This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case $$\begin{aligned} MT_{N}(a,\beta ) = \sup _{u\in W^{1,N}({\mathbb {R}}^N),\, \Vert \nabla u\Vert _N^a + \Vert u\Vert _N^N =1} \int _{{\mathbb {R}}^N}\Phi _N\left( (1-\beta /N)\alpha _N |u|^{\frac{N}{N-1}}\right) |x|^{-\beta } dx, \end{aligned}$$ M T N ( a , β ) = sup u ∈ W 1 , N ( R N ) , ‖ ∇ u ‖ N a + ‖ u ‖ N N = 1 ∫ R N Φ N ( 1 - β / N ) α N | u | N N - 1 | x | - β d x , where $$a>0$$ a > 0 , $$\beta \in [0,N)$$ β ∈ [ 0 , N ) , $$\Phi _N(t) = e^t -\sum _{k=0}^{N-2} \frac{t^k}{k!}$$ Φ N ( t ) = e t - ∑ k = 0 N - 2 t k k ! , $$\alpha _N = N \omega _{N-1}^{1/(N-1)}$$ α N = N ω N - 1 1 / ( N - 1 ) , and $$\omega _{N-1}$$ ω N - 1 denotes the surface area of the unit sphere in $${\mathbb {R}}^N$$ R N . More precisely, we study the effect of the parameter a to the attainability of $$MT_{N}(a,\beta )$$ M T N ( a , β ) . We will prove that for each $$\beta \in [0,N)$$ β ∈ [ 0 , N ) there exist the thresholds $$a_*(\beta )$$ a ∗ ( β ) and $$a^*(\beta )$$ a ∗ ( β ) such that $$MT_{N}(a,\beta )$$ M T N ( a , β ) is attained for any $$a \in (a_*(\beta ), a^*(\beta ))$$ a ∈ ( a ∗ ( β ) , a ∗ ( β ) ) and is not attained for $$a < a_*(\beta )$$ a < a ∗ ( β ) or $$a > a^*(\beta )$$ a > a ∗ ( β ) . We also give some qualitative estimates for $$a_*(\beta )$$ a ∗ ( β ) and $$a^*(\beta )$$ a ∗ ( β ) . Our results complete the recent studies on the sharp Moser–Trudinger type inequality under constraints due to do Ó, Sani and Tarsi (Commun Contemp Math 19:27, 2016), Lam (Proc Am Math Soc 145:4885–4892, 2017; Math Nachr 291(14–15):2272–2287, 2018) and Ikoma, Ishiwata and Wadade (Math Ann 373(1–2):831–851, 2019).

中文翻译：

---

约束条件下临界尖锐奇异 Moser-Trudinger 不等式的最大值存在的阈值

本文旨在研究临界情况下奇异 Moser-Trudinger supremum 的极大值存在性 $$\begin{aligned} MT_{N}(a,\beta ) = \sup _{u\in W^ {1,N}({\mathbb {R}}^N),\, \Vert \nabla u\Vert _N^a + \Vert u\Vert _N^N =1} \int _{{\mathbb {R }}^N}\Phi _N\left( (1-\beta /N)\alpha _N |u|^{\frac{N}{N-1}}\right) |x|^{-\beta } dx, \end{aligned}$$ MTN ( a , β ) = sup u ∈ W 1 , N ( RN ) , ‖ ∇ u ‖ N a + ‖ u ‖ NN = 1 ∫ RN Φ N ( 1 - β / N ) α N | 你| NN - 1 | × | - β dx , 其中 $$a>0$$ a > 0 , $$\beta \in [0,N)$$ β ∈ [ 0 , N ) , $$\Phi _N(t) = e^t - \sum _{k=0}^{N-2} \frac{t^k}{k!}$$ Φ N ( t ) = et - ∑ k = 0 N - 2 tkk ！, $$\alpha _N = N \omega _{N-1}^{1/(N-1)}$$ α N = N ω N - 1 1 / ( N - 1 ) , $$\omega _ {N-1}$$ ω N - 1 表示$${\mathbb {R}}^N$$ RN 中单位球体的表面积。更准确地说，我们研究了参数 a 对 $$MT_{N}(a,\beta )$$MTN ( a , β ) 可达到性的影响。我们将证明对于每个 $$\beta \in [0,N)$$ β ∈ [ 0 , N ) 存在阈值 $$a_*(\beta )$$ a ∗ ( β ) 和 $$a^ *(\beta )$$ a ∗ ( β ) 使得 $$MT_{N}(a,\beta )$$ MTN ( a , β ) 对于任何 $$a \in (a_*(\beta ) , a^*(\beta ))$$ a ∈ ( a ∗ ( β ) , a ∗ ( β ) ) 并且在 $$a < a_*(\beta )$$ a < a ∗ ( β ) 时不成立或 $$a > a^*(\beta )$$ a > a ∗ ( β ) 。我们还给出了 $$a_*(\beta )$$a ∗ ( β ) 和 $$a^*(\beta )$$ a ∗ ( β ) 的一些定性估计。我们的结果完成了最近对受 do Ó、Sani 和 Tarsi（Commun Contemp Math 19:27, 2016）、Lam（Proc Am Math Soc 145:4885–4892, 2017；Math Nachr 291(14–15):2272–2287,