Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics

Pages: 139 - 154, Volume 35, Issue 1, January 2015      doi:10.3934/dcds.2015.35.139

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Jean Dolbeault - Ceremade (UMR CNRS no. 7534), Université Paris Dauphine, Place de Lattre de Tassigny, 75775 Paris Cédex 16, France (email)
Robert Stańczy - Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (email)

Abstract: This paper is devoted to multiplicity results of solutions to nonlocal elliptic equations modeling gravitating systems. By considering the case of Fermi--Dirac statistics as a singular perturbation of Maxwell--Boltzmann statistics, we are able to produce multiplicity results. Our method is based on cumulated mass densities and a logarithmic change of coordinates that allow us to describe the set of all solutions by a non-autonomous perturbation of an autonomous dynamical system. This has interesting consequences in terms of bifurcation diagrams, which are illustrated by some numerical computations. More specifically, we study a model based on the Fermi function as well as a simplified one for which estimates are easier to establish. The main difficulty comes from the fact that the mass enters in the equation as a parameter which makes the whole problem non-local.

Keywords:  Gravitation, Fermi--Dirac statistics, Maxwell--Boltzmann statistics, Fermi function, cumulated mass density, mass constraint, bifurcation diagrams, nonlocal elliptic equations, dynamical system, singular perturbation.
Mathematics Subject Classification:  Primary: 35Q85, 70K05, 85A05; Secondary: 34E15, 37N05.

Received: September 2013;      Revised: May 2014;      Available Online: August 2014.