Hardy–Sobolev extremals, hyperbolic symmetry and scalar curvature equations

Abstract

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.

Poincaré–Sobolev equations in the hyperbolic space

“…1.1 is closely related to the study of Hardy-Sobolev-Mazya type equations and Grushin operators under partial symmetry of their solutions (See [7][8][9] …”

“…A point x ∈ R n is denoted as x = (y, z) ∈ R k × R n−k . One can see that u ∈ D 1,2 (R n ) is a cylindrically symmetric solutions of (1.2) (i.e., u(x) =ũ(|y|, z)) iff v = w • M solves (1.1) with dimension N = n − k + 1, p = p t and λ = η + (n−k) 2 −(k−2) 2 4 where w(r, z) = r n−2 2ũ (r, z) for (r, z) ∈ (0, ∞) × R n−k and M : B n−k+1 → (0, ∞) × R n−k is the standard isometry (see (6.29)) between the B N and the upper half space model of the hyperbolic space (see [8,9] for details). Note that when k = 2 and η = 0, then we have λ = ( N −1 2 ) 2 in (1.1).…”

Extremals for Sobolev and Moser Inequalities in Hyperbolic Space

“…Qualitative properties of extremals (and, in some case, identification) for (1.3) were established in [14], [13] ( see also [35] ) where we have shown L ∞ and Holder regularity and, in case μ ≥ 0, cylindrical symmetry of the extremals (notice however that for μ negative minimizers are not, in general, symmetric, see [17] ).…”

Existence and asymptotic behavior of solutions for Hénon type equations

“…was considered in [1][2][3]5] and references therein, where R n = R k × R n−k , 2 ≤ k < n, and a point x ∈ R n is denoted as x = (y, z) ∈ R k × R n−k . Various existence results were obtained in these papers.…”