Multiplicity of Solutions to the Brezis-Nirenberg Problem on Hyperbolic Spaces

This article investigates the multiplicity of solutions to the Brezis-Nirenberg problem on smooth bounded domains in the hyperbolic space ${\mathbb{B}}^N$ for $N\ge 4$. Specifically, we study the critical semilinear equation $-{\mathrm{\Delta }}_{{\mathbb{B}}^N}u=\lambda u+\mathrm{\mid }u{\mathrm{\mid }}^{2^{\ast }-2}u$ under Dirichlet boundary conditions for $\lambda >\frac{N(N-2)}{4}$. Overcoming the analytic challenges induced by the hyperbolic geometry and the intricate concentration profiles of Palais-Smale sequences, we establish the existence of multiple pairs of nontrivial solutions. Using the equivariant Ljusternik-Schnirelmann category, we obtain lower bounds on the number of solutions depending on the position of the parameter $\lambda$ relative to the Dirichlet spectrum of the Laplace-Beltrami operator.