On Poincaré-Sobolev level involving fractional GJMS operators on hyperbolic space

TL;DR

This work studies the Poincaré--Sobolev levels for fractional GJMS operators on hyperbolic space, introducing the auxiliary operator $\widetilde{\mathcal{P}}_s$ to recover a conformal intertwining with the Euclidean fractional Laplacian when $s$ is non-integer. It derives explicit Hardy-type lower bounds for both $\mathcal{P}_s$ and $\widetilde{\mathcal{P}}_s$, analyzes strict-gap regions via concentration-compactness, and proves the existence of Brezis--Nirenberg type solutions on $\mathbb{H}^n$ for both operators. The paper then characterizes the associated Poincaré--Sobolev levels $H_{n,s}$ and $\widetilde{H}_{n,s}$, including attainability, monotonicity, and gap behavior across a range of dimensions and orders. These results extend Euclidean and integer-order hyperbolic theories to the genuinely nonlocal fractional regime and provide sharp thresholds and energy asymptotics that illuminate the nonlinear spectral structure on hyperbolic space.

Abstract

This paper is devoted to a qualitative analysis of the Poincaré--Sobolev level associated with the fractional GJMS operators \(\mathcal{P}_s\) \(\bigl(s\in(0,\tfrac n2)\setminus\mathbb N\bigr)\) on the hyperbolic space \(\mathbb H^n\). In contrast to the integer-order case, when \(s\notin\mathbb N\) the operator \(\mathcal{P}_s\) does not enjoy the conformal covariance that allows one, in the upper half-space or ball model, to relate it to the Euclidean fractional Laplacian \((-Δ)^s\); this link is crucial for importing Euclidean theory. We therefore introduce \(\widetilde{\mathcal{P}}_s\) (\(s>0\)), which is conformally related to \((-Δ)^s\). Our purpose in the paper is to analyze the monotonicity, attainability, and strict-gap regions of the Poincaré--Sobolev levels associated with \(\mathcal{P}_s\) and with \(\widetilde{\mathcal{P}}_s\) with \(s\in(0,\tfrac n2)\setminus\mathbb N\). First, we reinterpret the Brezis--Nirenberg problem through the lens of Poincaré--Sobolev levels, connecting earlier results for the Euclidean Laplacian and for operators \(\mathcal{P}_k\) on \(\mathbb H^n\) with integer \(k\in(0,\tfrac n2)\). We then establish new, explicit lower bounds for the Hardy term in fractional Hardy--Sobolev--Maz'ya inequalities involving both \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). By applying the concentration--compactness principle together with a detailed analysis of the strict-gap regions for the Poincaré--Sobolev levels, we prove the existence of solutions to the Brezis--Nirenberg problem on \(\mathbb H^n\) for both operators. Finally, combining the Hardy lower bounds with criteria for attainability, we obtain a complete characterization of the Poincaré--Sobolev levels \(H_{n,s}\) and \(\widetilde H_{n,s}\).

Theorems & Definitions (37)

• Proposition 1.1
• Remark 1.1
• Proposition 1.2
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 1.3
• Theorem 1.6
• Theorem 1.7