Mathematische Annalen

, Volume 373, Issue 1–2, pp 831–851 | Cite as

Existence and non-existence of maximizers for the Moser–Trudinger type inequalities under inhomogeneous constraints

  • Norihisa Ikoma
  • Michinori Ishiwata
  • Hidemitsu WadadeEmail author


In this paper, we study the existence and non-existence of maximizers for the Moser–Trudinger type inequalities in \(\mathbb {R}^N\) of the form
$$\begin{aligned} D_{N,\alpha }(a,b):= \sup _{u\in W^{1,N}(\mathbb {R}^N),\,\Vert \nabla u\Vert _{L^N(\mathbb {R}^N)}^a+\Vert u\Vert _{L^N(\mathbb {R}^N)}^b=1} \int _{\mathbb {R}^N}\Phi _N\left( \alpha |u|^{N'}\right) dx. \end{aligned}$$
Here \(N\ge 2, N'=\frac{N}{N-1}, a,b>0, \alpha \in (0,\alpha _N]\) and \(\Phi _N(t):=e^t-\sum _{j=0}^{N-2}\frac{t^j}{j!}\) where \(\alpha _N:= N \omega _{N-1}^{1/(N-1)}\) and \(\omega _{N-1}\) denotes the surface area of the unit ball in \({\mathbb {R}}^{N}\). We show the existence of the threshold \(\alpha _*= \alpha _*(a,b,N) \in [0,\alpha _N]\) such that \(D_{N,\alpha }(a,b)\) is not attained if \(\alpha \in (0,\alpha _*)\) and is attained if \( \alpha \in (\alpha _*, \alpha _N)\). We also provide the conditions on (ab) in order that the inequality \(\alpha _*< \alpha _N\) holds.


Mathematics Subject Classification

47J30 46E35 26D10 



This work was supported by JSPS KAKENHI Grant Number JP16K17623. The authors would like to express their hearty thanks to the referees for their valuable comments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Norihisa Ikoma
    • 1
  • Michinori Ishiwata
    • 2
  • Hidemitsu Wadade
    • 3
    Email author
  1. 1.Department of Mathematics, Faculty of Science and TechnologyKeio UniversityYokohamaJapan
  2. 2.Department of Systems Innovation, Graduate School of Engineering ScienceOsaka UniversityToyonakaJapan
  3. 3.Faculty of Mechanical Engineering, Institute of Science and EngineeringKanazawa UniversityKanazawaJapan

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