [k(ξ0 )]− 2∗ −2 δθ−1 RN Qξ0 (y + ζ) uniformly with respect to ζ in K. 10 The following identity holds θ RN Proof. Qξ0 (y + ζ) dy = N (1 + |y|2 )N RN |y|2 − 1 + 2y · ζ dy. 6) with respect to t we have θtθ−1 Qξ0 (x) = ∇Qξ0 (tx) · x. In particular for t = 1 we get θQξ0 (x) = ∇Qξ0 (x) · x, Hence θ RN Qξ0 (y + ζ) dy = (1 + |y|2 )N RN for any x ∈ RN . y+ζ dy. (1 + |y|2 )N ∇Qξ0 (y + ζ) · Integrating by parts the right hand side of the above identity we obtain θ RN Qξ0 (y + ζ) dy = −N (1 + |y|2 )N RN Qξ0 (y + ζ) 1 − |y|2 − 2y · ζ dy (1 + |y|2 )N +1 which is the desired identity.

5] T. Aubin, Problemes isoperimetriques et espaces de Sobolev, J. Diff. Geom. 11 (1976), 573–598. [6] A. Bahri, Critical points at infinity in some variational problems, Pitman Research Notes in Mathematics Series 182 (1989), Longman. [7] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271–297. [8] V. Felli, M. Schneider, Perturbation results of critical elliptic equations of CaffarelliKohn-Nirenberg type, J.

4. 6 There holds [k(x) − k(ξ0 )] f ′ (Wδ,ξ ) Proof. 4. 30 = O |ξ − ξ0 |θ + δθ . RN 2 Wδ,ξ |x|2 1 2 . 19) if N = 3, and for any j = 1, . . , N and N ≥ 3 j Jλ′′ (Wδ,ξ + φλδ,ξ )ψδ,ξ Proof. λ = δ−1 O λδ + |ξ − ξ0 |θ + δθ + φλδ,ξ min{1,4/(N −2)} λ . 6. 13). 8 Assume ξ = ξ0 + δζ for ζ ∈ K ⊂⊂ RN . There holds ∗ RN 2∗ ∗ 2 [k(x) − k(ξ0 )] Wδ,ξ dx = α2N [k(ξ0 )]− 2∗ −2 δθ RN Qξ0 (y + ζ) dy + o(1) (1 + |y|2 )N uniformly with respect to ζ in K. Proof. 5) it follows ∗ RN ∗ 2∗ 2 [k(x) − k(ξ0 )] Wδ,ξ dx = α2N [k(ξ0 )]− 2∗ −2 RN ∗ [k(x) − k(ξ0 )] δN dx (δ2 + |x − ξ|2 )N 2∗ (setting CN (ξ0 ) = α2N [k(ξ0 )]− 2∗ −2 and y = δy + ξ) 1 [k(δy + ξ) − k(ξ0 )] = CN (ξ0 ) dy (1 + |y|2 )N RN 1 dy Qξ0 (δy + ξ − ξ0 ) = CN (ξ0 ) (1 + |y|2 )N {|δy+ξ−ξ0 |