This paper is addressed to study the existence of maximizers for the singular Moser–Trudinger supremum under constraints in the critical case MTN(a,β)=supu∈W1,N(RN),‖∇u‖Na+‖u‖NN=1∫RNΦN(1-β/N)αN|u|NN-1|x|-βdx,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} MT_{N}(a,\beta ) = \sup _{u\in W^{1,N}({\mathbb {R}}^N),\, \Vert \nabla u\Vert _N^a + \Vert u\Vert _N^N =1} \int _{{\mathbb {R}}^N}\Phi _N\left( (1-\beta /N)\alpha _N |u|^{\frac{N}{N-1}}\right) |x|^{-\beta } dx, \end{aligned}$$\end{document}where a>0\documentclass[12pt]{minimal}
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\begin{document}$$a>0$$\end{document}, β∈[0,N)\documentclass[12pt]{minimal}
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\begin{document}$$\beta \in [0,N)$$\end{document}, ΦN(t)=et-∑k=0N-2tkk!\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _N(t) = e^t -\sum _{k=0}^{N-2} \frac{t^k}{k!}$$\end{document}, αN=NωN-11/(N-1)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _N = N \omega _{N-1}^{1/(N-1)}$$\end{document}, and ωN-1\documentclass[12pt]{minimal}
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\begin{document}$$\omega _{N-1}$$\end{document} denotes the surface area of the unit sphere in RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document}. More precisely, we study the effect of the parameter a to the attainability of MTN(a,β)\documentclass[12pt]{minimal}
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\begin{document}$$MT_{N}(a,\beta )$$\end{document}. We will prove that for each β∈[0,N)\documentclass[12pt]{minimal}
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\begin{document}$$\beta \in [0,N)$$\end{document} there exist the thresholds a∗(β)\documentclass[12pt]{minimal}
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\begin{document}$$a_*(\beta )$$\end{document} and a∗(β)\documentclass[12pt]{minimal}
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\begin{document}$$a^*(\beta )$$\end{document} such that MTN(a,β)\documentclass[12pt]{minimal}
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\begin{document}$$MT_{N}(a,\beta )$$\end{document} is attained for any a∈(a∗(β),a∗(β))\documentclass[12pt]{minimal}
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\begin{document}$$a \in (a_*(\beta ), a^*(\beta ))$$\end{document} and is not attained for a<a∗(β)\documentclass[12pt]{minimal}
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\begin{document}$$a < a_*(\beta )$$\end{document} or a>a∗(β)\documentclass[12pt]{minimal}
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\begin{document}$$a > a^*(\beta )$$\end{document}. We also give some qualitative estimates for a∗(β)\documentclass[12pt]{minimal}
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\begin{document}$$a_*(\beta )$$\end{document} and a∗(β)\documentclass[12pt]{minimal}
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\begin{document}$$a^*(\beta )$$\end{document}. 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).
