On the maximizing problem associated with critical Sobolev inequality under inhomogeneous constraints

Abstract

In this paper, we study a maximizing problem associated with the critical Sobolev inequality under inhomogeneous constraints. The problem of this type was previously studied by Ishiwata and Wadade [On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in ${\mathbb{R}}^N$. Math Ann. 2016;364:1043–1068; On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934], Ishiwata [On variational problems associated with Sobolev type inequalities and related topics. Available from: http://www.rism.it/doc/Ishiwata.pdf] and Nguyen [Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math Ann. 2018;372(1–2):229–255]. Our results give a complete picture of the effect of the constraints on the attainability and non-attainability of the problem. The sharp Sobolev inequality plays a crucial role in our argument. Our method also provides a new and simple proof for the recent results of Ishiwata and Wadade [On the maximizing problem associated with Sobolev type embeddings under inhomogeneous constraints. Appl Anal. 2019;98(10):1916–1934] concerning the sub-critical Sobolev-type inequalities (or Gagliardo–Nirenberg inequalities).

Keywords:

• Sobolev-type embeddings
• maximizers
• attainability
• critical exponent

AMS Subject Classifications:

• 26D10
• 46E35

Acknowledgements

The author would like to thank the anonymous reviewers for the constructive and useful comments and suggestions which improve the organization and quality of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).