Fractional Semilinear Equations on Hyperbolic Spaces

TL;DR

This work addresses semilinear equations for the fractional Laplacian on hyperbolic space by deriving an explicit Green function via the Helgason-Fourier transform and translating the PDE into an integral equation with kernel $G_s$. A direct moving planes method is then employed in the integral form to obtain symmetry and nonexistence results in the subcritical regime and radial symmetry in the critical regime, supported by a suite of maximum principles on $\mathbb{H}^n$. The contributions include explicit expressions and sharp asymptotics for $G_s$, a rigorous equivalence between differential and integral formulations, and a hyperbolic-geometry–adapted moving-plane framework that extends classical Euclidean results to negatively curved spaces. These results enhance the understanding of nonlocal operators in curved manifolds and have potential implications for geometric analysis and fractional GJMS-type operators on hyperbolic spaces.

Abstract

We study a semilinear equation involving the fractional Laplacian on the hyperbolic space $\mathbb{H}^n$. Unlike in conformally compact Einstein manifolds, the fractional Laplacian on $\mathbb{H}^n$ does not enjoy conformal covariance. By employing Helgason-Fourier analysis, we explicitly derive the Green's function of the fractional Laplacian on $\mathbb{H}^n$ and study its asymptotic behaviors. We then apply a direct method of moving planes to the integral form of the equation, establishing symmetry of solutions and nonexistence of positive solutions in the critical and subcritical cases, respectively. In addition, we develop several maximum principles on hyperbolic space.

Theorems & Definitions (27)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 2.1
• Theorem 2.2
• Lemma 2.3: AS64, or Lemma 2.3 of BGS15
• Proposition 3.1: BGS15
• Proposition 3.2: BGS15
• Lemma 3.3: Banica07
• proof : Proof of Theorem \ref{['thm-asymptotics']}