Hardy–Sobolev inequalities and hyperbolic symmetry

Abstract

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.

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] …”

Hardy–Sobolev extremals, hyperbolic symmetry and scalar curvature equations

“…In this paper we continue the investigation initiated in [13] on positive extremals for HardySobolev-Maz'ya inequalities (see [18] and [10]) …”

Infinitely many solutions for a Hardy–Sobolev equation involving critical growth

“…If D R N , D 0, t D 1, the positive extremals have been completely identified in [20,21], and the non-degeneracy of the positive extremals has been proved in [22]. For more results, we can refer to [23][24][25][26][27][28][29][30]. However, for the case t ¤ 1, it seems that there are no results on the existence of solutions for problem (1.1) because in this case, the explicit form or the asymptotic properties of the positive extremals of (1.1) with D 0 in R N are unknown, which makes it impossible to verify that the Palais-Smale sequences corresponding to the functional I are precompact by the standard argument in [6].…”