Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 67, pp. 1-15.

Title: A Brezis-Nirenberg problem on hyperbolic spaces

Authors: Paulo Cesar Carriao (Univ. Federal de Minas Gerais, Belo Horizonte, MG, Brazil)
         Raquel Lehrer (Univ. Estadual do Oeste do Parana, Cascavel, PR, Brazil)
         Olimpio Hiroshi Miyagaki (Univ. Federal de Juiz de Fora, MG, Brazil)
         Andre Vicente (Univ. Estadual do Oeste do Parana, Cascavel, PR, Brazil)

Abstract:
We consider a Brezis-Nirenberg problem on the hyperbolic space $\mathbb{H}^n$.
 By using the  stereographic projection, the problem becomes a singular
 problem on the boundary of the open ball $B_1(0)\subset \mathbb{R}^n$.
 Thanks to the Hardy inequality, in a version due to Brezis-Marcus,
 the difficulty involving singularities can be overcame.
 We use the mountain pass theorem due to Ambrosetti-Rabinowitz and
 Brezis-Nirenberg arguments to obtain a nontrivial solution.

Submitted June 30, 2018. Published May 13, 2019.
Math Subject Classifications: 32Q45, 35A15, 35B38, 35B33.
Key Words: Variational method; critical point; critical exponent; hyperbolic manifold.