The thresholds of the existence of maximizers for the critical sharp singular Moser–Trudinger inequality under constraints

Abstract

This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case

$$\begin{aligned} MT_{N}(a,\beta ) = \sup _{u\in W^{1,N}({\mathbb {R}}^N),\, \Vert \nabla u\Vert _N^a + \Vert u\Vert _N^N =1} \int _{{\mathbb {R}}^N}\Phi _N\left( (1-\beta /N)\alpha _N |u|^{\frac{N}{N-1}}\right) |x|^{-\beta } dx, \end{aligned}$$

where $a>0$, $\beta \in [0,N)$, $\Phi _N(t) = e^t -\sum _{k=0}^{N-2} \frac{t^k}{k!}$, $\alpha _N = N \omega _{N-1}^{1/(N-1)}$, and $\omega _{N-1}$ denotes the surface area of the unit sphere in ${\mathbb {R}}^N$. More precisely, we study the effect of the parameter a to the attainability of $MT_{N}(a,\beta )$. We will prove that for each $\beta \in [0,N)$ there exist the thresholds $a_*(\beta )$ and $a^*(\beta )$ such that $MT_{N}(a,\beta )$ is attained for any $a \in (a_*(\beta ), a^*(\beta ))$ and is not attained for $a < a_*(\beta )$ or $a > a^*(\beta )$. We also give some qualitative estimates for $a_*(\beta )$ and $a^*(\beta )$. 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).