Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Symmetry for a quasilinear elliptic equation in Hyperbolic space

Ramya Dutta, Sandeep Kunnath

TL;DR

The paper studies radial symmetry, decay, and existence for a quasilinear $p$-Laplacian equation on hyperbolic space, viewing it as the Euler-Lagrange equation for the Poincaré-Sobolev inequality on $(\mathbb{H}^n,g)$. It develops a hyperbolic moving plane approach complemented by sharp decay estimates obtained through sub- and super-solution constructions and Picone identities, plus a classification of positive eigenfunctions for the associated eigenvalue problem on $\mathbb{H}^n$. Key contributions include the exponential decay of solutions and their gradients, the full hyperbolic symmetry of positive solutions, and existence/nonexistence results in subcritical and critical regimes, extending Euclidean symmetry theory to negatively curved spaces. These results provide sharp constants and extremal profiles for Poincaré-Sobolev inequalities in noncompact geometries, with potential implications for geometric analysis and nonlinear PDE in hyperbolic settings.$

Abstract

In this article we establish the radial symmetry of positive solutions of a p- Laplace equation in the Hyperbolic space, which is the Euler Lagrange equation of the p- Poincare Sobolev inequality in the Hyperbolic space. We will also establish the sharp decay of solution and its gradient and also investigate the question of existence of solution.

Symmetry for a quasilinear elliptic equation in Hyperbolic space

TL;DR

The paper studies radial symmetry, decay, and existence for a quasilinear -Laplacian equation on hyperbolic space, viewing it as the Euler-Lagrange equation for the Poincaré-Sobolev inequality on . It develops a hyperbolic moving plane approach complemented by sharp decay estimates obtained through sub- and super-solution constructions and Picone identities, plus a classification of positive eigenfunctions for the associated eigenvalue problem on . Key contributions include the exponential decay of solutions and their gradients, the full hyperbolic symmetry of positive solutions, and existence/nonexistence results in subcritical and critical regimes, extending Euclidean symmetry theory to negatively curved spaces. These results provide sharp constants and extremal profiles for Poincaré-Sobolev inequalities in noncompact geometries, with potential implications for geometric analysis and nonlinear PDE in hyperbolic settings.$

Abstract

In this article we establish the radial symmetry of positive solutions of a p- Laplace equation in the Hyperbolic space, which is the Euler Lagrange equation of the p- Poincare Sobolev inequality in the Hyperbolic space. We will also establish the sharp decay of solution and its gradient and also investigate the question of existence of solution.
Paper Structure (9 sections, 22 theorems, 188 equations)

This paper contains 9 sections, 22 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Decay estimate for solutions and its gradient
  4. Boundedness of solution
  5. Super-solution and sub-solutions to the equation
  6. Proof of sharp decay
  7. Proof of symmetry
  8. Strong comparison for Hyperbolic $p$-Laplace equation
  9. Existence and nonexistence

Key Result

Theorem 1.1

Let, $u$ be a solution of R21eqn1, then there is a point $O \in \mathbb{H}^n$ and a positive strictly monotone decreasing function $\Phi: [0,\infty) \to \mathbb{R}_+$ such that $u(x) = \Phi(\operatorname{dist}_{\mathbb{H}^n}(O,x))$. The function $\Phi(t)$ is $C^{1}$ at $t = 0$ with $\Phi'(0) = 0$ an

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 31 more