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Extra info for Hardy-Sobolev inequalities, hyperbolic symmetry and the Webster scalar curvature problem
Vol. 165 no. 1 (1999), pag. 117-149. E. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia of Mathematics and its applications Vol. 71 (1999), Cambridge University press.  W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. Vol. 129 (2001), pag. 1233-1246.  M. Badiale, E. Serra, Critical nonlinear elliptic equations with singularities and cylindrical symmetry, Rev. Mat. Iberoamericana Vol. 1 (2004), pag. 33-66. Tarantello, A Sobolev-Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics, Arch.
4) assures that Φε has a maximum or minimum and hence a critical point converging to (λ0 , ζ0 ) ∈ (0, ∞) × Rh as ε → 0. 1 we get multiplicity. Proof of Theorem B. We follow closely , where the Authors prove a similar result for the Webster scalar curvature problem on the CR sphere. The theorem can be established through the above finite dimensional reduction and a topological degree argument, namely proving that there exist an open set Ω ⊂ R+ × Rh such that deg(∇Γ, Ω, 0) = 0. We start proving that the critical points of the Melnikov function Γ(λ, ζ) = N −2 2(N − 1) RN −1) ϕ(λy, λz + ζ) 2(N U N −2 |y| lay in a bounded region λ2 + |ζ|2 ≤ R.
Non lin. Vol. 19 no 3 (2002), pag. 313-342.  A. Ambrosetti, J. Garcia Azorero, I. Peral, Perturbation of ∆u + u(N +2)/(N −2) = 0, the scalar curvature problem in RN , and related topics, J. Funct. Anal. Vol. 165 no. 1 (1999), pag. 117-149. E. Andrews, R. Askey, R. Roy, Special functions, Encyclopedia of Mathematics and its applications Vol. 71 (1999), Cambridge University press.  W. Beckner, On the Grushin operator and hyperbolic symmetry, Proc. Amer. Math. Soc. Vol. 129 (2001), pag. 1233-1246.