Global Compactness and Existence for Higher Order Critical Equations on Hyperbolic Spaces

TL;DR

This paper investigates higher-order critical Sobolev equations on hyperbolic spaces, establishing compactness results, analyzing bubble phenomena, and proving existence of solutions under certain conditions.

Contribution

It generalizes classical second-order results to higher-order operators on hyperbolic spaces, introducing a new profile decomposition and energy inequalities for sign-changing solutions.

Findings

Established a global compactness theorem for Palais--Smale sequences.

Identified two types of bubbles: conformal and escaping isometry bubbles.

Proved existence of positive solutions under potential concentration and smallness conditions.

Abstract

We study the higher-order Schr\"odinger equation with critical Sobolev exponent on the hyperbolic space $\mathbb{H}^n$: $$P_m u + a(x)\,u = |u|^{q-2}u, \quad u \in D^{m,2}(\mathbb{H}^n),$$ where $P_m$ is the GJMS operator of order $2m$, $q = \frac{2n}{n-2m}$ is the critical exponent, and $a(x) \geq 0$ is a potential in $L^{n/2m}(\mathbb{H}^n)$. This problem simultaneously generalizes the classical work of Benci--Cerami from second-order to arbitrary order and from Euclidean space to hyperbolic space. We establish a global compactness theorem (profile decomposition) for Palais--Smale sequences associated to this equation. The decomposition features two types of bubbles: concentrating bubbles arising from the conformal equivalence $\mathbb{H}^n \cong \mathbb{B}^n$, and isometry bubbles escaping to infinity. A key difficulty in the higher-order setting is that the classical positive/negative decomposition $u = u^+ + u^-$ fails in $W^{m,2}$ for $m \geq 2$. To overcome this, we employ the Moreau dual cone decomposition together with the positivity of the Green function of $P_m$ on $\mathbb{H}^n$, establishing an energy doubling inequality for sign-changing solutions: $I_\infty(u) \geq \frac{2m}{n}S^{n/2m}$. As an application, under a concentration condition on the potential $a(x)$ of Passaseo type, we prove that the equation admits at least one positive solution, and a second positive solution under a smallness condition on $\|a\|_{L^{n/2m}}$.