In this paper, we study the existence and non-existence of maximizers for the Moser–Trudinger type inequalities in RN\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {R}^N$$\end{document} of the form DN,α(a,b):=supu∈W1,N(RN),‖∇u‖LN(RN)a+‖u‖LN(RN)b=1∫RNΦNα|u|N′dx.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} D_{N,\alpha }(a,b):= \sup _{u\in W^{1,N}(\mathbb {R}^N),\,\Vert \nabla u\Vert _{L^N(\mathbb {R}^N)}^a+\Vert u\Vert _{L^N(\mathbb {R}^N)}^b=1} \int _{\mathbb {R}^N}\Phi _N\left( \alpha |u|^{N'}\right) dx. \end{aligned}$$\end{document}Here N≥2,N′=NN-1,a,b>0,α∈(0,αN]\documentclass[12pt]{minimal}
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\begin{document}$$N\ge 2, N'=\frac{N}{N-1}, a,b>0, \alpha \in (0,\alpha _N]$$\end{document} and ΦN(t):=et-∑j=0N-2tjj!\documentclass[12pt]{minimal}
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\begin{document}$$\Phi _N(t):=e^t-\sum _{j=0}^{N-2}\frac{t^j}{j!}$$\end{document} where 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 ball in RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^{N}$$\end{document}. We show the existence of the threshold α∗=α∗(a,b,N)∈[0,αN]\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _*= \alpha _*(a,b,N) \in [0,\alpha _N]$$\end{document} such that DN,α(a,b)\documentclass[12pt]{minimal}
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\begin{document}$$D_{N,\alpha }(a,b)$$\end{document} is not attained if α∈(0,α∗)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \in (0,\alpha _*)$$\end{document} and is attained if α∈(α∗,αN)\documentclass[12pt]{minimal}
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\begin{document}$$ \alpha \in (\alpha _*, \alpha _N)$$\end{document}. We also provide the conditions on (a, b) in order that the inequality α∗<αN\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _*< \alpha _N$$\end{document} holds.
