Electron. J. Differential Equations, Vol. 2019 (2019), No. 67, pp. 1-15.

A Brezis-Nirenberg problem on hyperbolic spaces

Paulo Cesar Carriao, Raquel Lehrer, Olimpio Hiroshi Miyagaki, Andre Vicente

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.

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Paulo César Carrião
Universidade Federal de Minas Gerais
- ICEX - DM, CEP: 31270-901
Belo Horizonte, MG, Brazil
email: pauloceca@gmail.com
Raquel Lehrer
Universidade Estadual do Oeste do Paraná - CCET
Rua Universitária, 2069
Jd. Universitário, CEP: 85819-110
Cascavel, PR, Brazil
email: rlehrer@gmail.com
Olíimpio Hiroshi Miyagaki
Universidade Federal de Juiz de Fora - DM
CEP: 36036-330
Juiz de Fora, MG, Brazil
email: ohmiyagaki@gmail.com
André Vicente
Universidade Estadual do Oeste do Paraná - CCET
Rua Universitária, 2069
Jd. Universitário
CEP: 85819-110
Cascavel, PR, Brazil
email: andre.vicente@unioeste.br

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