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

TL;DR

This paper proves the existence of multiple solutions to a critical semilinear PDE on hyperbolic spaces, revealing how the geometry influences solution multiplicity through topological methods.

Contribution

It introduces new multiplicity results for the Brezis-Nirenberg problem on hyperbolic spaces using topological tools, overcoming geometric and analytical challenges.

Findings

Multiple pairs of solutions are established for the problem.

Lower bounds on the number of solutions depend on the parameter's position relative to the spectrum.

The methods adapt Ljusternik-Schnirelmann theory to hyperbolic geometry.

Abstract

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 $-\Delta_{\mathbb{B}^N} u = \lambda u + |u|^{2^*-2}u$ 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.