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Elliptic Problems Involving Mixed Local-Nonlocal Operator in the Hyperbolic Space

Diksha Gupta, Konijeti Sreenadh

TL;DR

The article investigates the existence of solutions to nonlinear elliptic equations on hyperbolic space involving a mixed local-nonlocal operator, using a variational framework in H^1(B^N). It establishes subcritical existence via a Nehari-minimization and radial symmetry, and a critical-exponent result with a lower-order perturbation through a Mountain Pass approach in the radial setting, leveraging a continuous H^1-to-H^s embedding and a sharp Poincaré-Sobolev inequality. A key technical advance is the PV-based definition of the fractional operator with a radial kernel, which, together with a weak maximum principle for the mixed operator, helps control noncompactness arising from the hyperbolic geometry. The work extends Brezis-Nirenberg-type phenomena to hyperbolic spaces with nonlocal components and provides a robust variational framework for mixed local-nonlocal problems in curved geometries.

Abstract

This paper explores the existence of solutions to a class of nonlinear elliptic equations involving a mixed local-nonlocal operator of the form $-Δ_{\mathbb{B}^N} + (-Δ_{\mathbb{B}^N})^s$, with $0 < s < 1$, set in the hyperbolic space $\mathbb{B}^N$. By employing variational methods, we address both subcritical and critical nonlinearities, establishing the existence of weak solutions under appropriate conditions.

Elliptic Problems Involving Mixed Local-Nonlocal Operator in the Hyperbolic Space

TL;DR

The article investigates the existence of solutions to nonlinear elliptic equations on hyperbolic space involving a mixed local-nonlocal operator, using a variational framework in H^1(B^N). It establishes subcritical existence via a Nehari-minimization and radial symmetry, and a critical-exponent result with a lower-order perturbation through a Mountain Pass approach in the radial setting, leveraging a continuous H^1-to-H^s embedding and a sharp Poincaré-Sobolev inequality. A key technical advance is the PV-based definition of the fractional operator with a radial kernel, which, together with a weak maximum principle for the mixed operator, helps control noncompactness arising from the hyperbolic geometry. The work extends Brezis-Nirenberg-type phenomena to hyperbolic spaces with nonlocal components and provides a robust variational framework for mixed local-nonlocal problems in curved geometries.

Abstract

This paper explores the existence of solutions to a class of nonlinear elliptic equations involving a mixed local-nonlocal operator of the form , with , set in the hyperbolic space . By employing variational methods, we address both subcritical and critical nonlinearities, establishing the existence of weak solutions under appropriate conditions.
Paper Structure (14 sections, 9 theorems, 142 equations)

This paper contains 14 sections, 9 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Möbius Transformations and Convolution
  4. Hyperbolic Distance on $\mathbb{B}^N$
  5. Helgason Fourier Transform on the Hyperbolic Space
  6. A sharp Poincaré-Sobolev inequality
  7. Fractional Sobolev Spaces
  8. Existence of a Solution in the Subcritical case
  9. Variational Formulation
  10. Existence of a solution
  11. A critical exponent problem
  12. Existence of a Solution in the Critical Case with a Lower Order Perturbation
  13. Appendix
  14. Statements and Declarations

Key Result

Theorem 1.1

(Existence and symmetry). Let $s\in(0,1)$ be fixed and $1<p<2^*-1,\;\lambda < \frac{(N-1)^2}{4}$. Then,

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • ...and 13 more