On a version of Trudinger–Moser inequality with Möbius shift invariance

Abstract

The paper raises a question about the optimal critical nonlinearity for the Sobolev space in two dimensions, connected to loss of compactness, and discusses the pertinent concentration compactness framework. We study properties of the improved version of the TrudingerMoser inequality on the open unit disk B ⊂ R 2 , recently proved by G. Mancini and K. Sandeep [13]. Unlike the original Trudinger-Moser inequality, this inequality is invariant with respect to Möbius automorphisms of the unit disk, and as such is a closer analogy of the critical nonlinearity |u| 2 * in the higher dimension than the original Trudinger-Moser nonlinearity.

Improved Moser–Trudinger type inequalities in the hyperbolic spaceHn

“…. This improves the Moser-Trudinger inequality in hyperbolic spaces obtained recently by Mancini and Sandeep [29], by Mancini, Sandeep and Tintarev [30] and by Adimurthi and Tintarev [2] for λ = 0. In the limiting case λ = ( n−1 n ) n , we prove a Moser-Trudinger inequality with exact growth in H n , sup u∈C ∞ 0 (H n ) H n |∇gu| n g dVolg−( n−1 n ) n H n |u| n dVolg≤1 1 H n |u| n dVolg H n Φn(αn|u| n n−1 ) (1 + |u|)…”

Poincaré–Sobolev equations in the hyperbolic space

“…2 The Hardy-Sobolev-Mazya equation (1.2) admits a sequence v k of sign changing solutions such that ||∇v k || 2 → ∞ as k → ∞. and Theorem 1.…”

A subset of Caffarelli–Kohn–Nirenberg inequalities in the hyperbolic space $${{\mathbb {H}}}^N$$ H N

“…6 Moser-Trudinger inequality, or, more precisely, its refinement for hyperbolic spaces proved in [4,20], can be also identified as a two-dimensional case of a general inequality for N ≥ 2,…”