Revisiting Non-Rotating Star Models: Classical Existence and Uniqueness Theory and Scaling Relations

This paper presents a systematic study of the properties of non-rotating stellar models governed by the Euler-Poisson system under general equations of state, including the case of polytropic gaseous stars. We revisit and extend existence results by Auchmuty and Beals \cite{AB71}, adapt the uniqueness results from the quantum mechanical framework of Lieb and Yau \cite{LY87} to the classical Newtonian mechanical setting. The results are also synthesized in McCann \cite{McC06} but without proof. The second work we do is applying a scaling method to establish relations between solutions with different total masses. As the mass tends to zero, we analyze convergence properties of the density functions and identify precise rates for the contraction or extension of their supports.

💡 Research Summary

The paper conducts a thorough mathematical investigation of static, non‑rotating stellar configurations governed by the Euler–Poisson system. Starting from the full time‑dependent Euler–Poisson equations, the authors specialize to the stationary, zero‑velocity case, reducing the system to the elliptic balance equation ∇P(ρ) = ρ∇V, where V is the Newtonian potential generated by the density itself. The analysis is carried out for a very general class of equations of state P(ρ) that satisfy continuity, strict monotonicity, and appropriate growth conditions at zero and infinity (denoted (F1)–(F3)). Additional smoothness assumptions (F4 or the weaker F4′) are introduced when differentiability of the pressure is needed.

The central variational object is the energy functional
E₀(ρ) = U(ρ) – ½∬ ρ(x)ρ(y)/|x–y| dx dy,
with internal energy U(ρ)=∫A(ρ)dx, where A(s)=∫₀^s P(τ)τ⁻² dτ. The admissible class consists of non‑negative densities of prescribed total mass m that belong to L^{4/3}(ℝ³)∩L¹(ℝ³). The authors first prove the existence of a minimizer σ_m for each m≥0. Their proof follows the classic direct method but is carefully adapted to the present setting: they introduce a truncated, axially symmetric class W_R with uniform L^∞ bounds and compact support, establish weak continuity of the gravitational term G, and then pass to the limit R→∞ using a sequence of four lemmas that guarantee uniform bounds, negativity of the energy, confinement of the support, and finally global minimality. This yields Theorem 2.6, which collects all known existence properties: existence of a minimizer, strict monotonicity and convexity of the minimal energy curve e₀(m), spherical symmetry and radial monotonicity of σ_m, uniform L^∞ bounds, compact support within a radius R₀(m), continuity (and C¹ regularity under (F4)), satisfaction of the Euler–Lagrange condition A′(σ_m) =