On the global bifurcation diagram of the Gelfand problem

Bartolucci, Daniele; Jevnikar, Aleks

doi:10.2140/apde.2021.14.2409

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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

DOI: 10.2140/apde.2021.14.2409

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 $L^{\infty} (Ω)$ -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

Vol. 14, No. 8, 2021