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Existence and nonrelativistic limit of ground states to nonlinear Dirac equation

Pan Chen, Yanheng Ding, Qi Guo

TL;DR

This work analyzes ground states of nonlinear Dirac equations with power-type nonlinearities, proving the existence of energy ground states and establishing a precise nonrelativistic limit: as the speed of light $c$ tends to infinity, both energy and action ground states converge to their Schrödinger counterparts. The authors develop a robust variational framework on reduced Nehari manifolds, combine concentration-compactness with careful control of positive/negative spectral components, and obtain a convergence rate of $\mathcal{O}(c^{-2})$ for the ground-state energies. They also show that action and energy ground states are equivalent in the large-$c$ limit, strengthening the link between relativistic and nonrelativistic models. The results extend existing subcritical analyses to the Sobolev subcritical range $p\in(2,3)$ and provide a rigorous bridge between Dirac and Schrödinger ground states, with implications for relativistic quantum models and density functional contexts.

Abstract

This paper explores the existence and properties of ground states, including both energy and action ground states, for nonlinear Dirac equations with power-type potentials. \begin{equation*} -i c\sum\limits_{k=1}^3α_k\partial_k u +mc^2 β{u}- |{u}|^{p-2}{u}=ω{u}. \end{equation*} We establish the existence of energy ground states and demonstrate that as the speed of light approaches infinity, both energy and action ground states converge to their counterparts in the nonlinear Schrödinger equation. Furthermore, we characterize the convergence rate of the ground state energy and investigate the equivalence between action and energy ground states.

Existence and nonrelativistic limit of ground states to nonlinear Dirac equation

TL;DR

This work analyzes ground states of nonlinear Dirac equations with power-type nonlinearities, proving the existence of energy ground states and establishing a precise nonrelativistic limit: as the speed of light tends to infinity, both energy and action ground states converge to their Schrödinger counterparts. The authors develop a robust variational framework on reduced Nehari manifolds, combine concentration-compactness with careful control of positive/negative spectral components, and obtain a convergence rate of for the ground-state energies. They also show that action and energy ground states are equivalent in the large- limit, strengthening the link between relativistic and nonrelativistic models. The results extend existing subcritical analyses to the Sobolev subcritical range and provide a rigorous bridge between Dirac and Schrödinger ground states, with implications for relativistic quantum models and density functional contexts.

Abstract

This paper explores the existence and properties of ground states, including both energy and action ground states, for nonlinear Dirac equations with power-type potentials. \begin{equation*} -i c\sum\limits_{k=1}^3α_k\partial_k u +mc^2 β{u}- |{u}|^{p-2}{u}=ω{u}. \end{equation*} We establish the existence of energy ground states and demonstrate that as the speed of light approaches infinity, both energy and action ground states converge to their counterparts in the nonlinear Schrödinger equation. Furthermore, we characterize the convergence rate of the ground state energy and investigate the equivalence between action and energy ground states.

Paper Structure

This paper contains 12 sections, 56 theorems, 322 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Ground States
  3. Main Results
  4. Preliminary Results
  5. Existence of energy ground state
  6. Maximization problem
  7. Upper bound estimation for ground state energy
  8. Minimization problem
  9. Nonrelativistic Limits of energy ground state
  10. Nonrelativistic Limits of action ground state
  11. Equivalence of action and energy ground state
  12. Ground states of Nonlinear Schrödinger equation

Key Result

Theorem 1.3

Let $p\in (2,3)$, then there exists $c_0>0$, such that for $c>c_0$, the following results are valid.

Figures (2)

  • Figure 1.1: Diagram showing the relationships between the ground states. The double-headed arrows represent equivalences
  • Figure 6.1: Proof Roadmap for the Convergence of Energy Ground State

Theorems & Definitions (113)

  • Definition 1.1: Ground states of Dirac equation
  • Definition 1.2: Ground states of Schrödinger equation
  • Remark 1.2
  • Theorem 1.3: Existence and properties of energy ground states
  • Remark 1.4
  • Theorem 1.5: Nonrelativistic limit of energy ground states
  • Theorem 1.6: Nonrelativistic limit of action ground states
  • Remark 1.7
  • Theorem 1.8
  • Lemma 2.1
  • ...and 103 more