Critical GJMS Equations on ${\mathbb{H}}^n\times {\mathbb{S}}^m$

Abstract

Let $M={\mathbb{H}}^n\times {\mathbb{S}}^m$, where $n\ge 2$, $m\ge 1$, and $N=n+m$. Let $P_k$ be the order-$2k$ GJMS operator, with $1\le k<N/2$, and assume that $\hat{I}{›}_0=\inf \ddot{I}{ƒ}_{L^2(M)}(P_k)>0$. We study$$P_{k}U-\hat{I}»U=|U{|}^{q-2}U,q=\frac{2N}{N-2k},0<\hat{I}»\le \hat{I}{›}_0,$$and attainment of the associated critical quotient $S_{\hat{I}»,k}(M)$. Let $S_{N,k}$ be the Euclidean best Sobolev constant. For $0<\hat{I}»<\hat{I}{›}_0$, the inequality $S_{\hat{I}»,k}(M)<S_{N,k}$ implies attainment and a nontrivial weak solution. Localized Euclidean extremals establish this inequality when $N\ge 4k$, or when $2k+2\le N<4k$ and $\hat{I}»>\hat{I}{›}_{\mathrm{loc}}$, where $\hat{I}{›}_{\mathrm{loc}}$ is explicit. If $N\ge 2k+2$ and $S_{\hat{I}{›}_0,k}(M)<S_{N,k}$, attainment also holds at $\hat{I}»=\hat{I}{›}_0$ in the threshold form completion. At the threshold, $L^2$-coercivity fails precisely on the constant spherical eigenspace. We combine cocompactness for its hyperbolic coefficient with a profile decomposition relative to the critical transformations preserving $\mathcal{A}={\mathbb{R}}^{n-1}\times 0$. Under the threshold hypotheses above, the strict Euclidean inequality excludes concentration escaping $\mathcal{A}$ from normalized minimizing sequences. If $r_j^{(J)}$ denotes the remainder after the first $J$ extracted profiles, then$$\underset{J\to \infty }{\lim }\underset{j\to \infty }{\mathrm{limsup}}|r_j^{(J)}{|}_{L^q({\mathbb{R}}^N)}=0,$$which yields compactness modulo the axis-preserving transformations.

46 pages, 1 figure