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

Sekhar Ghosh, Vishvesh Kumar, Tapendu Rana

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 $-Δ_{\mathbb{B}^N} u = λu + |u|^{2^*-2}u$ under Dirichlet boundary conditions for $λ> \frac{N(N-2)}{4}$. 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 $λ$ relative to the Dirichlet spectrum of the Laplace-Beltrami operator.

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

Abstract

This article investigates the multiplicity of solutions to the Brezis-Nirenberg problem on smooth bounded domains in the hyperbolic space for . Specifically, we study the critical semilinear equation under Dirichlet boundary conditions for . 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 relative to the Dirichlet spectrum of the Laplace-Beltrami operator.
Paper Structure (15 sections, 14 theorems, 147 equations)

This paper contains 15 sections, 14 theorems, 147 equations.

Table of Contents

  1. Introduction
  2. Background on the Brezis-Nirenberg Problem
  3. The Brezis-Nirenberg Problem in Hyperbolic Spaces
  4. Main Result
  5. Geometric Framework and Conformal Reduction
  6. Ball Model of Hyperbolic Spaces
  7. Transformation of the Brezis-Nirenberg Problem on $\mathbb{B}^N$ to the Euclidean Laplacian
  8. Theory of Gradient Flow and Relative Equivariant Category
  9. Gradient Flow Associated with $\mathcal{I}_{\lambda, \varrho}$
  10. Equivariant Ljusternik-Schnirelmann Category
  11. Palais--Smale Decomposition of $\mathcal{I}_{\lambda, \varrho}$
  12. Energy Estimates and Proof of the Multiplicity Theorem
  13. Basic Notions and the Nehari Manifold:
  14. Topological Construction of Minimax Test Spaces
  15. Proof of the Multiplicity Result

Key Result

Theorem 1.1

Let $\Omega_{\mathbb{B}^N}$ be a bounded domain in $\mathbb{B}^N$ for $N \geq 4$. Then the following statements hold:

Theorems & Definitions (27)

  • Theorem 1.1: Sta03
  • Remark 1.2
  • Theorem 1.3
  • Remark 2.1
  • Theorem 2.2
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.5
  • ...and 17 more