关于\(\mathbb{H}^n\times\mathbb{S}^m\)上的临界GJMS方程
Critical GJMS Equations on $\mathbb{H}^n \times \mathbb{S}^m$
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中文总结 AI 辅助
研究\(\mathbb{H}^n\times\mathbb{S}^m\)上特定方程及临界商，通过分析不同条件下不等式关系，利用局部欧几里得极值等方法，探讨临界商可达性及非平凡弱解，得出相关结论并建立紧致性。
中文摘要 AI 辅助
设\(M = \mathbb{H}^n\times\mathbb{S}^m\)，其中\(n\geq2\)，\(m\geq1\)且\(N = n + m\)。令\(P_k\)为\(2k\)阶GJMS算子，\(1\leq k < N/2\)，假设\(\Lambda_0=\inf\sigma_{L^2(M)}(P_k)>0\)。研究方程\(P_kU - \lambda U = |U|^{q - 2}U\)（\(q=\frac{2N}{N - 2k}\)，\(0 < \lambda\leq\Lambda_0\)）及相关临界商\(S_{\lambda,k}(M)\)的可达性。对于\(0 < \lambda < \Lambda_0\)，不等式\(S_{\lambda,k}(M) < S_{N,k}\)意味着可达性和非平凡弱解。当\(N\geq4k\)或\(2k + 2\leq N < 4k\)且\(\lambda > \Lambda_{\mathrm{loc}}\)时，局部欧几里得极值建立此不等式。若\(N\geq2k + 2\)且\(S_{\Lambda_0,k}(M) < S_{N,k}\)，在阈值形式完备化中\(\lambda = \Lambda_..
英文摘要
Let $M=\mathbb{H}^n\times\mathbb{S}^m$, where $n\geq 2$, $m\geq 1$, and $N=n+m$. Let $P_k$ be the order-$2k$ GJMS operator, with $1\leq k<N/2$, and assume that $Λ_0=\infσ_{L^2(M)}(P_k)>0$. We study$$P_kU-λU=|U|^{q-2}U,\qquad q=\frac{2N}{N-2k},\qquad 0<λ\leqΛ_0,$$and attainment of the associated critical quotient $S_{λ,k}(M)$. Let $S_{N,k}$ be the Euclidean best Sobolev constant. For $0<λ<Λ_0$, the inequality $S_{λ,k}(M)<S_{N,k}$ implies attainment and a nontrivial weak solution. Localized Euclidean extremals establish this inequality when $N\geq4k$, or when $2k+2\leq N<4k$ and $λ>Λ_{\mathrm{loc}}$, where $Λ_{\mathrm{loc}}$ is explicit. If $N\geq2k+2$ and $S_{Λ_0,k}(M)<S_{N,k}$, attainment also holds at $λ=Λ_0$ 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$$\lim_{J\to\infty}\limsup_{j\to\infty}|r_j^{(J)}|_{L^q(\mathbb{R}^N)}=0,$$which yields compactness modulo the axis-preserving transformations.
