Overview
This study investigates the behavior of critical GJMS equations on hyperbolic space, specifically focusing on the phenomenon of concentration at the conformal boundary of the ball model. It precisely characterizes this concentration as an escape to infinity along hyperbolic isometries for critical equations driven by the GJMS operator $P_m$ of order $2m$. The research establishes a two-channel global compactness theorem for Palais–Smale sequences, differentiating between interior Euclidean concentrating bubbles and boundary half-space profiles. Additionally, the work introduces an abstract nodal-energy principle for sign-changing critical points and applies it within the context of the $P_m$ operator.
Research Context
The research addresses critical equations involving the GJMS operator $P_m$ of order $2m$, defined as $P_m u+a(x)u=|u|^{\frac{4m}{n-2m}}u$, where $u\in\mathcal D^{m,2}(\mathbb H^n)$ and $n \geq 2m$. A non-negative potential $a$ is considered within the critical class $L^{n/(2m)}(\mathbb H^n)$. A key aspect of the context is the understanding that on hyperbolic space, concentration at the conformal boundary of the ball model represents escape to infinity via hyperbolic isometries. The study builds upon existing methods, particularly for $m\geq2$, where it aims to avoid reliance on the second-order Pohozaev exclusion of boundary profiles.
Approach
The research employs a multi-faceted approach to analyze the critical GJMS equations. Initially, it focuses on defining the phenomenon of concentration at the conformal boundary on hyperbolic space as escape to infinity along hyperbolic isometries. This forms the basis for the subsequent development of a two-channel global compactness theorem for Palais–Smale sequences. Within this theorem, the interior profiles are identified as Euclidean concentrating bubbles, while the boundary half-space profiles are characterized as solutions of the limiting hyperbolic equation transported by diverging isometries, through an explicit conformal lift.
A further methodological component involves addressing the limitations of truncation into positive and negative parts in $W^{m,2}$. To overcome this, the study introduces and proves an abstract nodal-energy principle. This principle applies to Hilbert spaces continuously embedded in $L^q$ where the inverse Riesz map is positivity preserving, or equivalently, whose positive cone has its polar cone contained within the negative cone. When applied, this principle asserts that every sign-changing critical point carries at least twice the one-bubble energy. The applicability of this principle to $P_m$ is enabled by the positivity of its Green kernel.
The research further incorporates a ball-model nonexistence argument, which, when combined with the nodal-energy principle, establishes a Palais–Smale window below the two-bubble threshold. Finally, an abstract two-level barycenter–concentration criterion is isolated and applied to Passaseo-type concentrated potentials. This application aims to demonstrate the existence of a nontrivial solution for sufficiently large concentration parameters, and a second solution at a strictly higher normalized level under a critical $L^{n/(2m)}$-smallness condition.
Findings
- Concentration at the conformal boundary of the ball model on hyperbolic space is precisely identified as escape to infinity along hyperbolic isometries for critical GJMS equations.
- A two-channel global compactness theorem for Palais–Smale sequences was established. This theorem identifies interior profiles as Euclidean concentrating bubbles and boundary half-space profiles as solutions of the limiting hyperbolic equation transported by diverging isometries via an explicit conformal lift.
- For $m\geq2$, the established theorem avoids reliance on the second-order Pohozaev exclusion of boundary profiles.
- An abstract nodal-energy principle was proven for Hilbert spaces continuously embedded in $L^q$ with a positivity-preserving inverse Riesz map. This principle states that every sign-changing critical point carries at least twice the one-bubble energy.
- Application of the nodal-energy principle to $P_m$, facilitated by the positivity of its Green kernel, combined with a ball-model nonexistence argument, yields a Palais–Smale window below the two-bubble threshold.
- An abstract two-level barycenter–concentration criterion was isolated and applied to Passaseo-type concentrated potentials, resulting in a nontrivial solution for all sufficiently large concentration parameters.
- Under a critical $L^{n/(2m)}$-smallness condition, the application of the barycenter–concentration criterion also yields a second solution at a strictly higher normalized level.