A Brezis-Nirenberg problem on hyperbolic spaces

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.

Keywords

• Variational method
• critical point
• critical exponent
• hyperbolic manifold