临界奇异各向异性Moser-Trudinger不等式的极值和阈值

中文摘要 AI 辅助

设$N\ge2$，$q>1$，且$0<\beta<N$，我们在约束条件\\[ \lVert{F(\nabla u)}\rVert_N^a+\lVert{u}\rVert_q^b\le1,\qquad a>0,\qquad 0<b\le N \\]下研究临界奇异各向异性Moser--Trudinger泛函的极值。对于$b<N$，上确界可以达到。在$b=N$时，令\\[ q_-:=\frac{N^2(N-2)}{(N-1)(N-\beta)},\qquad q_+:=\frac{N^2}{N-\beta}。\\] 对于$q<q_-$，对每个$a>0$上确界都可以达到。对于$q_-\le q<q_+$，存在$a_c\in(N,\infty]$使得当$0<a<a_c$时上确界可以达到，而当$a_c<\infty$且$a>a_c$时上确界不能达到。该论证无法确定在$a=a_c$处是否达到。径向爆破分析给出了用非线性格林函数表示的浓度界。第一个非零Taylor项决定了严格比较，精确的欧几里得约化表明阈值与$F$无关，一个单独的补充部分使用经典格林函数和紧Taylor项给出了$q=N$时的另一种证明。

英文摘要

For $N\ge2$, $0<β<N$ and $q>1$, we study maximizers of the critical singular anisotropic Moser--Trudinger integral \[ \int_{\mathbb{R}^N} \frac{Φ_{N,q,β} (λ_N(1-β/N)|u|^{N/(N-1)})} {F^o(x)^β}\,\mathrm{d}x, \qquad \|F(\nabla u)\|_N^a+\|u\|_q^b\le 1. \] where $a>0$, $0<b\le N$ and $Φ_{N,q,β}$ is the integrable Taylor remainder. The supremum is attained for every $a>0$ when $b<N$. For $b=N$ and $1<q<q_+$, the first retained Taylor term separates two regimes, with \[ q_-:=\frac{N^2(N-2)}{(N-1)(N-β)},\qquad q_+:=\frac{N^2}{N-β}. \] Attainment holds for every $a>0$ if $1<q<q_-$, while for $q>1$, $q_-\le q<q_+$, there is a finite threshold $a_c>N$ and attainment holds if and only if $0<a\le a_c$. An exact Euclidean reduction shows that the threshold is independent of the anisotropy. The proof combines critical--subcritical scaling with a nonlinear Green-function concentration bound. Radial flux and Pohozaev identities determine the first correction to the subcritical supremum, and refined Green tests give the strict comparison needed for attainment at the finite threshold, including in dimension two.

发表机构

• Shandong University（山东大学）
• Chongqing Normal University（重庆师范大学）

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