Stationary periodic solutions to Nonlinear Dirac equations with non-coercive potentials

TL;DR

The paper proves the existence of nontrivial stationary periodic solutions to a nonlinear Dirac equation on the three-torus with a non-coercive nonlinearity. By introducing a coercive perturbation, the authors restore the Palais–Smale condition and establish a uniform local linking structure, enabling a min–max construction for the perturbed problems. Uniform a priori bounds are derived for the perturbed solutions, and a blow-up analysis rules out concentration, allowing passage to the limit as the perturbation vanishes to obtain a nontrivial periodic solution to the original equation. The approach hinges on a detailed spectral decomposition of the Dirac operator and a fractional Sobolev space framework, yielding a rigorously justified existence result for spatially periodic, stationary Dirac fields.

Abstract

We obtain periodic solutions for nonlinear Dirac equations with a nonlinear term that is not necessarily coercive. This amounts to study the equation on a three-dimensional torus. The Palais-Smale condition is enhanced by involving a coercive perturbation. Uniform estimates for the critical levels as well as the Sobolev norms for the perturbed solutions are obtained, making it possible to pass to a limit which gives a nontrivial solution.

Theorems & Definitions (19)

• Theorem 1.1
• Proposition 1.2
• Remark 2.1
• Lemma 2.2
• proof
• Lemma 4.1
• proof : Proof of Lemma \ref{['lemma:local control']}
• Lemma 4.2
• Lemma 4.3
• proof : Proof of Lemma \ref{['lemma:negativity on boundary-1']}