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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. doi:10.1007/s00526-021-01929-3.

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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
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URLURL TypeDescription Paper
Frank, Rupert L.0000-0001-7973-4688
Kovařík, Hynek0000-0003-3647-8447
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 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.
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Studienstiftung des Deutschen VolkesUNSPECIFIED
Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro ApplicazioniUNSPECIFIED
Issue or Number:2
Classification Code:Mathematics Subject Classification: 35A15; 35J20; 35B44
Record Number:CaltechAUTHORS:20210222-100903531
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Official Citation:Frank, R.L., König, T. & Kovařík, H. Energy asymptotics in the three-dimensional Brezis–Nirenberg problem. Calc. Var. 60, 58 (2021).
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108138
Deposited By: George Porter
Deposited On:23 Feb 2021 15:31
Last Modified:16 Nov 2021 19:09

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