# Group analysis of ODEs and the invariance principle in by Ibragimov N.Kh.

By Ibragimov N.Kh.

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Extra resources for Group analysis of ODEs and the invariance principle in mathematical physics (Russ.Math.Surv. 47, n.4, 89-156)

Example text

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Talenti, Best constants in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353– 372. [15] S. Terracini, On positive entire solutions to a class of equations with a singular coefficient and critical exponent, Adv. Differential Equations 1 (1996), no. 2, 241–264.

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 |