Vol. 14, No. 8, 2021

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On the global bifurcation diagram of the Gelfand problem

Daniele Bartolucci and Aleks Jevnikar

Vol. 14 (2021), No. 8, 2409–2426
Abstract

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 L(Ω)-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.

Keywords
global bifurcation, Gelfand problem, Rabinowitz continuum, mean field equation
Mathematical Subject Classification 2010
Primary: 35B45, 35J60, 35J99
Milestones
Received: 20 January 2019
Revised: 30 April 2020
Accepted: 15 September 2020
Published: 19 December 2021
Authors
Daniele Bartolucci
Department of Mathematics
University of Rome “Tor Vergata”
Rome
Italy
Aleks Jevnikar
Department of Mathematics, Computer Science and Physics
University of Udine
Udine
Italy