Energy asymptotics in the three-dimensional Brezis–Nirenberg problem

Frank, Rupert L. and König, Tobias and Kovařík, Hynek (2021) Energy asymptotics in the three-dimensional Brezis–Nirenberg problem. Calculus of Variations and Partial Differential Equations, 60 (2). Art. No. 58. ISSN 0944-2669. https://resolver.caltech.edu/CaltechAUTHORS:20210222-100903531

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Abstract

For a bounded open set Ω⊂ℝ³ we consider the minimization problem S(a + V) = inf|[0≡u∈H¹₀(Ω) [∫_Ω(|∇u|² + (a + ϵV)|u|²) dx]/(∫_Ω u⁶ dx)^(1/3)] involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of S(a+V)−S as ϵ → 0+, where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have S(a + ϵV) < S for all sufficiently small ϵ > 0.

Item Type:

Article

Related URLs:

URL	URL Type	Description
https://doi.org/10.1007/s00526-021-01929-3	DOI	Article
https://arxiv.org/abs/1908.01331	arXiv	Discussion Paper

ORCID:

Author	ORCID
Frank, Rupert L.	0000-0001-7973-4688
Kovařík, Hynek	0000-0003-3647-8447

Additional Information:

© The Author(s) 2021. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Received 09 October 2020. Accepted 25 January 2021. Published 19 February 2021. This paper may be reproduced, in its entirety, for non-commercial purposes. Partial support through US National Science Foundation Grants DMS-1363432 and DMS-1954995 (R.L.F.), Studienstiftung des deutschen Volkes (T.K.) and Gruppo Nazionale per Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) (H.K.) is acknowledged. Communicated by Andrea Malchiodi.

Funders:

Funding Agency	Grant Number
NSF	DMS-1363432
NSF	DMS-1954995
Studienstiftung des Deutschen Volkes	UNSPECIFIED
Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni	UNSPECIFIED

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Classification Code:

Mathematics Subject Classification: 35A15; 35J20; 35B44

Record Number:

CaltechAUTHORS:20210222-100903531

Persistent URL:

https://resolver.caltech.edu/CaltechAUTHORS:20210222-100903531

Official Citation:

Frank, R.L., König, T. & Kovařík, H. Energy asymptotics in the three-dimensional Brezis–Nirenberg problem. Calc. Var. 60, 58 (2021). https://doi.org/10.1007/s00526-021-01929-3

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ID Code:

108138

Collection:

CaltechAUTHORS

Deposited By:

George Porter

Deposited On:

23 Feb 2021 15:31

Last Modified:

23 Feb 2021 15:31

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