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

We study the higher-order Schrödinger equation with critical Sobolev exponent on the hyperbolic space ${\mathbb{H}}^n$: $$P_{m}u+a(x)u=\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u,u\in D^{m,2}({\mathbb{H}}^n),$$ where $P_m$ is the GJMS operator of order $2m$, $q=\frac{2n}{n-2m}$ is the critical exponent, and $a(x)\ge 0$ is a potential in $L^{n\mathrm{/}2m}({\mathbb{H}}^n)$. 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 $m\ge 2$. 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: $I_{\mathrm{\infty }}(u)\ge \frac{2m}{n}{S}^{n\mathrm{/}2m}$. 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 $\mathrm{\parallel }a{\mathrm{\parallel }}_{L^{n\mathrm{/}2m}}$.