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

By Ibragimov N.Kh.

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Pure Appl. Math. 42 (1989), 271–297. [8] V. Felli, M. Schneider, Perturbation results of critical elliptic equations of CaffarelliKohn-Nirenberg type, J. Differential Equations 191 (2003), no. 1, 121–142. [9] J. Garcia Azorero, I. Peral, Hardy Inequalities and some critical elliptic and parabolic problems, J. Diff. Equations 144 (1998), no. 2, 441–476. [10] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc.

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 |

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