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

TL;DR

This paper investigates the multiplicity of solutions for nonlocal elliptic equations modeling gravitating systems with Fermi-Dirac statistics, using bifurcation analysis and numerical illustrations to understand solution structures.

Contribution

It introduces a novel approach combining mass density analysis and coordinate transformations to study bifurcations in nonlocal elliptic equations with Fermi-Dirac statistics.

Findings

Multiple solution branches identified through bifurcation diagrams.

Numerical computations illustrate the complex solution structure.

The mass parameter significantly influences the solution multiplicity.

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 one, we are able to produce multiplicity results. Our method is based on cumulated mass densities and a logarithmic change of coordinates that allows 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 a 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.