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On the global
bifurcation diagram of the Gelfand problem
Daniele Bartolucci and Aleks Jevnikar |
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Vol. 14 (2021), No. 8, 2409–2426
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Abstract |
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For domains of first kind we describe the qualitative behavior of the global bifurcation diagram of the unbounded branch of solutions of the Gelfand problem crossing the origin. At least to our knowledge this is the first result about the exact monotonicity of the branch of nonminimal solutions which is not just concerned with radial solutions and/or with symmetric domains. Toward our goal we parametrize the branch not by the -norm of the solutions but by the energy of the associated mean field problem. The proof relies on a refined spectral analysis of mean-field-type equations and some surprising properties of the quantities triggering the monotonicity of the Gelfand parameter. |
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Keywords
global bifurcation, Gelfand problem, Rabinowitz continuum,
mean field equation
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Mathematical Subject Classification 2010
Primary: 35B45, 35J60, 35J99
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Milestones
Received: 20 January 2019
Revised: 30 April 2020
Accepted: 15 September 2020
Published: 19 December 2021
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Authors |
| Daniele Bartolucci | |
| Department of Mathematics University of Rome “Tor Vergata” Rome Italy |
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| Aleks Jevnikar | |
| Department of Mathematics, Computer
Science and Physics University of Udine Udine Italy |
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