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

Rupert L. Frank, Tobias König and Hynek Kovařík

Vol. 17 (2024), No. 5, 1633–1692
Abstract

We describe the asymptotic behavior of positive solutions u𝜀 of the equation Δu + au = 3u5𝜀 in Ω 3 with a homogeneous Dirichlet boundary condition. The function a is assumed to be critical in the sense of Hebey and Vaugon, and the functions u𝜀 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 Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation Δu + (a + 𝜀V )u = 3u5 in Ω.

Keywords
blow-up, critical elliptic equations
Mathematical Subject Classification
Primary: 35B44, 35J60
Milestones
Received: 11 August 2021
Revised: 27 July 2022
Accepted: 23 September 2022
Published: 20 June 2024
Authors
Rupert L. Frank
Mathematisches Institut
Ludwig-Maximilians-Universitat München
München
Germany
Department of Mathematics
Caltech
Pasadena, CA
United States
Tobias König
Institut de Mathématiques de Jussieu
Paris Rive Gauche Université de Paris
Campus des Grands Moulins
Paris
France
Institut für Mathematik
Goethe University Frankfurt
Frankfurt am Main
Germany
Hynek Kovařík
DICATAM, Sezione di Matematica
Università degli studi di Brescia
Brescia
Italy

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