MSC:primary 35J60 secondary 35B05, 35A15We prove nondegeneracy of extremals for some Hardy-SobolevMaz'ya inequalities and present applications to scalar curvaturetype problems, including the Webster scalar curvature equation in a cylindrically symmetric setting. The main theme is hyperbolic symmetry.

In this article we classify all positive finite energy solutions of the equation − u= u n n−2 |y| in R n where R n = R k × R n−k , n > k 2 and a point x ∈ R n is denoted as x = (y, z) ∈ R k × R n−k . As a consequence we obtain the best constant and extremals of a related Hardy-Sobolev inequality.

We discuss uniqueness and nondegeneracy of extremals for some wheighted Sobolev inequalities and give some applications to Grushin and scalar curvature type equations. The main theme is hyperbolic symmetry.

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Given Ω a smooth bounded domain of R n , n 3, we consider functions u ∈ H 2 2,0 (Ω) that are weak solutions to the equationwhere 2 := 2(n−s) n−2 , s ∈ [0, 2) and a, f ∈ C ∞ (Ω). In this article, we prove the maximal regularity of solutions to the above equation, depending on the value of s ∈ [0, 2) and the relative position of Ω with respect to the origin. In particular, the solutions are in C 4 (Ω) when s = 0.