Maximizers for the variational problems associated with Sobolev type\n inequalities under constraints

2017/05/23 by Van Hoang Nguyen, Nguyen, Van Hoang
Computer Science · Engineering · Mathematics · #26D10 #46E35 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1705.08434

openalex publication_date 2017/05/23 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

We propose a new approach to study the existence and non-existence of\nmaximizers for the variational problems associated with Sobolev type\ninequalities both in the subcritical case and critical case under the\nequivalent constraints. The method is based on an useful link between the\nattainability of the supremum in our variational problems and the attainablity\nof the supremum of some special functions defined on (0,\∞). Our approach\ncan be applied to the same problems related to the fractional Laplacian\noperators. Our main results are new in the critical case and in the setting of\nthe fractional Laplacian operator which was left open in the work of Ishiwata\nand Wadade citeIshiwata,Ishiwata1. In the subcritical case, our approach\nprovides a new and elementary proof of the results of Ishiwata and Wadade\n citeIshiwata,Ishiwata1.\n

Citations

• Hitchhikerʼs guide to the fractional Sobolev spaces
• A relation between pointwise convergence of functions and convergence of functionals
• Extremal functions for the Moser--Trudinger inequality of Adimurthi--Druet type in W^1,N(\mathbb R^N)

Related

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• Rigidity theorems for best Sobolev inequalities