Two-channel global compactness and existence for critical GJMS equations on hyperbolic space

Abstract

On hyperbolic space, concentration at the conformal boundary of the ball model is not an exceptional phenomenon to be excluded: it is escape to infinity along hyperbolic isometries. We make this precise for critical equations driven by the GJMS operator $P_m$ of order $2m$, \[ P_m u+a(x)u=|u|^{\frac{4m}{n-2m}}u, \qquad u\in\mathcal D^{m,2}(\mathbb H^n), \qquad n>2m, \] with a nonnegative potential $a$ in the critical class $L^{n/(2m)}({\mathbb{H}}^n)$, and prove a two-channel global compactness theorem for Palais--Smale sequences: interior profiles are the usual Euclidean concentrating bubbles, while boundary half-space profiles are identified, through an explicit conformal lift, with solutions of the limiting hyperbolic equation transported by diverging isometries. For $m\ge 2$ this avoids relying on the second-order Pohozaev exclusion of boundary profiles. Since truncation into positive and negative parts is unavailable in $W^{m,2}$, we also prove an abstract nodal-energy principle: in every Hilbert space continuously embedded in $L^q$ whose inverse Riesz map is positivity preserving, equivalently whose positive cone has polar cone contained in the negative cone, every sign-changing critical point carries at least twice the one-bubble energy. Applied to $P_m$ through the positivity of its Green kernel, and combined with a ball-model nonexistence argument, this yields a Palais--Smale window below the two-bubble threshold. Finally we isolate an abstract two-level barycenter--concentration criterion and apply it to Passaseo-type concentrated potentials, obtaining a nontrivial solution for all sufficiently large concentration parameters and a second one, at a strictly higher normalized level, under a critical $L^{n/(2m)}$-smallness condition.

28 pages