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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang


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Volume 12, Issue 2

Issues

Maximal solutions for the ∞-eigenvalue problem

João Vitor da Silva
  • Departamento de Matemática, FCEyN, Universidad de Buenos Aires; and IMAS – CONICET, Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n., Buenos Aires, Argentina
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/ Julio D. Rossi
  • Corresponding author
  • Departamento de Matemática, FCEyN, Universidad de Buenos Aires; and IMAS – CONICET, Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n., Buenos Aires, Argentina
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/ Ariel M. Salort
  • Departamento de Matemática, FCEyN, Universidad de Buenos Aires; and IMAS – CONICET, Ciudad Universitaria, Pabellón I (1428) Av. Cantilo s/n., Buenos Aires, Argentina
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Published Online: 2017-07-06 | DOI: https://doi.org/10.1515/acv-2017-0024

Abstract

In this article we prove that the first eigenvalue of the -Laplacian

{min{-Δv,|v|-λ1,(Ω)v}=0in Ω,v=0on Ω,

has a unique (up to scalar multiplication) maximal solution. This maximal solution can be obtained as the limit as 1 of concave problems of the form

{min{-Δv,|v|-λ1,(Ω)v}=0in Ω,v=0on Ω.

In this way we obtain that the maximal eigenfunction is the unique one that is the limit of the sub-homogeneous problems as happens for the usual eigenvalue problem for the p-Laplacian for a fixed 1<p<.

Keywords: Maximal solutions; eigenvalue problems; degenerate fully nonlinear elliptic equations,infinity-Laplacian operator

MSC 2010: 35B27; 35J60; 35J70

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About the article


Received: 2017-05-01

Revised: 2017-06-19

Accepted: 2017-06-23

Published Online: 2017-07-06

Published in Print: 2019-04-01


This work has partially been supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET-Argentina, PIP GI No. 11220150100036CO).


Citation Information: Advances in Calculus of Variations, Volume 12, Issue 2, Pages 181–191, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2017-0024.

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