By Charles Hutton
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Extra resources for A Course of Mathematics. For the Use of Academies as Well as Private Tuition. Volume I (of Two)
River Edge, NJ, 2002.  S. Stapelkamp, Das Brezis-Nirenberg Problem im Hn, PhD thesis, Universitat Basel, 2003. ; Tintarev, K, On existence of minimizers for the HardySobolev-Maz’ya inequality, Ann. Mat. Pura Appl. (4) 186 (2007), no. 4, 645–662. Gianni Mancini Dipartimento di Matematica, Universit`a degli Studi ”Roma Tre” Largo S. Leonardo Murialdo, 1 - 00146 Roma, Italy. in Research by G.
1, 339–363.  Maz’ja, Vladimir G, Sobolev spaces, Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985.  R. Musina Ground state solutions of a critical problem involving cylindrical weights, Nonlinear Analysis TMA, to appear  R. Monti Sobolev inequalities for weighted gradients, Comm. Part. Diff. , 31 (2006), no. 10-12, 1479–1504.  R. Monti, D. Morbidelli, Kelvin transform for Grushin operators and critical semilinear equations, Duke Math. J. Vol. 131 no. 1 (2006), pag.
Also, we see that the special solutions of (Eqλ ) described above, rewrite in the ball model as √ 2 n−1+ 1 − |ξ| u(ξ) = c(λ, n) 2 8 √ (n−1)2 −4λ 2 2 , u˜(ξ) = n−1+ 1 − |ξ| c˜(λ, n) 1 + |ξ|2 (n−1)2 −4λ 2 Appendix B. 5). We will use here Rn+ , as a model for Hn , n ≥ 2. 1. Isometric Sobolev spaces and Poincar´ e inequality in Hn Let u ∈ C0∞ (Hn ), k ∈ N and N := k + (n − 1). Define v(y, z) = (Tk u)(y, z) := |y|− N −2 2 u(|y|, z), y ∈ Rk , z ∈ Rn−1 Thus Tk u ∈ C0∞ (RN ) , has cylindrical symmetry and is compactely supported 1 (RN ), by Rk × Rn−1 \ Rn−1 .
A Course of Mathematics. For the Use of Academies as Well as Private Tuition. Volume I (of Two) by Charles Hutton