Adv. Nonlinear Anal. 2 (2013), 309 – 324 DOI 10.1515 / anona-2013-0001 © de Gruyter 2013 Trudinger–Moser inequality in the hyperbolic space H N Gianni Mancini, Kunnath Sandeep and Cyril Tintarev Abstract. We prove a version of the Trudinger–Moser inequality in the hyperbolic space H N , which gives a sharper version of the Trudinger–Moser inequality on the Euclid- ean unit ball, as well as a hyperbolic space version of the Onofri inequality, and prove the existence of extremal functions to some related problems. Keywords. Trudinger–Moser inequality, elliptic problems in critical dimension, concentration compactness, weak convergence, Palais–Smale sequences, hyperbolic space, Poincaré disk, Hardy inequalities. 2010 Mathematics Subject Classification. Primary 35J20, 35J60; secondary 46E35, 47J30, 58J70. 1 Introduction The classical Trudinger–Moser inequality states that for any bounded domain  in R N sup u2W 1;N 0 ./; kruk N 1 Z  e ˛juj N N1 dx< 1; (1.1) iff ˛  ˛ N D N! 1 N1 N 1 where ! N 1 denotes the surface measure of the (N  1)-dimensional unit sphere in R N . This inequality appears as a limiting case of the Sobolev embedding of W 1;N 0 ./ and the credit for the proof is shared by Yudovich [27], Peetre [22], Pohozhaev [23], Trudinger [25] and Moser [20]. The optimal constant ˛ N is due to Moser [20]. In the same work Moser also established the analogous sharp in- equality on the Euclidean sphere, with the aim of studying the the problem of prescribing the Gaussian curvature on the sphere. This inequality has been gener- alized to higher order Sobolev spaces ([1]) and to compact Riemannian and sub- Riemannian manifolds. Another important question associated with this inequality is about the exis- tence of extremal functions for (1.1). Existence of extremal functions when  is