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Blow-up of solutions of critical elliptic equation in three dimensions

Frank, Rupert L. and König, Tobias and Kovařík, Hynek (2021) Blow-up of solutions of critical elliptic equation in three dimensions. . (Unpublished) https://resolver.caltech.edu/CaltechAUTHORS:20211004-232817904

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Abstract

We describe the asymptotic behavior of positive solutions uϵ of the equation −Δu + au = 3u^(5−ϵ) in Ω ⊂ ℝ³ with a homogeneous Dirichlet boundary condition. The function a is assumed to be critical in the sense of Hebey and Vaugon and the functions u_ϵ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brézis and Peletier (1989). Similar results are also obtained for solutions of the equation −Δu +(a+ϵV)u = 3u⁵ in Ω.


Item Type:Report or Paper (Discussion Paper)
Related URLs:
URLURL TypeDescription
http://arxiv.org/abs/2102.10525arXivDiscussion Paper
ORCID:
AuthorORCID
Frank, Rupert L.0000-0001-7973-4688
König, Tobias0000-0002-8808-897X
Kovařík, Hynek0000-0003-3647-8447
Additional Information:© 2021 by the authors. 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.) and through ANR BLADE-JC ANR-18-CE40-002 (T.K.) is acknowledged. T.K. thanks Paul Laurain for several useful discussions.
Funders:
Funding AgencyGrant Number
NSFDMS-1363432
NSFDMS-1954995
Agence Nationale pour la Recherche (ANR)ANR-18-CE40-002
Record Number:CaltechAUTHORS:20211004-232817904
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20211004-232817904
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:111204
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:07 Oct 2021 19:10
Last Modified:07 Oct 2021 19:10

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