# American Institute of Mathematical Sciences

January  2015, 35(1): 139-154. doi: 10.3934/dcds.2015.35.139

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

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

Received  September 2013 Revised  May 2014 Published  August 2014

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.
Citation: Jean Dolbeault, Robert Stańczy. Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi--Dirac statistics. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 139-154. doi: 10.3934/dcds.2015.35.139
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##### References:
 [1] Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic and Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014 [2] Giuseppe Maria Coclite, Helge Holden. Ground states of the Schrödinger-Maxwell system with dirac mass: Existence and asymptotics. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 117-132. doi: 10.3934/dcds.2010.27.117 [3] Simone Paleari, Tiziano Penati. Equipartition times in a Fermi-Pasta-Ulam system. Conference Publications, 2005, 2005 (Special) : 710-719. doi: 10.3934/proc.2005.2005.710 [4] Alexander Pankov. Traveling waves in Fermi-Pasta-Ulam chains with nonlocal interaction. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2097-2113. doi: 10.3934/dcdss.2019135 [5] Antonio Giorgilli, Simone Paleari, Tiziano Penati. Local chaotic behaviour in the Fermi-Pasta-Ulam system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 991-1004. doi: 10.3934/dcdsb.2005.5.991 [6] Yemin Chen. Analytic regularity for solutions of the spatially homogeneous Landau-Fermi-Dirac equation for hard potentials. Kinetic and Related Models, 2010, 3 (4) : 645-667. doi: 10.3934/krm.2010.3.645 [7] Kousuke Kuto, Tohru Tsujikawa. Bifurcation structure of steady-states for bistable equations with nonlocal constraint. Conference Publications, 2013, 2013 (special) : 467-476. doi: 10.3934/proc.2013.2013.467 [8] Richard Sharp, Anastasios Stylianou. Statistics of multipliers for hyperbolic rational maps. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1225-1241. doi: 10.3934/dcds.2021153 [9] Carmen Cortázar, Manuel Elgueta, Jorge García-Melián, Salomé Martínez. Finite mass solutions for a nonlocal inhomogeneous dispersal equation. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1409-1419. doi: 10.3934/dcds.2015.35.1409 [10] Santiago Cano-Casanova. Decay rate at infinity of the positive solutions of a generalized class of $T$homas-Fermi equations. Conference Publications, 2011, 2011 (Special) : 240-249. doi: 10.3934/proc.2011.2011.240 [11] Francisco Guillén-González, Mamadou Sy. Iterative method for mass diffusion model with density dependent viscosity. Discrete and Continuous Dynamical Systems - B, 2008, 10 (4) : 823-841. doi: 10.3934/dcdsb.2008.10.823 [12] Shishuo Fu, Zhicong Lin, Yaling Wang. Refined Wilf-equivalences by Comtet statistics. Electronic Research Archive, 2021, 29 (5) : 2877-2913. doi: 10.3934/era.2021018 [13] Jacopo De Simoi. Stability and instability results in a model of Fermi acceleration. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 719-750. doi: 10.3934/dcds.2009.25.719 [14] Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic and Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1 [15] Gisèle Ruiz Goldstein, Jerome A. Goldstein, Naima Naheed. A convexified energy functional for the Fermi-Amaldi correction. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 41-65. doi: 10.3934/dcds.2010.28.41 [16] Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic and Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009 [17] Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129 [18] Xinliang An, Avy Soffer. Fermi's golden rule and $H^1$ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013 [19] Takahisa Inui, Nobu Kishimoto, Kuranosuke Nishimura. Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6299-6353. doi: 10.3934/dcds.2019275 [20] Andrew Comech, David Stuart. Small amplitude solitary waves in the Dirac-Maxwell system. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1349-1370. doi: 10.3934/cpaa.2018066

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