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 . 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).